Essays on Minimal Supersolutions of BSDEs and on Cross ...

Essays on Minimal Supersolutions of BSDEs and onCross Hedging in Incomplete Markets

D I S S E R T A T I O N

zur Erlangung des akademischen Grades

doctor rerum naturalium(Dr. Rer. Nat.)

im Fach Mathematik

eingereicht an derMathematisch-Naturwissenschaftlichen Fakultät II

der Humboldt-Universität zu Berlin

vonHerrn Dipl.-Math. Gregor Heyne

Präsident der Humboldt-Universität zu Berlin:Prof. Dr. Jan-Hendrik Olbertz

Dekan der Mathematisch-Naturwissenschaftlichen Fakultät II:Prof. Dr. Elmar Kulke

Gutachter:1. Prof. Dr. Ulrich Horst2. Prof. Dr. Peter Imkeller3. Prof. Dr. Bruno Bouchard

Tag der Verteidigung: 30.08.2012

Abstract

In this thesis we study supersolutions of backward stochastic differential equa-tions (BSDEs) and a specific hedging problem in mathematical finance.In the first part of the thesis we analyze BSDEs with generators that are mono-

tone in y, convex in z, jointly lower semicontinuous, and bounded below by anaffine function of the control variable. We consider the set of all supersolutionswith respect to a given generator and a given terminal condition. We prove sev-eral properties of this set such as stability under pasting and a certain downwarddirectedness. The first central result establishes existence and uniqueness of a min-imal supersolution. We show further that our setting allows to derive importantproperties of the minimal supersolution such as the flow property and the projec-tivity, with time consistency as a special case. Next we investigate the stabilityof the minimal supersolution with respect to pertubations of the generator or theterminal condition. We find that, for instance, the functional which maps theterminal condition to the infimum over all value processes evaluated at time zerois not only defined on the same domain as the original expectation operator, butalso shares some of its main properties such as monotone convergence and Fatou’sLemma. Moreover, this leads to lower semincontinuity and dual representationsof the functional. Finally, we demonstrate the scope of our method by giving asolution of the problem of maximizing expected exponential utility.In the second part of the thesis we investigate quadratic hedging of contingent

claims with basis risk.We first show how to optimally cross-hedge risk when the logspread between the

hedging instrument and the risk is asymptotically stationary. At the short end,the optimal hedge ratio is close to the cross-correlation of the log returns, whereasat the long end, the optimal hedge ratio equals one. For linear risk positions wederive explicit formulas for the hedge error, while for non-linear positions swiftsimulation analysis is possible. Finally, we demonstrate that even in cases withno clear-cut decision concerning the asymptotic stationarity of the logspread it isbetter to allow for mean reversion of the spread rather than to neglect it.Secondly, we study a model where the correlation between the hedging instru-

ment and the underlying of the contingent claim is random itself. We assume thatthe correlation is a process which evolves according to a stochastic differentialequation with values between the boundaries −1 and 1. We keep the correlationdynamics general and derive an integrability condition on the correlation processand its first variation process that allows to describe and compute the quadratichedge by means of a simple hedging formula. Furthermore we show that ourconditions are fulfilled by a large class of correlation dynamics. We give variousnon-trivial explicit examples.

ii

Zusammenfassung

In dieser Arbeit untersuchen wir Superlösungen von stochastischen Rückwärts-differentialgleichungen (BSDEs) und ein spezifisches Absicherungsproblem der Fi-nanzmathematik.

Im ersten Teil der Arbeit analysieren wir BSDEs mit Generatoren, die mono-ton in y, convex in z, gemeinsam unterhalbstetig und von unten durch eine affineFunktion der Kontrollvariable beschränkt sind. Wir betrachten die Menge aller Su-perlösungen für einen fixen Generator und eine fixe Endbedingung. Wir beweisenmehrere Eigenschaften dieser Menge, wie zum Beispiel Stabilität bei Verklebenund eine bestimmte Gerichtetheit nach unten. Das erste Hauptresultat ist derNachweis der Existenz und Eindeutigkeit einer minimalen Superlösung. Wir zei-gen weiterhin, dass für die minimale Superlösung wichtige Eigenschaften, wie zumBeispiel die Flusseigenschaft und die Projektivität, mit Spezialfall Zeitkonsistenz,gelten. Danach untersuchen wir die Stabilität der minimalen Superlösung bezüg-lich Störungen des Generators oder der Endbedingung. Es stellt sich zum Beispielheraus, dass das Funktional welches die Endbedingung auf das Infimum über alleWertprozesse zur Zeit null abbildet nicht nur den gleichen Definitionsbereich wieder Erwartungswert hat, sondern auch einige seiner wichtigsten Eigenschaften, wiemonotone Konvergenz und Fatou’s Lemma teilt. Das führt im Weiteren zur Un-terhalbstetigkeit und zu dualen Darstellungen dieses Funktionals. Schlussendlichdemonstrieren wir die Bandbreite unserer Methode, indem wir eine Lösung desNutzenmaximierungsproblems für die Exponentialnutzenfunktion herleiten.

Im zweiten Teil der Arbeit untersuchen wir die quadratische Absicherung vonfinanziellen Risikopositionen unter Basisrisiko.

Zuerst zeigen wir wie optimal abgesichert wird, wenn die Differenz der Loga-rithmen von Absicherungsinstrument und Risiko asymptotisch stationär ist. Amkurzen Ende ist die optimale hedge ratio nahe der Korrelation der logarithmiertenRenditen, wohingegen am langen Ende die optimale hedge ratio gleich eins ist. Fürlineare Risikopositionen leiten wir explizite Formeln für den Absicherungsfehler herund zeigen, dass für nichtlineare Positionen eine schnelle Simulation möglich ist.Schlussendlich, demonstrieren wir, dass es im Falle von Modellunsicherheit besserist, mean reversion der logarithmischen Differenz von Absicherungsinstrument undRisiko anzunehmen.

Zweitens untersuchen wir ein Modell in dem die Korrelation zwischen Absiche-rungsinstrument und Basiswert stochastisch ist. Wir nehmen an, dass die Korrela-tion ein Prozess ist, der sich gemäß einer stochastischen Differentialgleichung mitWerten zwischen −1 und 1 entwickelt. Wir halten die Korrelationsdynamik allge-mein und leiten eine Integrabilitätsbedingung bezüglich des Korrelationsprozessesund seines Prozesses erster Variation her, die uns erlaubt die optimale quadra-tische Absicherung durch eine einfache Formel zu beschreiben und zu berechnen.Weiterhin zeigen wir, dass unsere Bedingungen von einer großen Klasse von Korre-lationsdynamiken erfüllt werden. Wir führen einige nichttriviale explizite Beispieleauf.

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Contents

Introduction 1

I. Minimal Supersolutions of Convex BSDEs 15

1. Minimal Supersolutions of Convex BSDEs 171.1. Setting and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2. Minimal Supersolutions of BSDEs . . . . . . . . . . . . . . . . . . . . . 19

1.2.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.2.2. General Properties of A (·, g) and Eg . . . . . . . . . . . . . . . . 19

1.3. Existence, Uniqueness and Stability . . . . . . . . . . . . . . . . . . . . 261.3.1. Existence and Uniqueness of Minimal Supersolutions . . . . . . . 261.3.2. Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.3.3. Non positive generators . . . . . . . . . . . . . . . . . . . . . . . 391.3.4. Expected exponential utility maximization . . . . . . . . . . . . 41

1.4. Helly’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

II. Cross Hedging 47

2. Futures Cross-hedging with a Stationary Spread 492.1. The Continuous-time Model with a Stationary Spread . . . . . . . . . . 49

2.1.1. Model Specification . . . . . . . . . . . . . . . . . . . . . . . . . 492.1.2. An Empirical Illustration . . . . . . . . . . . . . . . . . . . . . . 52

2.2. Optimal Variance Hedging with Futures Contracts . . . . . . . . . . . . 542.3. Standard Deviation of the Hedge Error . . . . . . . . . . . . . . . . . . . 59

2.3.1. Linear Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3.2. Non-linear Positions . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4. The Performance of Suboptimal Hedging Strategies . . . . . . . . . . . . 672.4.1. The Costs of Ignoring a Long-term Relationship or Falsely As-

suming a Long-term Relationship . . . . . . . . . . . . . . . . . . 672.4.2. The Costs of Using a Static Hedge . . . . . . . . . . . . . . . . . 74

2.5. Including Directional Views and Stochastic Volatility . . . . . . . . . . . 802.6. Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3. Stochastic correlation 833.1. A brief review of local risk minimization . . . . . . . . . . . . . . . . . . 83

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Contents

3.2. The model and the main results . . . . . . . . . . . . . . . . . . . . . . . 853.3. Derivation of the hedge formula . . . . . . . . . . . . . . . . . . . . . . . 89

3.3.1. Deriving the FS decomposition with BSDEs . . . . . . . . . . . . 903.3.2. Differentiability with respect to the initial conditions . . . . . . . 923.3.3. Differentiability of ψ . . . . . . . . . . . . . . . . . . . . . . . . . 953.3.4. The hedge as variational derivative . . . . . . . . . . . . . . . . . 98

3.4. A class of correlation dynamics which fulfill the main assumptions . . . 1013.5. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.5.1. Modelling correlation directly . . . . . . . . . . . . . . . . . . . . 1053.5.2. Modelling correlation with Ornstein-Uhlenbeck processes . . . . 106

vi

Introduction

In recent years the theory of backward stochastic differential equations (BSDEs) hasemerged as an active field of research in probability theory. In mathematical terms thesolution of a BSDE on a Brownian probability space is in principle a pair of adaptedprocesses (Y, Z), which fulfills

Ys −t∫s

gu(Yu, Zu)du+t∫s

ZudWu = Yt, YT = ξ, (I)

for all 0 ≤ s ≤ t ≤ T , where the FT -measurable random variable ξ is the so-calledterminal condition and the measurable function g the generator. The process Y isusually referred to as the value process, while Z is the control process.Ever since their inception the interest in these equations has been spurred not only by

an intrinsic mathematical motivation but also because BSDEs appear naturally in a va-riety of problems in applied probability theory. There are, for instance, rich connectionsto partial differential equations, control theory and mathematical finance. Especiallyin the latter BSDEs are a powerful tool for providing solutions for example to ques-tions on optimal hedging, utility maximization and stochastic equilibria. To illustratethe connection of BSDEs with mathematical finance think of the Bachelier model, seeBachelier [1900], with drift zero and volatility one, on a one-dimensional Brownian prob-ability space. It is well-known that this model describes a complete financial market,where every contingent claim ξ is attainable. In the terminology of BSDEs we coulddescribe the same fact by stating that the BSDE with generator g = 0 and terminal con-dition ξ has a solution (Y,Z). More precisely, Z yields the replicating strategy, whereasY models the corresponding value process. This simple example of a zero generatoradditionally provides an insight into the relation of BSDEs and textbook stochasticanalysis. Indeed, under the assumption that ξ is square integrable the unique solutionof (I) is given by (Y,Z) = (E[ξ | F·], Z), where Z is obtained from the martingalerepresentation theorem.Let us give a brief and selective introduction to the development of BSDE theory.

Stochastic backward differential equations with non-zero, linear generator made theirfirst appearance in stochastic control theory as the equation satisfied by the adjointprocess, see for example Bismut [1973]. The first systematic study was given in Par-doux and Peng [1990], who proved existence and uniqueness for Lipschitz continuousgenerators and square integrable terminal conditions. Following this work research onBSDEs increased considerably. One of the best known papers of this time is probablyEl Karoui et al. [1997b]. It presents several new results and collects existing knowledge

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INTRODUCTION

into a comprehensive treatment. Moreover, the authors describe in great detail possibleapplications of Lipschitz BSDE theory in stochastic control theory, and in particularin mathematical finance. However, in many applications more sophisticated generatorsthan Lipschitz continuous functions are required. Consequently, considerable work hasbeen done to relax the assumptions on g and ξ, and to find conditions which guaranteeexistence and uniqueness of solutions, stability properties, and comparison theorems.With regards to this, a major milestone in BSDE theory is due to Kobylanski [2000]. Inher work she gave existence, uniqueness and stability of BSDE with generators that havea quadratic growth in the control variable and a bounded terminal condition. Briandand Hu [2008] then proved that also unbounded terminal conditions with certain expo-nential integrability lead to unique solutions. These results opened the door for furtherapplications, since problems arising in stochastic control often lead to BSDEs with gen-erators with quadratic growth. For instance, preference based hedging in incompletemarkets as considered in Hu et al. [2005] is a typical example. Another interestingbut somewhat precluding result in this context is given in Delbaen et al. [2010]. ForBSDEs with superquadratic growth in the control variable, it is essentially shown that,for bounded terminal conditions, in general a bounded solution does not need to existand even if it exists it need not be unique.In the first part of this thesis, we take the previous discussion, in particular the

results in Delbaen et al. [2010], as a starting point. Instead of trying to find anotherset of conditions on the generator which allows for existence and uniqueness of BSDEsolutions, we want to find a weaker concept than solutions of BSDEs which allowssuch theorems with less restrictions on the generator. We are especially interestedin discontinuous generators with non-Lipschitz growth in the value variable, possiblysuperquadratic growth in the control variable and more general unbounded terminalconditions. It turns out that the notion of minimal supersolution of a BSDE is verywell suited for this problem. In contrast to a solution of a BSDE a supersolution isa pair (Y,Z) of adapted process, which fulfills, for a given generator g and terminalcondition ξ,

Ys −t∫s

gu(Yu, Zu)du+t∫s

ZudWu ≥ Yt, YT ≥ ξ, (II)

for all 0 ≤ s ≤ t ≤ T . Moreover, the value process Y of a supersolution is required tobe càdlàg. A supersolution is called minimal if its value process is, at any time, smallerthan or equal to the value process of any other supersolution. The central questionsregarding the set of supersolutions of a BSDE are as follows. Does there exist a mini-mal supersolution? Is it unique? How about stability of the minimal supersolution withrespect to pertubations of the input data? Let us mention that supersolutions wereintroduced in El Karoui et al. [1997b], but no existence and uniqueness of the minimalsupersolution was given. This was first done by Peng [1999], who, however, consid-ered only Lipschitz continuous constrained generators and square integrable terminalconditions.The main mathematical contribution of the first part of this thesis is to formulate a

2

INTRODUCTION

setting, which allows to extend the theory of supersolutions of BSDEs beyond Lipschitzcontinuous generators and to work with terminal conditions that are only integrable.More precisely, we consider generators that are convex with respect to z, monotone iny, jointly lower semicontinuous, and bounded below by an affine function of the controlvariable z. Given these assumptions we derive several new and important results. Inparticular we show that there exists a unique minimal supersolution and that it isstable with respect to pertubations of the terminal condition or the generator. Fora more detailed and technical description of our novel approach, the results, and theprecise mathematical contribution of this chapter we refer to page 5. Let us finallymention that the setting developed in the first part of this thesis is robust enough toallow even further progress in the theory of supersolutions. It is, for instance, possibleto obtain existence and uniqueness results even if the convexity and the monotonicityassumptions are replaced by a mild normalization condition, see Heyne et al. [2012].

In the second part of this thesis we focus on a specific hedging problem in mathe-matical finance. Namely, we want to investigate how to hedge optimally if a hedginginstrument is not perfectly correlated with the risk to be hedged, that is when a non-hedgeable risk, called basis risk remains. A typical example for such a situation is anairline company that wants to protect itself against changing kerosene prices. Sincethere is no liquid kerosene futures market the airline company may fall back on fu-tures on less refined oil, such as crude oil futures, for hedging its kerosene risk. Thisis a reasonable approach, if the price evolvements of kerosene and of crude oil are verysimilar. Figure 0.1 illustrates the close comovement of the two price series at the In-tercontinentalExchange (ICE). There is a rich literature about optimal hedging with

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Figure 0.1.: Time evolvement of the daily price of crude oil in US$/BBL (dashed line) and for jetkerosene in US$/BBL (solid line) from 1995/01/02 until 2010/06/30 (resulting in 4043observations).

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INTRODUCTION

basis risk, see for instance Duffie and Richardson [1991] and Schweizer [1992], who de-rive cross-hedging strategies minimizing quadratic objective functions, or Davis [2006]and Henderson [2002], who provide cross-hedging strategies maximizing the hedger’sexpected utility.One of the goals, when studying cross hedging problems, is to derive strategies that

allow for tractable representations. More precisely, in Markovian diffusion settings,hedging a European plain vanilla contingent claim, one often seeks to obtain character-izations of the optimal hedge ξ given by, at time t,

ξt = A>t ∇ψ(t,Xt). (III)

Here X represents a vector of stochastic processes describing the financial model, ψ(t, x)is a value function, typically an expectation of some functional of XT , and A, oftenreferred to as the hedge ratio, is a vector-valued stochastic process given by a functionof the coefficients of the SDE describingX. Moreover, ∇ψ denotes the gradient of ψ withrespect to the initial value of X and the components of this vector are commonly knownas the Greeks. Now, for Formula (III) to be tractable it is essential that the Greeks canbe computed easily. There exist a variety of methods, which allow for computation of∇ψ(t, x), and often the particular choice of a method depends on the choice of modeland vice versa. Popular approaches include, for instance, the finite difference method,the finite element method, the integration by parts method of Malliavin calculus, Fourieranalysis based on affine model structure, and representations based on first variationprocesses.The aim of the second part of this thesis is to motivate and to study two models

that extend contemporary hedging literature in two different aspects. More precisely,we first investigate and interprete the effect of cointegration between risk and hedginginstrument on the components of Formula (III) and the hedge error distribution. Sec-ondly, we derive a tractable version of Formula (III) in a model where the correlationbetween risk and hedging instrument is random.In Chapter 2 an empirical study shows that the correlation between the log prices in

the kerosene and crude oil example above strongly depends on the sampling frequency.More precisely, the short-term correlations are considerably lower than the long-termcorrelations, pointing towards a long-term relationship with potential short-term devi-ations. However, this behaviour is not reflected by the widely studied models, whereboth processes, the risk source and the hedging instrument, are modeled as geomet-ric Brownian motions. On the contrary, in these models, compare for example Duffieand Richardson [1991] and Schweizer [1992], and Davis [2006] and Henderson [2002],the variance of the spread of the log prices is increasing in time. Motivated by theseobservations we set up a new model for cross hedging, whose main feature is a meanreverting, or asymptotic stationary, spread between the log prices. In order to get aprecise understanding of the influence an asymptotic stationary spread exerts on thehedging strategy we keep the model deliberately simple. By this the representation cor-responding to Formula (III) can be calculated explicitly, which allows a rigorous studyof the effect of a long-term relationship on optimal cross-hedging strategies. Moreover,

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INTRODUCTION

our model allows an efficient calculation of the basis risk entailed by the optimal cross-hedges. A more detailed introduction, further motivation for the choice of our model,and the contributions of this chapter are given on page 8.In Chapter 3 we change the focus from a model with asymptotic stationary spread

to a model where both the risk source and the tradable asset are modeled as geometricBrownian motions. In all the related hedging literature, see above and also Musielaand Zariphopoulou [2004] and Monoyios [2004], the authors consider models where thecorrelation between the Brownian motions underlying both processes is assumed to beconstant. The central aim of this chapter is to relax this restriction and to derive theexistence of an optimal hedging strategy, when the correlation is allowed to be a ran-dom process with values between −1 and 1. More precisely, we will assume that thecorrelation process is the solution of a general stochastic differential equation (SDE),and we will work in the setting of local risk minimization, see Schweizer [2001] for anintroduction. Given this framework, we want to find a representation of the locally riskminimizing strategy analogous to (III) and moreover explicitly characterize the corre-sponding Greeks. To this end, and here lie the first main mathematical contributions ofthis chapter, we prove differentiability of certain expectations, we prove explicit char-acterizations of the respective derivatives, and we prove an explicit representation forthe control process of a particular BSDE. Our proofs require certain integrability as-sumptions on the correlation process and its first variation process. Further, in orderto simplify the verification of these conditions for specific correlation models we provesufficient and simple to check characterizations based directly on the coefficients of theSDE modeling the correlation. We find, in particular, that we may consider non-trivialcorrelation processes whose absolute value is not uniformly bounded away from one. Amore detailed introduction to this chapter, its mathematical contribution and additionalrelated literature are given on page 11.The content of this thesis is strongly based on Drapeau et al. [2011], Ankirchner et al.

[2011] and Ankirchner and Heyne [2012].Let us give in the following more detailed descriptions of the results in this thesis.

Introduction to Chapter 1

On a filtered probability space, where the filtration is generated by a d-dimensionalBrownian motion W , we consider the process Eg(ξ) given by

Egt (ξ) = ess infYt ∈ L0

t : (Y,Z) ∈ A(ξ, g), t ∈ [0, T ] ,

where A(ξ, g) is the set of all pairs of càdlàg value processes Y and control processes Z,in some suitable space, such that

Ys −t∫s

gu(Yu, Zu)du+t∫s

ZudWu ≥ Yt and YT ≥ ξ, (IV)

5

INTRODUCTION

for all 0 ≤ s ≤ t ≤ T . Here the terminal condition ξ is a random variable, the generatorg a measurable function of (y, z) and the pair (Y,Z) is a supersolution of the BSDE(IV).The main objective of this chapter is to state a new and general set of assumptions

on the generator and the terminal condition which guarantees that there exists a uniqueminimal supersolution. More precisely, we show that under our assumptions the pro-cess Eg(ξ) = lims↓·,s∈Q Egs (ξ) is well-defined, is a modification of Eg(ξ), and equals thevalue process of the unique minimal supersolution, that is, there exists a unique controlprocess Z such that (Eg(ξ), Z) ∈ A(ξ, g). Before we prove this central result we deriveseveral properties of the set of supersolutions and the process Eg(ξ), such as stabilityunder pasting, downward directedness, and a comparison principle. Furthermore, weprove important properties such as the flow property and the projectivity, with timeconsistency as a special case. The existence theorem immediately yields a comparisontheorem for minimal supersolutions. We also study the stability of the minimal super-solution with respect to pertubations of the terminal condition or the generator. Ourresults show that the mapping ξ 7→ Eg0 (ξ), which can be viewed as a nonlinear expecta-tion, is not only defined on the same domain as the original expectation operator E[·],but also shares some of its main properties such as monotone convergence and Fatou’sLemma. These properties allow us to conclude that Eg0 (·) is L1-lower semicontinuousand to obtain dual representation. Finally, we demonstrate the scope of our method bygiving a solution of the problem of maximizing expected exponential utility.Before we give further details on the specific choice of our spaces and assumptions, let

us recall that related problems have been investigated throughout the literature before.Most notably, nonlinear expectations have been a prominent topic in mathematicaleconomics since Allais famous paradox, see [Föllmer and Schied, 2004, Section 2.2].Typical examples are the monetary risk measures introduced by Artzner et al. [1999]and Föllmer and Schied [2002], Peng’s g-expectations, see Peng [1997], the variationalpreferences by Maccheroni et al. [2006], and the recursive utilities by Duffie and Epstein[1992]. Especially the g-expectation, which is defined as the initial value of the solu-tion of a BSDE, is closely related to Eg0 (·), since each pair (Y,Z) that solves the BSDEcorresponding to (IV) is also a supersolution and hence an element of A(ξ, g). The con-cept of a supersolution of a BSDE appears already in El Karoui et al. [1997b, Section2.2]. For further references see Peng [1999], who derives monotonic limit theorems forsupersolutions of BSDEs and proves the existence of a minimal constrained supersolu-tion. Another related concept are stochastic target problems, which were introducedand studied by Soner and Touzi [2002], by means of controlled stochastic differentialequations and dynamic programming methods.Our main contribution is to provide a setting where we relax the usual Lipschitz

requirements for the generator g. As already mentioned, we suppose that g is convexwith respect to z, monotone in y, jointly lower semicontinuous, and bounded below byan affine function of the control variable z. In order to provide an intuition on howthese assumptions contribute toward the existence and uniqueness of a control processZ such that (Eg(ξ), Z) ∈ A(ξ, g), let us suppose for the moment that g is positive.

6

INTRODUCTION

Given an adequate space of control processes, the value process of each supersolutionand the process Eg(ξ) are in fact supermartingales. By suitable pasting, we may nowconstruct a decreasing sequence (Y n) of supersolutions, whose pointwise limit is againa supermartingale and equal to Eg(ξ) on all dyadic rationals. Since the generator gis positive, it can be shown that Eg(ξ) lies below Eg(ξ), P -almost surely, at any time.This suggests to consider the càdlàg supermartingale Eg(ξ) as a candidate for the valueprocess of the minimal supersolution. However, it is not clear a priori that the sequence(Y n) converges to Eg(ξ) in some suitable sense. Yet, taking into account the additionalsupermartingale structure we can prove, by using Helly’s theorem, that (Y n) convergesP ⊗ dt-almost surely to Eg(ξ). It remains to obtain a unique control process Z suchthat (Eg(ξ), Z) ∈ A(ξ, g). To that end, we prove that, for monotone sequences of super-solutions, a positive generator yields, after suitable stopping, a uniform L1-bound forthe sequence of supremum processes of the associated sequence of stochastic integrals.This, along with a result by Delbaen and Schachermayer [1994], and standard com-pactness arguments and diagonalization techniques yield the candidate control processZ as the limit of a sequence of convex combinations. Now, joint lower semicontinuityof g, positivity, and convexity in z allow us to use Fatou’s Lemma to verify that thecandidate processes (Eg(ξ), Z) are a supersolution of the BSDE. Thus, Eg(ξ) is in factthe value process of the minimal supersolution and a modification of Eg(ξ). Finally, theuniqueness of Z follows from the uniqueness of the Doob-Meyer decomposition of thecàdlàg supermartingale Eg(ξ).

Let us give further reference of related assumptions and methods in the existing lit-erature. Delbaen et al. [2010] consider superquadratic BSDEs with generators that arepositive and convex in z but do not depend on y. However, their principal aim andtheir method differ from ours. Indeed, they primarily study the well-posedness of su-perquadratic BSDEs by establishing a dual link between cash additive time-consistentdynamic utility functions and supersolutions of BSDEs. To view supersolutions as su-permartingales is one of the key ideas in our approach and we make ample use of therich structure supermartingales provide. Note that the idea to use classical limit theoryof supermartingales in the theory of BSDEs appears already in El Karoui and Quenez[1995], who study the problem of option pricing in incomplete financial markets. How-ever, the analysis is done via dual formulations and only for linear generators that donot depend on y. The construction of solutions of BSDEs by monotone approximationsis also a classical tool, see for example Kobylanski [2000] for quadratic generators andBriand and Hu [2008] for generators that are in addition convex in z. The idea to ap-ply compactness theorems such as Delbaen and Schachermayer [1994, Lemma A1.1], orDelbaen and Schachermayer [1996, Theorem A], in order to derive existence of BSDEsis new to the best of our knowledge. Usually existence proofs rely on a priori estimatescombined with a fixed point theorem, see for example El Karoui et al. [1997b], or onconstructing Cauchy sequences in complete spaces, see for example Briand and Con-fortola [2008] or Ankirchner et al. [2007]. As already mentioned, Peng [1999] studiesthe existence and uniqueness of minimal supersolutions. However, he assumes a Lip-schitz continuous constrained generator, a square integrable terminal condition, and

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INTRODUCTION

employs a very different approach. It is based on a monotonic limit theorem, [Peng,1999, Theorem 2.4], the penalization method introduced in El Karoui et al. [1997a], andit leads to monotone increasing sequences of supersolutions. Parallel to us, Cheriditoand Stadje [2012] have investigated existence and stability of supersolutions of BSDEs.They consider generators that are convex in z and Lipschitz in y. However, their set-ting and methods are quite different from ours. Namely, they approximate by discretetime BSDEs and work with terminal conditions that are bounded lower semicontinuousfunctions of the Brownian motion. Finally, given our local L1-bounds, the compactnessunderlying the construction of the candidate control process is a special case of resultsobtained by Delbaen and Schachermayer [1996].

Our second contribution is to allow for local supersolutions. This happens to be par-ticularly adequate to establish monotone continuity properties of the minimal superso-lution with respect to the terminal condition or the generator. We call a supersolution(Y, Z) of the BSDE local, if the stochastic integral of Z is well defined and thus acontinuous local martingale. In order to avoid so-called “doubling strategies”, presenteven for the most simple driver g ≡ 0, see Dudley [1977] or Harrison and Pliska [1981,Section 6.1], we require in addition that

∫ZdW is a supermartingale. In this setting,

the stochastic integral of the candidate control process in the proof of the existencetheorem is only a local martingale. However, we may once again use our assumptionson the generator to prove the supermartingale property. In addition, similar argumentsallow us to formulate theorems such as montone convergence and Fatou’s lemma forthe non-linear operator Eg0 (·) on the same domain as the standard expectation E[·] andto obtain its L1-lower semicontinuity. To complete the picture, we point out that ourapproach neither needs nor provides much integrability for the control processes. Theunderlying reason is that the compactness arguments in our proof are based on L1 ratherthan H1-bounds for the stochastic integrals. Yet, given some additional integrability onthe terminal condition, we obtain a candidate control process, whose stochastic integralbelongs to H1 and therefore is a true martingale. However, monotone stability for anincreasing sequence of terminal conditions does not hold without the additional assump-tion that A(ξ, g), where ξ is the limit terminal condition, is not empty. This guaranteesthe required integrability of the limit pair (Y , Z). In contrast, such an assumptionis not necessary in our initial setting, where, in order to obtain suitable bounds andto construct the dominating candidate supermartingale, it is enough to know that themonotone limit of the minimal supersolutions at time zero is finite.

Replacing the positivity assumption with the condition that the generator is boundedbelow by an affine function of the control variable, it is obvious that the value and controlprocesses of our supersolutions are supermartingales under another measure closelylinked to the generator g. In fact, for a positive generator we have supermartingales withrespect to the initial probability measure P , while for a non-positive generator, whichis bounded below in the above sense, we consider supermartingales under the measuregiven by the corresponding Girsanov transform. This yields a generator dependentconcept of admissibility. The implication thereof is illustrated by giving a minimalsupersolution based approach to the well known problem of exponential expected utility

8

INTRODUCTION

maximization, where this admissibility is related in a natural way to the market priceof risk.The chapter is organized as follows. In Section 1.1 we fix our notations and the

setting. A precise definition of minimal supersolutions, some of our main conditionsand first structural properties of Eg(ξ) are given in Section 1.2. Our main results, thatis, existence and stability theorems, are given in Section 1.3, which concludes with anexample on maximizing expected exponential utility.

Introduction to Chapter 2The correlation between the price changes is the crucial determinant of an optimal cross-hedge. A common approach in the literature and in practice is to obtain the optimalhedge ratio by using the most frequent returns or price increments being available,irrespective of the time to maturity. This is a valid approach if the correlation (betweenthe returns or price increments) and the ratio of the standard deviations are constantwith respect to the sampling frequency, such as for correlated (geometric) Brownianmotions. However, in many cases the correlation depends strongly on the selected timeinterval. For example in our empirical illustration the correlation of the daily log returnsof kerosene and crude oil is only 0.52, which seems unexpectedly low given the strongcomovement in the price series. The correlations of the weekly, monthly and yearlylog returns in contrast are at 0.72, 0.84 and 0.98, respectively. Thus, the short-termcorrelation is considerably lower than the long-term correlation, pointing towards along-term relationship with potential short-term deviations. This property is closelyrelated to the concept of cointegration. It dates back to Engle and Granger [1987] andGranger [1981] and assumes that a set of time series share a long-term relationship withtemporary deviations from this “equilibrium”. More precisely, consider two integratedtime series (of order one). They are cointegrated if a linear combination of them isstationary. This is supported for our example in Figure 0.2, which shows on the lowerpanel a clear mean reverting behavior of the spread between the logarithmic prices ofkerosene and crude oil. Note that we do not use an estimated cointegration vectorbut rather assume that the spread between the log prices is stationary. This is morerestrictive, but empirically supported by the p-value of the augmented Dickey-Fullertest (which is ≤ 0.001) indicating that the null hypothesis of a non-stationary spread isrejected. Kerosene and crude oil, however, is only one example for a pair of cointegratedprocesses and there are many studies pointing towards a cointegration relation betweenasset prices and corresponding hedging instruments, see e.g. Alexander [1999], Baillie[1989], Brenner and Kroner [1995], Lien and Luo [1993] and Ng and Pirrong [1996] andthe references therein.The long-term relationship between the kerosene price and the crude oil price leads

to the observed increasing correlation in our example so that the optimal hedge ratiosare not constant, but depend on time to maturity. Intuitively, for long-term hedges it islikely that the two assets are in their equilibrium relationship, whereas in the short-termthe dynamics are dominated by noisy fluctuations.

9

INTRODUCTION

40

80

160

96 98 00 02 04 06 08 10-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-0.5

-0.45

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

95 96 97 98 99 00 01 02 03 04 05 06 07 08 09 10 11

time

time

price

spread

oflogprices

logreturnskerosene


Figure 0.2.: The upper left panel depicts the time evolvement of the daily price of crude oil inUS$/BBL (dashed line) and for jet kerosene in US$/BBL (solid line) from 1995/01/02until 2010/06/30 (resulting in 4043 observations). The upper right panel exhibits thescatter plot of the corresponding daily log returns and shows that there is positive correla-tion among the two series as already mentioned in the text (with a correlation coefficientof 0.52). The lower panel depicts the time evolvement of the spread of the log prices.

10

INTRODUCTION

The aim of this chapter is to set up a model that allows a rigorous study of the effectof a long-term relationship on optimal cross-hedging strategies, and at the same timeallows an efficient calculation of the basis risk entailed by the optimal cross-hedges.We reproduce the long-term relationship of the prices by describing the logspread as anOrnstein-Uhlenbeck process, and by modeling the futures price as a geometric Brownianmotion (GBM). Noteworthy, our model differs from the widely studied models whereboth processes, the risk source and the hedging instrument, are GBMs. Such models areconsidered for example in Duffie and Richardson [1991], Schweizer [1992], who derivecross-hedging strategies minimizing quadratic objective functions, and in Ankirchneret al. [2010], Davis [2006], Monoyios [2004], Musiela and Zariphopoulou [2004], whoprovide cross-hedging strategies maximizing the hedger’s expected utility. In thesemodels the spread of the log prices is not asymptotically stationary since the varianceof the spread is proportional to time. We further show that these models underhedge therisk when cointegration is present (see Section 2.4). Our model, in contrast, explicitlyaccounts for an asymptotic stationary logspread. Furthermore, it is easy to estimate andit is still tractable enough to allow for a quick calculation of the hedge error standarddeviation under different trading strategies. In particular, we are able to derive time-consistent strategies that minimize the variance of the hedge error.To this end, we first solve the optimization problem of finding the dynamic strategy

that minimizes the variance of the hedge error. Variance optimal hedging strategies havebeen first described in Föllmer and Sondermann [1986]. We make use of their methodand transfer it to the specific case of cross-hedging risk with futures contracts withinour Markovian model. The optimal hedging strategy can be expressed in terms of therisk’s Greeks and a hedge ratio decaying with time to maturity. Moreover, for linearrisk positions we are able to derive a closed-form formula for the hedge error standarddeviation.The chapter is structured as follows. Section 2.1 introduces our model and presents

some empirical evidence, while Section 2.2 briefly reviews hedging with futures contractsand derives the variance optimal hedging strategy for our model. Section 2.3 developsthe implied hedge errors within our model for linear and non-linear positions and Sec-tion 2.4 compares the hedge errors between different models and (suboptimal) hedgingstrategies emphasizing the importance of allowing for an asymptotic stationary spread.An extension of our model to account for stochastic volatility is given in Section 2.5.Section 2.6 concludes.

Introduction to Chapter 3In this chapter we assume that the price of the tradable asset and the value of thenon-tradable index evolve according to geometric Brownian motions. However, we willassume that the correlation between the driving Brownian motions is not constant, buta random process with values between −1 and 1. More precisely, we will assume thatthe correlation process is the solution of a stochastic differential equation (SDE).We consider European options on the non-tradable index and derive the asset hedging

11

INTRODUCTION

strategy that locally minimizes the quadratic hedging error, the so-called locally riskminimizing strategy. Essentially, the optimal hedge can be described by the followingfactors: the asset hedge ratio, defined as

ρtindex volaasset vola ,

where ρt is the correlation process, and the correlation hedge ratio, defined as

γcorrelation vola

asset vola ,

where γ is the correlation between the asset and ρt. The derivative with respect tothe asset (resp. the correlation) of the expected value of the option under the so-calledminimal equivalent local martingale measure will be called asset delta (resp. correlationdelta). We will show that the optimal hedge is the asset hedge ratio multiplied with theasset delta plus the correlation hedge ratio multiplied with the correlation delta, that is

optimal hedge = asset hedge ratio× asset delta+ correlation hedge ratio× correlation delta.

In order to obtain this characterization we first show that in our setting the locallyrisk minimizing strategy may be expressed in terms of the solution (Y,Z) of a certainlinear BSDE. In fact, this is an observation made before in El Karoui et al. [1997b].More precisely, the locally risk minimizing strategy depends on the control process Zof the corresponding BSDE. Now, in order to obtain an explicit representation of thestrategy one has to explicitly characterize Z. In El Karoui et al. [1997b] it is shown thatin Markovian settings, that is, when the randomness in the terminal condition and in thegenerator of a BSDE is induced by a forward diffusion, in principle such representationsare possible. More precisely, it can be shown that under suitable regularity assumptionson the coefficients of the forward diffusion process, the terminal condition, and thegenerator the control process of a BSDE can be expressed as a function of the firstvariation process of the value process Y and the volatility coefficient of the forwardprocess. However, the coefficients of the volatility matrix of the forward processes inour model do not satisfy the prerequisites of El Karoui et al. [1997b, Proposition 5.9].In particular, our coefficients do not have uniformly bounded derivatives, and thereforethese results are not directly applicable in our setting.With this in mind there are two main mathematical contributions in this chapter.

Firstly, we prove that the value process of our linear BSDE can be differentiated withrespect to the initial values of our forward processes and that the gradient can be explic-itly written as a vector of expectations based on first variation processes of the forwarddiffusions. In general, by assuming a stochastic correlation, there is no closed formulafor the asset delta, but it is straightforward to show that it has a representation interms of a simple expectation based on first variation processes. The major effort, how-ever, lies in showing that the correlation delta can be expressed as a simple expectation

12

INTRODUCTION

as well. Here the main difficulty is that the dynamics of the non-tradable asset con-tains a term which may induce the non-differentiability of the flow of the non-tradableasset with respect to the initial value of the correlation process and which has to becontrolled when interchanging expectation and differentiation. In order to fix this, werequire that the absolute value of the correlation process is always strictly below one andwe introduce a natural integrability condition on the correlation and its first variationprocess. Given these conditions various technical arguments yield that expectation anddifferentiation may be interchanged. Secondly, we recover in our setting the classicalrepresentation of Z, despite the lack of regularity of our coefficients. This requires toprove a characterization of the control process of our particular BSDEs in the spiritof El Karoui et al. [1997b, Proposition 5.9]. To that end we rely on a technical proofwhich is based on an approximation and mollification procedure, and on argumentsinvolving Malliavin calculus. Both steps combined yield the explicit characterization ofthe optimal strategy.Given the existence and the representation of the locally risk minimizing strategy

a natural question is then which correlation processes fulfill our main assumptions?Based on boundary theory for diffusions and integrability properties of solutions oflinear SDEs we prove that our conditions can be sufficiently characterized by conditionsbased directly on the coefficients of the SDE modeling the correlation. We use thischaracterization to provide several examples of correlation processes which fulfill ourassumptions. In particular, we may consider non-trivial correlation processes whoseabsolute value is not uniformly bounded away from one.We want to point out to two papers that allow for stochastic correlation in pricing

contingent claims. In van Emmerich [2006] quanto options are priced by assuming thatthe exchange rate is stochastically correlated with the underlying. Frei and Schweizer[2008] deal with exponential utility indifference valuation of contingents claims based onrisk sources that are stochastically correlated with assets traded on financial markets.However, both only give results on the value of the optimization problem but do nothave explicit expressions for the optimal hedging strategy, compare, for instance, theremark at the end of Section 3.1 in Frei and Schweizer [2008].The chapter is organised as follows. In Section 3.1 we give a short introdution into

local risk minimization. Section 3.2 introduces our model and gives an overview on themain results we obtained. The details we use to derive our hedge formula are providedin Section 3.3. We continue in Section 3.4 by analyzing the boundary behaviour andintegrability properties of correlation processes. We conclude with Section 3.5 by givingsome explicit examples of correlation processes for which our main results hold.

13

INTRODUCTION

AcknowledgmentsI thank my supervisor, Ulrich Horst, for his guidance, support and continuous encour-agement during my time as a PhD student. Special thanks go to Stefan Ankirchner forthe fruitful collaboration on cross hedging and for many intense discussions. I also owespecial thanks to Michael Kupper for sharing his research enthusiasm and creativitywith me. Moreover, I am very grateful to Samuel Drapeau for his relentless pursuit ofan optimal presentation of stochastic target problems. I thank Georgi Dimitroff andChristian Pigorsch for sharing their interest on cross hedging with stationary spread.I also thank Markus Mocha, Anthony Réveillac, Mikhail Urusov, and Nicholas Westray

for many helpful discussions.Parts of my research were done while being employed at the Humboldt Universität zu

Berlin and the Quantitative Products Laboratory, a joint initiative by Deutsche BankAG, Humboldt Universität zu Berlin and Technische Universität Berlin. I gratefullyacknowledge their financial support which enabled me to concentrate on my research.

14

Part I.

Minimal Supersolutions ofConvex BSDEs

15

1. Minimal Supersolutions of ConvexBSDEs

On a filtered probability space, where the filtration is generated by a d-dimensionalBrownian motion W , we consider the process Eg(ξ) given by

Egt (ξ) := ess infYt ∈ L0

t : (Y, Z) ∈ A(ξ, g), t ∈ [0, T ] ,

where A(ξ, g) is the set of all pairs of càdlàg value processes Y and control processes Zsuch that

Ys −t∫s

gu(Yu, Zu)du+t∫s

ZudWu ≥ Yt and YT ≥ ξ, (1.1)

for all 0 ≤ s ≤ t ≤ T . Here the terminal condition ξ is a random variable, the generatorg a measurable function of (y, z) and the pair (Y, Z) is a supersolution of the backwardstochastic differential equation (BSDE) (1.1).The main objective of this chapter is to state conditions which guarantee that there

exists a unique minimal supersolution. More precisely, we show that the process Eg(ξ) =lims↓·,s∈Q Egs (ξ) is a modification of Eg(ξ) and equals the value process of the unique min-imal supersolution, that is, there exists a unique control process Z such that (Eg(ξ), Z) ∈A(ξ, g). The existence theorem immediately yields a comparison theorem for minimalsupersolutions. We also study the stability of the minimal supersolution with respectto pertubations of the terminal condition or the generator. Our results show that themapping ξ 7→ Eg0 (ξ), which can be viewed as a nonlinear expectation, fulfills a montoneconvergence theorem and Fatou’s Lemma on the same domain as the expectation oper-ator E[·]. These properties allow us to conclude that Eg0 (·) is L1-lower semicontinuous.The chapter is organized as follows. In Section 1.1 we fix our notations and the

setting. A precise definition of minimal supersolutions, some of our main conditionsand first structural properties of Eg(ξ) are given in Section 1.2. Our main results,that is, existence and stability theorems, are given in Section 1.3. In Section 1.3.4 weillustrate the scope of our method with an example on maximizing expected exponentialutility. We conclude this chapter with a version of Helly’s theorem in Section 1.4.

1.1. Setting and NotationsWe consider a fixed time horizon T > 0 and a filtered probability space(Ω,F , (Ft)t∈[0,T ] , P ), where the filtration (Ft) is generated by a d-dimensional Brow-

17

1. Minimal Supersolutions of Convex BSDEs

nian motion W and fulfills the usual conditions. We further assume that F = FT .The set of F-measurable and Ft-measurable random variables is denoted by L0 andL0t , respectively, where random variables are identified in the P -almost sure sense. The

sets Lp and Lpt denote the set of random variables in L0 and L0t , respectively, with

finite p-norm, for p ∈ [1,+∞]. Throughout this chapter, inequalities and strict in-equalities between any two random variables or processes X1, X2 are understood in theP -almost sure or in the P ⊗ dt-almost sure sense, respectively, that is, X1 ≤ (<)X2 isequivalent to P

[X1 ≤ (<)X2] = 1 or P ⊗ dt

[X1 ≤ (<)X2] = 1, respectively. Given

a process X and t ∈ [0, T ] we denote X∗t := sups∈[0,t] |Xs|. We denote by T theset of stopping times with values in [0, T ] and hereby call an increasing sequence ofstopping times (τn), such that P [

⋃n τn = T] = 1, a localising sequence of stopping

times. By S := S (R) we denote the set of all càdlàg progressively measurable pro-cesses Y with values in R and further denote with Prog the σ-algebra on Ω × [0, T ]generated by all progessively measurable processes. For p ∈ [1,+∞[, we further de-note by Lp := Lp (W ) the set of progressively measurable processes Z with values inR1×d, such that ‖Z‖Lp := E[(

∫ T0 Z2

sds)p/2]1/p < +∞. For any Z ∈ Lp, the stochasticintegral (

∫ t0 ZsdWs)t∈[0,T ] is well defined, see [Protter, 2004], and is by means of the

Burkholder-Davis-Gundy inequality a continuous martingale. For the Lp-norm, the setLp is a Banach space, see [Protter, 2004]. We further denote by L := L (W ) the set ofprogressively measurable processes with values in R1×d, such that there exists a local-ising sequence of stopping times (τn) with Z1[0,τn] ∈ L1, for all n ∈ N. Here again, thestochastic integral

∫ZdW is well defined and is a continuous local martingale.

For adequate integrands a, Z, we generically write∫ads or

∫ZdW for the respective

integral processes (∫ t

0 asds)t∈[0,T ] and (∫ t

0 ZsdWs)t∈[0,T ]. Finally, given a sequence (xn)in some convex set, we say that a sequence (yn) is in the asymptotic convex hull of (xn),if yn ∈ conv xn, xn+1, . . ., for all n.Normal integrands have been introduced by Rockafellar [1976] and are particularly

adequate to model integral problems with constraints. In our setting, a normal integrandis a function g : Ω× [0, T ]× R× R1×d → ]−∞,+∞], such that

• (y, z) 7→ g(ω, t, y, z) is lower semicontinuous for all (ω, t) ∈ Ω× [0, T ];

• (ω, t) 7→ g(ω, t, y, z) is progressively measurable for all (y, z) ∈ R× R1×d.

It is shown in [Rockafellar and Wets, 1998, Chapter 14.F], that for all progressivelymeasurable processes Y,Z, the process g (Y, Z) is itself progressively measurable andso, the integral

∫g (Y, Z) ds is well defined P -almost surely under the assumption that

+∞−∞ = +∞. The section theorem as well as the Fubini, Tonelli theorem [Kallen-berg, 2002, Lemma 1.26 and Theorem 1.27] extend to that context. Finally, the lowersemicontinuity yields an extended Fatou’s lemma, that is,∫

lim infn

gs (Y ns , Zns ) ds ≤ lim infn

∫gs (Y ns , Zns ) ds,

for any sequence Y n, Zn of progressively measurable processes, if g ≥ 0.

18

1.2. Minimal Supersolutions of BSDEs


1.2.1. Definitions

Given a normal integrand g, henceforth called generator, and a terminal condition ξ ∈L0, a pair (Y, Z) ∈ S×L is a supersolution of a BSDE, if, for all s, t ∈ [0, T ], with s ≤ t,holds

Ys −t∫s

gu(Yu, Zu)du+t∫s

ZudWu ≥ Yt and YT ≥ ξ. (1.2)

For such a supersolution (Y,Z), we call Y the value process and Z its control process.Due to the càdlàg property, Relation (1.2) holds for all stopping times 0 ≤ σ ≤ τ ≤ T ,in place of s and t, respectively. Note that the formulation in (1.2) is equivalent to theexistence of a càdlàg increasing process K, with K0 = 0, such that

Yt = ξ +T∫t

gu(Yu, Zu)du+ (KT −Kt)−T∫t

ZudWu, t ∈ [0, T ]. (1.3)

Although the notation in (1.3) is standard in the literature concering supersolutionsof BSDEs, see for example El Karoui et al. [1997b], Peng [1999], we will keep with(1.2) since the proofs of our main results exploit this structure. We consider only thosesupersolutions (Y,Z) ∈ S × L of a BSDE where Z is admissible, that is, where thecontinuous local martingale

∫ZdW is a supermartingale. We are then interested in the

setA(ξ, g) = (Y,Z) ∈ S × L : Z is admissible and (1.2) holds (1.4)

and the process

Egt (ξ) = ess infYt ∈ L0

t : (Y,Z) ∈ A(ξ, g), t ∈ [0, T ] . (1.5)

By Eg we mean the functional mapping terminal conditions ξ ∈ L0 to the processEg (ξ). Since the set A (ξ, g) and therefore Eg (ξ) depends on the time horizon T , weindicate this by writing AT (ξ, g) and Eg·,T (ξ, g), if necessary. Note that the essentialinfima in (1.5) can be taken over those (Y,Z) ∈ A(ξ, g), where YT = ξ. A pair (Y,Z)is called minimal supersolution, if (Y,Z) ∈ A(ξ, g), and if for any other supersolution(Y ′, Z ′) ∈ A(ξ, g), holds Yt ≤ Y ′t , for all t ∈ [0, T ].

1.2.2. General Properties of A (·, g) and Eg

In this section we collect various statements regarding the properties of A (·, g) andEg. The first lemma ensures that the set of admissible control processes is stable underpasting and that we may concatenate elements of A(ξ, g) along stopping times andpartitions of our probability space.

19


Lemma 1.1. Fix a generator g, a terminal condition ξ ∈ L0, a stopping time σ ∈ T ,and (Bn) ⊂ Fσ a partition of Ω.

1. Let (Zn) ⊂ L be admissible. Then Z = Z11[0,σ] +∑n≥1 Z

n1Bn1]σ,T ] is admissible.

2. Let ((Y n, Zn)) ⊂ A(ξ, g) such that Y 1σ 1Bn ≥ Y nσ 1Bn , for all n ∈ N. Then (Y , Z) ∈

A (ξ, g), where

Y = Y 11[0,σ[ +∑n≥1

Y n1Bn1[σ,T ] and Z = Z11[0,σ] +∑n≥1

Zn1Bn1]σ,T ]. (1.6)

Proof. 1. Let Mn and M denote the stochastic integrals of the Zn and Z, respec-tively. It follows from (Zn) ⊂ L and from (Bn) being a partition that Z ∈ L and that∫ t∨σs∨σ ZudWu =

∑1Bn

∫ t∨σs∨σ Z

nudWu. Now observe that the admissibility of all Zn yields

E[Mt − Ms | Fs

]= E

[M1

(t∧σ)∨s −M1s | Fs

]+ E

∑n≥1

1BnE [Mnt∨σ −Mn

s∨σ | Fs∨σ] | Fs

≤ 0 ,

for 0 ≤ s ≤ t ≤ T .

2. Z is admissible by Item 1. Since Y 1σ 1Bn ≥ Y nσ 1Bn , for all n ∈ N, it follows on the

set s < σ ≤ t that

Y 1s −

σ∫s

gu(Y 1u , Z

1u)du+

σ∫s

Z1udWu −

t∫σ

gu(Yu, Zu)du+t∫

σ

ZudWu

≥ Y 1σ −

∑n≥1

1Bn

t∫σ

gu(Y nu , Znu )du−t∫

σ

ZnudWu

≥∑n≥1

1Bn

Y nσ − t∫σ

gu(Y nu , Znu )du+t∫

σ

ZnudWu

≥∑n≥1

1BnY nt .

Hence,

Ys−t∫s

gu(Yu, Zu)du+t∫s

ZudWu ≥ 1σ>tY 1t +

∑n≥1

1Bn(1σ≤sY nt + 1s<σ≤tY nt

)= Yt

and thus (Y , Z) ∈ A(ξ, g).

In the following, some properties of the generator are key, and therefore, we say that agenerator g is

(Pos) positive, if g (y, z) ≥ 0, for all (y, z) ∈ R× R1×d.

20


(Inc) increasing, if g (y, z) ≥ g (y′, z), for all y, y′ ∈ R with y ≥ y′, and all z ∈ R1×d.

(Dec) decreasing, if g (y, z) ≤ g (y′, z), for all y, y′ ∈ R with y ≥ y′, and all z ∈ R1×d.

The next proposition addresses how A (ξ, g) depends on the terminal condition andthe generator and which impact they have on Eg (ξ). The first two properties arecrucial in the proof of the existence and uniqueness theorem in Section 1.3. The thirditem concerns the monotonicity of Eg (ξ) with respect to ξ and g. Combined with theexistence theorem, this yields in fact a comparison principle for minimal supersolutionsof BSDEs. We will illustrate its scope in the proof of our stability results and in theexample on utility maximization in Section 1.3.4. Finally, the last item concerns thecash (super/sub) additivity of the functional Eg(ξ).

Proposition 1.2. For t ∈ [0, T ], generators g, g′ and terminal conditions ξ, ξ′ ∈ L0, itholds

1. the set Yt : (Y,Z) ∈ A(ξ, g) is directed downwards.

2. assuming (Pos), ξ− ∈ L1 and A (ξ, g) 6= ∅, then for all ε > 0, there exists(Y ε, Zε) ∈ A (ξ, g) such that Egt (ξ) ≥ Y εt − ε.

3. (monotonicity) if ξ′ ≤ ξ and g′ (y, z) ≤ g (y, z), for all y, z ∈ R × R1×d, thenA (ξ′, g′) ⊃ A (ξ, g) and Eg

′

t (ξ′) ≤ Egt (ξ).

4. (convexity) if (y, z) 7→ g(y, z) is jointly convex, then A (λξ + (1− λ) ξ′, g) ⊃λA (ξ, g) + (1− λ)A (ξ′, g), for all λ ∈ (0, 1), and so

Egt (λξ + (1− λ) ξ′) ≤ λEgt (ξ) + (1− λ) Egt (ξ′) .

5. for m ∈ L0t ,

• (cash superadditivity) assuming (Inc) andm ≥ 0, then Egt (ξ +m) ≥ Egt (ξ)+m.

• (cash subadditivity) assuming (Dec), m ≥ 0, and the existence of (Y,Z) ∈A(ξ, g), such that At(Yt +m, g) 6= ∅, then Egt (ξ +m) ≤ Egt (ξ) +m.

• (cash additivity) assuming that g does not depend on y, the existence of(Y, Z) ∈ A(ξ, g), such that At(Yt +m+, g) 6= ∅, and the existence of (Y,Z) ∈A(ξ +m, g), such that At(Yt +m−, g) 6= ∅, then Egt (ξ +m) = Egt (ξ) +m.

Proof. 1. Given (Y i, Zi) ∈ A(ξ, g), for i = 1, 2, we have to construct (Y , Z) ∈ A(ξ, g),such that Yt ≤ min

Y 1t , Y

2t

. To this end, we define the stopping time

τ = infs > t : Y 1

s > Y 2s

∧ T

and set Y = Y 11[0,τ [ + Y 21[τ,T [, YT = ξ, and Z = Z11[0,τ ] + Z21]τ,T ]. Since Y 1τ ≥ Y 2

τ ,Lemma 1.1 yields (Y , Z) ∈ A(ξ, g) and by definition holds Yt = min

Y 1t , Y

2t

.

21


2. In view of the first assertion, there exists a sequence ((Y n, Zn)) ⊂ A(ξ, g) such that(Y nt ) decreases to Egt (ξ). Set Y n = Y 11[0,t) + Y n1[t,T ] and Zn = Z11[0,t] + Zn1(t,T ].From Lemma 1.1 follows that ((Y n, Zn)) ⊂ A(ξ, g) and (Y nt ) decreases to Egt (ξ) byconstruction. Relation (1.2), Y n0 = Y 1

0 , and g positive yield

t∫0

Zns dWs ≥ ξ −T∫t

Zns dWs +T∫

0

gs (Y ns , Zns ) ds− Y n0 ≥ −ξ− −T∫t

Zns dWs − Y 10 .

Taking conditional expectation with respect to Ft and using the supermartingal prop-erty of

∫ZndW yield

t∫0

Zns dWs ≥ −E[ξ−∣∣∣ Ft]− Y 1

0 .

Given ε > 0, since A (ξ, g) 6= ∅, the sets Bn = An \ An−1 ∈ Ft, where An = Egt (ξ) ≥Y nt − ε and A0 = ∅, form a partition of Ω. Since (Y nt ) is decreasing, it follows thatY 1t 1Bn ≥ Y nt 1Bn , for all n ∈ N. Consequently, by means of Lemma 1.1, (Y , Z), defined

as in (1.6), is an element of A(ξ, g) and Egt (ξ) ≥ Yt − ε by construction.

3. Straightforward inspection.

4. The joint convexity of g yields (λY + (1 − λ)Y ′, λZ + (1 − λ)Z ′) ∈ A(λξ + (1 −λ)ξ′, g), for all (Y,Z) ∈ A(ξ, g), all (Y ′, Z ′) ∈ A(ξ′, g), and all λ ∈ (0, 1). Hence,λA (ξ, g)+(1− λ)A (ξ′, g) ⊃ A (λξ + (1− λ) ξ′, g) and in particular, Egt (λξ+(1−λ)ξ′) ≤λEgt (ξ) + (1− λ)Egt (ξ′).

5. Let us show the cash superadditivity. For m ∈ L0t with m ≥ 0, given (Y, Z) ∈

A(ξ +m, g), and 0 ≤ s ≤ t′ ≤ T , it follows from (1.2) and (Inc) that

Ys −m1[t,T ] (s)−t′∫s

gu(Yu −m1[t,T ](u), Zu)du+t′∫s

ZudWu

≥ Ys −m1[t,T ] (s)−t′∫s

gu(Yu, Zu)du+t′∫s

ZudWu ≥ Yt′ −m1[t,T ] (t′) .

Hence, (Y − m1[t,T ], Z) ∈ A(ξ, g) and thus Egt (ξ + m) − m ≥ Egt (ξ). For the cashsubadditivity the same argument yields

Ys +m1[t,T ] (s)−t′∫s

gu(Yu +m1[t,T ](u), Zu)du+t′∫s

ZudWu ≥ Yt′ +m1[t,T ] (t′) ,

for all t ≤ s ≤ t′ ≤ T , and all (Y,Z) ∈ A(ξ, g). Given (Y , Z) ∈ At(Yt + m, g)our usual pasting argument yields (Y , Z) ∈ A(ξ + m, g), with Yt + m = Yt and thusEgt (ξ) +m ≥ Egt (ξ+m). The cash additivity in case where g is independent of y follows

22


from Egt (ξ) +m = Egt (ξ+m+)−m− = Egt (ξ+m+m−)−m− = Egt (ξ+m), since (Dec)and (Inc) are simultaneously fulfilled.

The following Lemma states that under the assumption of a positive generator the valueprocess of a supersolution is a supermartingale. To view supersolutions as supermartin-gales is one of the key ideas in our approach. We will make ample use of the richstructure supermartingales provide throughout this chapter.

Lemma 1.3. Let g be a generator fulfilling (Pos), and ξ ∈ L0 be a terminal conditionsuch that ξ− ∈ L1. Let (Y,Z) ∈ A(ξ, g). Then ξ ∈ L1, Y is a supermartingale, Z isunique and Y has the unique decomposition

Y = Y0 −A+M, (1.7)

where M denotes the supermartingale∫ZdW and A is a predictable, increasing, càdlàg

process with A0 = 0.

Proof. Relation (1.2), positivity of g, admissibility of Z and ξ− ∈ L1 imply E[|Yt|] <+∞, for all t ∈ [0, T ]. Since −ξ− ≤ ξ ≤ YT , we deduce that ξ ∈ L1. Again, from (1.2),admissibility of Z and positivity of g we derive by taking conditional expectation, thatYs ≥ E[Yt | Fs]. Thus Y is a supermartingale with Yt ≥ E [ξ | Ft]. Relation (1.2) impliesfurther that there exist an increasing and càdlàg processK, with K0 = 0, such that (1.7)holds with A =

∫g(Y,Z)ds+K. Note that A is optional and therefore predictable due

to the Brownian filtration, see [Revuz and Yor, 1999, Corollary V.3.3]. Since Y is acàdlàg supermartingale the Doob-Meyer theorem, see [Protter, 2004, Theorem III.3.13],implies the unique decomposition Y = Y0 +M−A, where M is a local martingale and Ais an increasing process which is predictable, and M0 = A0 = 0. In our filtration everylocal martingale is continuous, see [Protter, 2004, Corollary IV.3.1, p. 187] and thusA is càdlàg. Hence A and A and M and M are indistinguishable. Moreover, from thepredictable representation property of local martingales and from P (

⋃nτn = T) = 1,

for τn = inft ≥ 0||Mt| ≥ n ∧ T , we obtain the P ⊗ dt-almost sure uniqueness of Z.

We now prove that for a positive generator Eg (ξ) is in fact a supermartingale, which,in addition, dominates its right hand limit process. This is crucial for the proof of theexistence and uniqueness theorem.

Proposition 1.4. Let g be a generator fulfilling (Pos), and ξ ∈ L0 be a terminalcondition such that ξ− ∈ L1. Suppose that A (ξ, g) 6= ∅, then Eg (ξ) is a supermartingale,

Egs (ξ) := limt↓s,t∈Q

Egt (ξ), for all s ∈ [0, T ), EgT (ξ) := ξ,

is a well-defined càdlàg supermartingale, and

Egs (ξ) ≥ Egs (ξ), for all s ∈ [0, T ]. (1.8)

Proof. Note first that Eg(ξ) is adapted by definition. Furthermore, given (Y,Z) ∈A(ξ, g) 6= ∅, Lemma 1.3 implies ξ ∈ L1 and Yt ≥ E [ξ | Ft]. Hence Yt ≥ Egt (ξ) ≥ E[ξ | Ft]

23


and Egt (ξ) ∈ L1. As for the supermartingale property and (1.8), fix 0 ≤ s ≤ t ≤ T . Inview of item 2 of Proposition 1.2, for all ε > 0, there exists (Y ε, Zε) ∈ A(ξ, g) such thatEgs (ξ) ≥ Y εs − ε. Due to (1.2) it follows

Egt (ξ) ≤ Y εt ≤ Y εs −t∫s

gu(Y εu , Zεu)du+t∫s

ZεudWu

≤ Egs (ξ)−t∫s

gu(Y εu , Zεu)du+t∫s

ZεudWu + ε ≤ Egs (ξ) +t∫s

ZεudWu + ε. (1.9)

Taking conditional expectation on both sides with respect to Fs and the supermartingaleproperty of

∫ZεdW yields Egs (ξ) ≥ E[Egt (ξ) | Fs], and so Eg (ξ) is a supermartingale.

That Eg(ξ) is well-defined càdlàg supermartingale follows from Karatzas and Shreve[2004, Proposition 1.3.14]. Finally, (1.8) follows directly from (1.9) and the definitionof Eg (ξ).

Remark 1.5. The previous proposition suggests to consider the càdlàg supermartingaleEg(ξ) as a candidate for the value process of the minimal supersolution. Note furtherthat, if Eg (ξ) is the value process of the minimal supersolution it is a modification ofEg (ξ) by definition. Hence, in the following, whenever we assume that Eg (ξ) is minimal,if we make P -almost sure statements, we may write either of them.

The final result of this Section shows that our setup allows to derive various propertiesthat are important in the context of non-linear expectations and dynamic risk measures.In particular, we prove that Eg (ξ), if it is the value process of the minimal supersolution,fulfills the flow-property and, under the additional assumption g(y, 0) = 0, for all y ∈R, we show projectivity, with time-consistency as a special case. In the context ofBSDE solutions such properties were first established in Peng [1997], for the case ofLipschitz generators. For dynamic risk measures the (strong) time-consistency has beeninvestigated in discrete time in [Cheridito et al., 2006, Föllmer and Penner, 2006] aswell as in continuous time in [Bion-Nadal, 2009, Delbaen, 2006], for instance.

Proposition 1.6. For t ∈ [0, T ], generator g and terminal condition ξ ∈ L0, it holds

1. Egs,T (ξ) ≤ Egs,t(Egt,T (ξ)), for all 0 ≤ s ≤ t. Suppose that Eg (ξ) is a minimal

supersolution, then the flow-property holds, that is

Egs,T (ξ) = Egs,t(Egt,T (ξ)), for all 0 ≤ s ≤ t. (1.10)

2. if g (y, 0) = 0, for all y ∈ R, then Egs(Egt (ξ)

)≤ Egs (ξ), for all 0 ≤ s ≤ t. Assuming

(Pos), and supposing that Eg (ξ) is a minimal supersolution, then Eg (ξ) is time-consistent, that is

Egs (Egt (ξ)) = Egs (ξ) , for all 0 ≤ s ≤ t. (1.11)

24


3. assuming (Pos), g (y, 0) = 0, for all y ∈ R, and Eg (ξ) is a minimal supersolution,then the projectivity holds, that is

Egs (1AEgt (ξ)) = Egs (1Aξ) , for all 0 ≤ s ≤ t and A ∈ Ft. (1.12)

Proof. 1. Fix 0 ≤ s ≤ t. Obviously, (Ys, Zs)s∈[0,t] ∈ At(Egt,T (ξ) , g), for all (Y,Z) ∈

AT (ξ, g). Hence Egs,t(Egt,T (ξ)) ≤ Egs,T (ξ). Suppose now that Eg·,T (ξ) is a minimal su-

persolution with corresponding admissible control process Z ∈ L. For all (Y,Z) ∈At(Egt,T (ξ) , g), holds Yt ≥ Egt,T (ξ) and, with the same argumentation as in Lemma 1.1,we can paste in a monotone way to show that (Y , Z) ∈ AT (ξ, g), where Y = Y 1[0,t[ +Eg·,T (ξ) 1[t,T ] and Z = Z1[0,t] + Z1]t,T ]. Thus, by definition, Egs,t(E

gt,T (ξ)) ≥ Egs,T (ξ).

2. Given (Y, Z) ∈ A (ξ, g), we define Y = Y 1[0,t[ + Egt (ξ) 1[t,T ] and Z = Z1[0,t]. SinceYt ≥ Egt (ξ) and g (y, 0) = 0, it is straightforward to verify that (Y , Z) ∈ A(Egt (ξ) , g).From Ys ≥ Ys, for all s ∈ [0, t], follows that Egs (Egt (ξ)) ≤ Egs (ξ), for all s ∈ [0, t]. Thecase where Eg (ξ) is a minimal supersolution and Assumption (Pos) holds, follows from(1.12) for A = Ω.

3. Fix A ∈ Ft. Suppose that Eg (ξ) is a minimal supersolution with correspondingcontrol process Z. Given (Y, Z) ∈ A (1AEgt (ξ), g), it follows from Proposition 1.4 belowthat Yt ≥ E [1AEgt (ξ) | Ft] = 1AEgt (ξ). Since g (y, 0) = 0, it is straightforward to checkthat Y = Y 1[0,t[ + Egt (ξ) 1A1[t,T ], and Z = Z1[0,t] is such that (Y , Z) ∈ A (1AEgt (ξ) , g).We can henceforth assume that Ys = 1AEgt (ξ), for all s ≥ t. Now, we define Y =Y 1[0,t[ + Eg (ξ) 1A1[t,T ] and Z = Z1[0,t] + Z1A1]t,T ]. For 0 ≤ s < t ≤ t′ ≤ T holds

Ys −t′∫s

g(Yu, Zu

)du+

t′∫s

ZudWu ≥ Yt +

− t′∫t

gu

(Egu (ξ) , Zu

)du+

t′∫t

ZudWu

1A

≥

Egt (ξ)−t′∫t

gu

(Egu (ξ) , Zu

)du+

t′∫t

ZudWu

1A ≥ Egt′ (ξ) 1A.

Hence, for all 0 ≤ s ≤ t′ ≤ T , holds

Ys −t′∫s

g(Yu, Zu

)du+

t′∫s

ZudWu ≥ Yt′1t′<t + Egt′ (ξ) 1A1t≤t′ = Yt′

and YT = 1Aξ, which implies that (Y , Z) ∈ A (1Aξ, g). Since Ys = Ys, for all s ≤ t, wededuce Egs (1Aξ) ≤ Egs (1AEt (ξ)).On the other hand, consider (Y, Z) ∈ A (1Aξ, g). From Yt ≥ E [1Aξ | Ft] = 1AE [ξ | Ft],we obtain Yt1Ac ≥ 0. Since Eg (ξ) is a minimal supersolution, it follows that Yt ≥Egt (ξ) 1A. Indeed, let B = Yt < Egt (ξ) 1A, then Yt1Ac ≥ 0 implies B ⊂ A. Con-sequently, by similar arguments as in Lemma 1.1, the processes Y = Eg (ξ) (1[0,t[ +1Bc1[t,T ]) + Y 1B1[t,T ] and Z = Z(1[0,t[ + 1Bc1[t,T ]) + Z1B1[t,T ] are such that (Y , Z) ∈

25


A (ξ, g), which implies P [B] = 0. It is also straightforward to check that Y = Y 1[0,t[ +Eg (ξ) 1A1[t,T ] and Z = Z1[0,t] + Z1(t,T ]1A are such that (Y , Z) ∈ A (1Aξ, g). Thus wecan assume that Yt = 1AEgt (ξ). Defining Y = Y 1[0,t[ + Egt (ξ) 1A1[t,T ] and Z = Z1[0,t],it holds (Y , Z) ∈ A (1AEgt (ξ) , g). Thus Egs (1AEgt (ξ)) ≤ Egs (1Aξ), since Ys = Ys, for alls ≤ t.

1.3. Existence, Uniqueness and StabilityIn this section, we give conditions, which guarantee the existence and uniqueness ofa minimal supersolution. We show that the corresponding value process is given byEg(ξ). Moreover, we analyze the stability of Eg(ξ) with respect to perturbations of theterminal condition or the generator. In addition to the assumptions (Pos) and (Inc)or (Dec) introduced above, we require convexity of g in the control variable. To thatend, we say that a generator g is

(Con) convex, if g (y, λz + (1− λ) z′) ≤ λg (y, z) + (1− λ) g (y, z′), for all y ∈ R, allz, z′ ∈ R1×d and all λ ∈ (0, 1).

1.3.1. Existence and Uniqueness of Minimal SupersolutionsThe following theorem on existence and uniqueness of a minimal supersolution is thefirst main result of this chapter. Note, that we require that A(ξ, g) 6= ∅. In the context offinding minimal elements in some set this is quite a standard assumption, see Peng [1999]for an example in the context of minimal supersolutions. However, let us point out thatin many applications A(ξ, g) 6= ∅ might be guaranteed by specific model assumptions,see for instance the example on utility maximization in Section 1.3.3. It might also beautomatically granted under further assumptions, see Cheridito and Stadje [2012], or forinstance if the BSDE Yt−

∫ Ttgs (Ys, Zs) ds+

∫ TtZsdWs = ξ has a solution (Y, Z) ∈ S×L,

such that Z is admissible. In the latter case, A (ξ, g) 6= ∅, for all ξ ∈ L0 such thatξ− ∈ L1, with ξ ≥ ξ.

Theorem 1.7. Let g be a generator fulfilling (Pos), (Con) and either (Inc) or (Dec)and ξ ∈ L0 be a terminal condition, such that ξ− ∈ L1. If A(ξ, g) 6= ∅, then there existsa unique minimal supersolution (Y , Z) ∈ A(ξ, g). Moreover, Eg (ξ) is the value processof the minimal supersolution, that is (Eg (ξ) , Z) ∈ A(ξ, g).

Note that Remark 1.5 implies that under the assumptions of Theorem 1.7 the processEg (ξ) is in fact a modification of Eg (ξ).

Proof. Step 1: Uniqueness. Given Z ∈ L such that (Eg(ξ), Z) ∈ A(ξ, g), the definitionof Eg(ξ) implies that for any other supersolution (Y, Z ′) ∈ A(ξ, g) holds Egt (ξ) ≤ Yt, forall t ∈ [0, T ]. The uniqueness of Z follows as in Lemma 1.3.

The remainder of the proof provides existence of Z ∈ L such that (Eg(ξ), Z) ∈ A(ξ, g).

26

1.3. Existence, Uniqueness and Stability

Step 2: Construction of an approximating sequence. For any n, i ∈ N, let tni = iT/2n.There exist ((Y n, Zn)) ⊂ A(ξ, g) such that

Egtni(ξ) ≥ Y ntn

i− 1/n, for all n ∈ N, and all i = 0, . . . , 2n − 1, (1.13)

andY nt ≥ Y n+1

t , for all t ∈ [0, T ], and all n ∈ N. (1.14)

Indeed, by means of Proposition 1.2.2, for each n ∈ N, we may select a family((Y n,i, Zn,i))i=0,...,2n−1 in A(ξ, g), such that Egtn

i(ξ) ≥ Y n,itn

i− 1/n, i = 0, . . . , 2n− 1. We

suitably paste this family in order to obtain (1.13). We start with

Y n,0 = Y n,0, Zn,0 = Zn,0,

and continue by recursively setting, for i = 1, . . . , 2n − 1,

Y n,i = Y n,i−11[0,τni

[ + Y n,i1[τni,T [, Y n,iT = ξ,

Zn,i = Zn,i−11[0,τni

] + Zn,i1]τni,T ],

where τni are stopping times given by τni = inft > tni : Y n,i−1t > Y n,it ∧ T . From the

definition of these stopping times and Lemma 1.1 follows that the pairs (Y n,i, Zn,i),i = 0, . . . , 2n − 1, are elements of A(ξ, g). Hence, the sequence

((Y n, Zn) := (Y n,2n−1, Zn,2

n−1))

fulfills (1.13) by construction. Note that ((Y n, Zn)) is not necessarily monotone in thesense of (1.14). However, this can be achieved by pasting similarly. More precisely, wechoose

Y 1 = Y 1, Z1 = Z1,

and continue by recursively setting, for n ∈ N,

Y n =2n−1∑i=0

(Y n1[tn

i,τni

[ + Y n−11[τni,tni+1[

), Y nT = ξ,

Zn =2n−1∑i=0

(Zn1]tn

i,τni

] + Zn−11]τni,tni+1]

),

where τni are stopping times given by τni = inft > tni : Y nt > Y n−1t ∧ tni+1, for

i = 0, . . . , 2n − 1. By construction ((Y n, Zn)) fulfills both (1.13) and (1.14), and((Y n, Zn)) ⊂ A(ξ, g) with Lemma 1.1.

Step 3: Bound on∫ZndW . We now take the sequence ((Y n, Zn)) fulfilling (1.13) and

(1.14) and provide an inequality which will enable us to use compactness arguments for

27


(Zn) later in the proof. More precisely, we argue that, for all n ∈ N, holds∣∣∣∣∣∣t∫

0

Zns dWs

∣∣∣∣∣∣ ≤ Bnt :=∣∣Y 1t

∣∣+ E[ξ−∣∣∣ Ft]+ E

[ξ−]

+∣∣Y 1

0∣∣+Ant , (1.15)

for all t ∈ [0, T ], where Ant is the positive increasing process defined in Lemma 1.3.Moreover, it holds

E[AnT ] ≤ Y 10 − E[ξ].

Indeed, by the same arguments as in the proof of Lemma 1.2.2, recall Y n0 ≤ Y 10 , follows

t∫0

Zns dWs ≥ −E[ξ−∣∣∣ Ft]− Y 1

0 . (1.16)

On the other hand, from Y nt ≤ Y 1t and −Y n0 ≤ E[ξ−], recall Lemma 1.3, it follows

t∫0

Zns dWs ≤ Y 1t +Ant − Y n0 ≤ Y 1

t +Ant + E[ξ−]. (1.17)

Combining (1.17) and (1.16) yields (1.15). The L1 bound on An follows from Y n0 −AnT +

∫ T0 Zns dWs = ξ, Y 1

0 ≥ Y n0 , and the supermartingal property of∫ZndW .

Note that if (Bn,∗T ) in (1.15) were bounded in L1, then, by means of the Burkholder-Davis-Gundy inequality, (Zn) would be a bounded sequence in L1 and we could apply[Delbaen and Schachermayer, 1996, Theorem A] to find a sequence in the asymptoticconvex hull of (Zn), which converges in L1 and P⊗dt-almost surely along some localizingsequence of stopping times to some limit Z ∈ L1. Here, even if (An,∗T ) = (AnT ) isuniformly bounded, this is however not necessarily the case for Y 1,∗

T and (E[ξ− | F·])∗T ,and this is the reason why we introduce the following localization.

Step 4: First localization. Due to our Brownian setting and since ξ− ∈ L1, we knowthat the martingale E[ξ− | F·], has a continuous version, see [Revuz and Yor, 1999,Theorem V.3.5]. Moreover, Y 1 is a càdlàg supermartingale and thus we may take thelocalising sequence

σk = inft > 0 :

∣∣Y 1t

∣∣+ E[ξ−∣∣∣ Ft] > k

∧ T, k ∈ N, (1.18)

which is independent of n ∈ N. For a fixed k ∈ N, Inequality (1.15) yields(∫Zn1[0,σk]dW

)∗T

≤ Bk,n, for all n ∈ N, (1.19)

28


where Bk,n =∣∣Y 1

0∣∣+ E[ξ−] + k +AnT . Due to E[AnT ] ≤ Y 1

0 − E[ξ] we have

supn∈N

E[Bk,n

]<∞. (1.20)

Since (Bk,n)n∈N is a sequence of positive random variables we may apply [Delbaen andSchachermayer, 1994, Lemma A1.1]. It provides a sequence (Bk,n)n∈N in the asymptoticconvex hull of (Bk,n)n∈N, which converges almost surely to a random variable Bk ≥ 0.The Bk,n inherit the integrability of the Bk,n and we can conclude with Fatou’s lemmathat

E[Bk] <∞. (1.21)

Let Zk,n be the convex combination of (Zn) corresponding to Bk,n so that(∫Zk,n1[0,σk]dW

)∗T

≤ Bk,n, for all n ∈ N. (1.22)

Step 5: Second localization. The next two steps follow some known compactness argu-ments, which, in the case of L1, can be found in [Delbaen and Schachermayer, 1996].For the sake of completeness we develop the argumentation. Given an m ∈ N, we startby taking a fast subsequence (Bk,m,n)n∈N of (Bk,n)n∈N converging in probability to Bk.More precisely, we choose (Bk,m,n)n∈N such that

P[∣∣Bk,m,n − Bk∣∣ ≥ 1

]≤ 2−n

m. (1.23)

Consider now the stopping time τk,m given by

τk,m = inft ≥ 0 :

(∫Zk,m,n1[0,σk]dW

)∗t

≥ m, for some n ∈ N∧ T, (1.24)

where the sequence (Zk,m,n1[0,σk])n∈N is the subsequence of (Zk,n1[0,σk])n∈N corre-sponding to the fast subsequence (Bk,m,n)n∈N. The definition of τk,m as well as theBurkholder-Davis-Gundy inequality imply that the sequence of processes(Zk,m,n1[0,σk]1[0,τk,m])n∈N is bounded in L2. The Alaoglu-Bourbaki theorem and theEberlein-Šmulian theorem in the Banach space L2 imply the existence of Zk,m ∈ L2,such that, up to a subsequence, (Zk,m,n1[0,σk]1[0,τk,m])n∈N converges weakly to Zk,m. Asa consequence of the Hahn-Banach theorem, there exists a sequence in the asymptoticconvex hull of (Zk,m,n1[0,σk]1[0,τk,m])n∈N, again denoted with (Zk,m,n1[0,σk]1[0,τk,m])n∈N,which converges in L2 to Zk,m. By taking another subsequence we also have the P ⊗dt-almost sure convergence.

29


Step 6: (τk,m)m∈N is a localizing sequence of stopping times. We estimate as follows

P[τk,m = T

]= P

[(∫Zk,m,n1[0,σk]dW

)∗T

< m , for all n ∈ N]

≥ 1− P[Bk,m,n ≥ m , for some n ∈ N

]≥ 1− P

[∣∣Bk,m.n − Bk∣∣ ≥ 1 , for some n ∈ N∪Bk > m− 1

]≥ 1−

∑n

P[∣∣Bk,m,n − Bk∣∣ ≥ 1

]− P

[Bk > m− 1

]≥ 1− 1

m−E[Bk + 1

]m

−−−−→m→∞

1,

where we used (1.22) in the second line and (1.23), the Markov inequality and the factthat E[Bk] <∞ in the last one.

Step 7: Construction of the candidate Z. For given k,m > 0, we constructed inStep 5 the process Zk,m as the L2 and P ⊗ dt-almost sure limit of a sequence inthe asymptotic convex hull of (Zk,m,n1[0,σk]1[0,τk,m])n∈N. With (Bk,m,n)n∈N we de-note the corresponding subsequence of convex combinations of (Bk,m,n)n∈N and notethat (

∫Zk,m,n1[0,σk]dW )∗T ≤ Bk,m,n, for all n ∈ N, as in (1.22). Hence, by the same

procedure as in Step 5, we can find, for m′ > m, a fast subsequence (Zk,m′,n1[0,σk])n∈Nin the asymptotic convex hull of (Zk,m,n1[0,σk])n∈N and a Zk,m′ ∈ L2 such that(Zk,m′,n1[0,σk]1[0,τk,m′ ])n∈N converges in L2 and P ⊗ dt-almost surely to Zk,m′ . We it-erate this procedure and define (Zk,n)n∈N as the diagonal sequence Zk,n = Zk,n,n andZk as

Zk0 = 0, Zk =∞∑m=1

Zk,m1]τk,m−1,τk,m]. (1.25)

From Zk,m′1[0,τk,m] = Zk,m, for m′ > m, follows that (Zk,n1[0,σk]1[0,τk,n])n∈N converges

in L2 and P ⊗ dt-almost surely to Zk. With the sequence (Zk,n)n∈N and the processZk at hand, we now diagonalize our program above with respect to k and n. As before,we get a diagonal sequence Zn = Zn,n, and a process Z given by

Z0 = 0, Z =∞∑k=1

1]σk−1,σk]Zk, (1.26)

such thatZn1[0,τn]

P⊗dt-almost surely−−−−−−−−−−−−→n→∞

Z, (1.27)

for τn = σn ∧ τn,n, where σn and τn,n are as in (1.18) and (1.24), respectively. Forlater reference, note that by construction holds Zk′,m1[0,σk]1[0,τk,m] = Zk,m, as soon ask′ ≥ k and also Z1[0,σk]1[0,τk,m] = Zk,m. Likewise (Zn1[0,τl])n∈N converges in L2 andP ⊗ dt-almost surely to Zl,l. This yields, via the Burkholder-Davis-Gundy inequality,

30


up to a subsequence,

t∧τl∫0

Zns dWs −−−−−→n→+∞

t∧τl∫0

ZsdWs, for all t ∈ [0, T ], P -almost surely. (1.28)

Hence, diagonalizing yields

t∫0

Zns dWs −−−−−→n→+∞

t∫0

ZsdWs, for all t ∈ [0, T ], P -almost surely. (1.29)

Step 8: Monotone convergence to Eg(ξ). Let Yt = limn Ynt , for t ∈ [0, T ], be the point-

wise monotone limit of the sequence (Y n). By monotone convergence Y is a super-martingale and, since our filtration is right-continuous, by standard arguments we maydefine the càdlàg supermartingale Y by setting Yt = lims↓t,s∈Q Ys, for all t ∈ [0, T ), andYT = ξ. By construction Ytin = Egtin (ξ). Hence, Yt = Egt (ξ), for all t ∈ [0, T ], and

Y nt ≥ Yt ≥ Egt (ξ) ≥ Egt (ξ) ≥ E

[ξ∣∣∣ Ft] , (1.30)

where the third inequality follows from Proposition 1.4. Now, the process Eg(ξ) is thenatural candidate for the value process of the minimal supersolution for two reasons. Itis càdlàg and it is dominated by Eg(ξ) as (1.30) shows. However, it is not clear a priorithat the sequence (Y n) converges to Eg(ξ) in some suitable sense. Taking into accountthe additional structure provided by the supermartingale property of the Y n we canprove nonetheless

Eg(ξ) = Y = limn→∞

Y n, P ⊗ dt-almost surely. (1.31)

Indeed, recall the decomposition Y nt = Y n0 −Ant +Mnt , for all t ∈ [0, T ], and the L1-bound

on (AnT ) given in Step 3. We now consider the sequence ((Y n, Zn)) in the asymptoticconvex hull of (Y n, Zn), which corresponds to the sequence (Zn) constructed in Step 7.From the decomposition of the Y n, see Lemma 1.3, we obtain that Y nt = Y n0 −Ant +Mn

t ,for all t ∈ [0, T ]. Moreover, the (AnT ) inherit the L1-bound given in Step 3. By means ofHelly’s theorem, see Lemma 1.25, we obtain the existence of a subsequence (An) in theasymptotic convex hull of (An) and the existence of an increasing positive integrableprocess A, such that limk→∞ Ankt = At, for all t ∈ [0, T ], P -almost surely. Consequently,by monotonicity of (Y n) and (1.29), Yt = Y0 − At + Mt, for all t ∈ [0, T ]. Hence, thejumps of Y are given by the countably many jumps of the increasing process A, whichimplies

Yt = Y0 − lims↓t,s∈Q

As + Mt, for all t ∈ [0, T ), YT = ξ.

Moreover, the jumptimes of the càdlàg process Y are exhausted by a sequence of stop-ping times (ρj) ⊂ T , which coincide with the jumptimes of A. Therefore, Y = Y ,

31


P ⊗ dt-almost surely, which implies (1.31).

Step 9: Verification. Let us now show that (Eg(ξ), Z) ∈ A (ξ, g), which, by means of(1.30), would end the proof. We start with the verification of (1.2) under the Assumption(Inc). Due to (1.31) there exists a Lebesgue nullset I ⊂ [0, T ], such that Egt (ξ) =limn→∞ Y nt , P -almost surely, for all t ∈ Ic. Let s, t ∈ Ic with s ≤ t. By using (1.29),the P ⊗ dt-almost sure convergence of Zn1[0,τn] to Z, and Fatou’s lemma we obtain

Egs −t∫s

gu(Egu, Zu)du+t∫s

ZudWu

≥ lim supn

Y ns − t∫s

gu(Egu, Znu1[0,τn](u))du+t∫s

ZnudWu

, (1.32)

where Y n denotes the convex combination of (Y n) corresponding to Zn. We denoteby λ

(n)i , n ≤ i ≤ M (n), λ(n)

i ≥ 0,∑i λ

(n)i = 1 the convex weights of Zn. Since our

generator fullills (Con), and since, for n large enough, we have Znu1[0,τn](u) = Znu , forall s ≤ u ≤ t, we may further estimate the above by

Egs −t∫s

gu(Egu, Zu)du+t∫s

ZudWu

≥ lim supn

M(n)∑i=n

λ(n)i

Y is − t∫s

gu(Egu, Ziu)du+t∫s

ZiudWu

.

Since Y it ≥ Egt (ξ) ≥ Egt (ξ), for all t ∈ [0, T ], and i ∈ N, we use (Inc) and the fact that

the (Y n, Zn) are supersolutions to conclude

Egs −t∫s

gu(Egu, Zu)du+t∫s

ZudWu

≥ lim supn

M(n)∑i=n

λ(n)i

Y is − t∫s

gu(Y iu, Ziu)du+t∫s

ZiudWu

≥ lim sup

n

M(n)∑i=n

λ(n)i Y it = lim sup

nY nt = lim sup

nY nt = Egt . (1.33)

As for the case of s, t ∈ I, with s ≤ t, we approximate them both from the right bysome sequences (sn) ⊂ Ic and (tn) ⊂ Ic, such that sn ↓ s, tn ↓ t, sn ≤ tn. For eachsn and tn holds (1.33). Passing to the limit by using the right-continuity of Eg and thecontinuity of −

∫g(Eg, Z)du +

∫ZdW we deduce that (1.33), holds for all s, t ∈ [0, T ]

with s ≤ t.

32


It remains to show admissibility of Z. By means of (1.33), (1.30), and positivity of g itholds

t∫0

ZsdWs ≥ E[ξ∣∣∣ Ft]− E0. (1.34)

Being bounded from below by a martingale, the continuous local martingale∫ZdW is

by Fatou’s lemma a supermartingale and thus Z is admissible. Hence, the proof underAssumptions (Pos), (Con) and (Inc) is completed.

The proof under (Dec) replacing (Inc) only differs in the verification of (1.2). Indeed,instead of only approximating Z in the Lebesgue integral we approximate Eg(ξ) P ⊗dt-almost surely with the sequence (Y n) as well, that is (1.32) becomes, by means of (1.31)and Fatou’s lemma,

Egs −t∫s

gu(Egu, Zu)du+t∫s

ZudWu

≥ lim supn

Y ns − t∫s

gu(Y nu , Znu1[0,τn](u))du+t∫s

ZnudWu

.

This entails, by monotonicity of the sequence (Y n) and the fact that the convex com-binations in Zn consist of elements of (Zi) with index greater or equal than n, that wemay write −

∫ tsgu(Y nu , Ziu)du ≥ −

∫ tsgu(Y iu, Ziu)du in (1.33) and this ends the proof.

Theorem 1.7 ensures the existence and uniqueness of the minimal supersolution whichis càdlàg. The following proposition provides a condition under which Eg(ξ) is in factcontinuous.

Proposition 1.8. Let g be a generator fulfilling (Pos), (Con) and either (Inc) or(Dec) and ξ ∈ L0 be a terminal condition, such that ξ− ∈ L1. Suppose that A(ξ, g) 6= ∅.Assume that for any ζ ∈ L∞ (Fτ ), τ ∈ T , there exist Y ∈ S and an admissible Z ∈ L,which solve the backward stochastic differential equation

Yt −τ∫t

gs (Ys, Zs) ds+τ∫t

ZsdWs = ζ, for all t ∈ [0, τ ].

Then Eg (ξ) is continuous.

Proof. In view of Theorem 1.7, there exists Z ∈ L such that (Eg, Z) ∈ A(ξ, g). Hence,Eg can only have negative jumps. Assume that Eg has a negative jump, that is P [τ ≤T ] > 0, for the stopping time τ = inft > 0 : ∆Egt < 0. We then fix m big enoughsuch that the stopping time τm = inft > 0 : |Egt | > m ∧ τ satisfies P [−m < ∆Egτm <

0 ∩ τm = τ] > 0. Since Eg is continuous on [0, τ [ and Eg has only negative jumps,Egτm ∨ −m ∈ L∞ (Fτm). By assumption there exist Y ∈ S and an admissible Z ∈ L

33


such that

Ys +τm∫s

gu(Yu, Zu

)−

τm∫s

ZudWu = Egτm ∨ −m, for all s ∈ [0, τm].

Similar to Lemma 1.1, we derive (Y 1[0,τm[ + Eg1[τm,T ], Z1[0,τm] + Z1]τm,T ]) ∈ A (ξ, g).Hence, by optimality of Eg in A(ξ, g) holds Eg ≤ Y 1[0,τm[ + Eg1[τm,T ]. Moreover, wehave

Egτm− > Egτm ∨ −m = Yτm = Yτm− on the set −m < ∆Egτm < 0 ∩ τm = τ .

Hence, for the stopping time τ = inft > 0 : Egt > Yt ∧ τm we deduce P [τ < τm] > 0,since the processes Eg and Y are continuous on [0, τm[. But then Eg Y on [0, τm[,which is a contradiction.

Under the assumptions of Theorem 1.7, Eg is the value process of the minimal su-persolution with a control process Z in L which defines a supermartingale. Next weaddress the question whether and under which conditions some stronger assumptionson the control process can be obtained. More precisely, that the corresponding controlprocess Z belongs to some Lp, for p ≥ 1, and therefore defines a true martingale insteadof a supermartingale. Defining

Ap (ξ, g) := (Y, Z) ∈ A (ξ, g) : Z ∈ Lp , (1.35)

this means that (Eg (ξ) , Z) ∈ Ap (ξ, g). Peng [1999] provides a positive answer to thatquestion in the case where p = 2, the terminal condition ξ ∈ L2 and the generator is notnecessarily positive but Lipschitz. Compare also with Cheridito and Stadje [2012] forsupersolutions of BSDEs where the control process is in BMO, if the terminal conditionis a bounded lower semicontinuous function of the Brownian motion and the generatoris convex in z and Lipschitz and increasing in y. Here, we provide an answer to the casewhere p = 1 in the context of Section 1.2. Given a terminal condition ξ, obtaining Eg (ξ)as a minimal solution with a control process within L1 comes at two costs. Indeed, astronger integrability condition on the terminal value is required, that is, we imposethat (E[ξ− | F·])∗T ∈ L1. As for the second cost, A1 (ξ, g) 6= ∅ is also required, which,in view of A1 (ξ, g) ⊂ A (ξ, g), is also a stronger assumption.

Theorem 1.9. Suppose that the generator g fulfills (Pos), (Con) and either (Inc) or(Dec). Let ξ ∈ L0 be a terminal condition, such that (E[ξ− | F·])∗T ∈ L1. If A1 (ξ, g) 6=∅, then there exists a unique minimal supersolution (Y , Z) ∈ A1(ξ, g). Moreover, Eg (ξ)is the value process of the minimal supersolution, that is (Eg (ξ) , Z) ∈ A1(ξ, g).

Remark 1.10. As in Section 1.2, note that for (Y,Z) ∈ A1 (ξ, g), the value process Y isa supermartingale with terminal value greater or equal than ξ. Moreover, we have Y ∗T ∈L1. Indeed, by using the decomposition (1.7), we derive Y ∗t ≤ |Y0|+ AT +

(∫ZdW

)∗T.

34


We further have AT ≤ Y0 +∫ T

0 ZsdWs−ξ and thus E [|AT |] ≤ Y0 +E [ξ−]. Consequently

E [Y ∗T ] ≤ |Y0|+ E[ξ−]

+ Y0 + E

∫ ZdW

∗T

<∞.

Proof (of Theorem 1.9). Since A1 (ξ, g) ⊂ A(ξ, g), the assumption A1 (ξ, g) 6= ∅ impliesthe existence of Z ∈ L such that (Eg(ξ), Z) ∈ A (ξ, g). We are left to show that Z ∈ L1.SinceA1 (ξ, g) 6= ∅, we can suppose in the proof of Theorem 1.7 that (Y 1, Z1) ∈ A1 (ξ, g).Since (1.15) holds for (Eg (ξ) , Z), instead of (Y n, Zn), we have(∫

ZdW

)∗T

≤∣∣Y 1

0∣∣+E

[ξ−]+ AT +(Y 1)∗T +

(E[ξ−∣∣∣ F·])∗

T, for all n ∈ N, (1.36)

where 0 ≤ E[AT ] ≤ E[ξ] − Y 10 . Since (E[ξ− | F·])∗T ∈ L1, by means of Remark 1.10,

the right hand side of (1.36), is in L1. Thus, by means of the Burkholder-Davis-Gundyinequality, Z belongs to L1.

1.3.2. Stability ResultsIn this section we address the stability of Eg(·) with respect to perturbations of theterminal condition or the generator. First we show that the functional Eg0 is not onlydefined on the same domain as the usual expectation, but also shares some of its mainproperties, such as Fatou’s lemma as well as a monotone convergence theorem.

Theorem 1.11. Suppose that the generator g fulfills (Pos), (Con) and either (Inc)or (Dec). Let (ξn) be a sequence in L0, such that ξn ≥ η, for all n ∈ N, where η ∈ L1.

• Monotone convergence: If (ξn) is increasing P -almost surely to ξ ∈ L0, then Eg0 (ξ) =limn Eg0 (ξn).

• Fatou’s lemma: Eg0 (lim infn ξn) ≤ lim infn Eg0 (ξn).

Proof. Monotone convergence: From Proposition 1.2 and by monotonicity, it followsthat Eg(ξn) ≤ Eg(ξn+1) ≤ · · · ≤ Eg(ξ). Hence, we may define Y0 = limn Eg0 (ξn). Notethat Y0 ≤ Eg0 (ξ). If Y0 = +∞, then also Eg0 (ξ) = +∞ and there is nothing to prove.Suppose now that Y0 <∞. This implies that A(ξn, g) 6= ∅, for all n ∈ N. Since ξn ≥ η,Proposition 1.4 yields (ξn) ⊂ L1 and (Eg(ξn)) is a well-defined increasing sequence ofcàdlàg supermartingales. We define Yt = limn Egt (ξn), for all t ∈ [0, T ]. Note thatY0 = Y0. We show that Y is a càdlàg supermartingale.To this end, note that the sequence

(Eg(ξn)− Eg(ξ1)

)is positive and increases to

Y − Eg(ξ1). Therefore monotone convergence yields

0 ≤ E[Yt − Egt (ξ1)] = limnE[Egt (ξn)− Egt (ξ1)].

35


The supermartingale property of Eg(ξn) implies that E[Egt (ξn)] ≤ Eg0 (ξn) ≤ Y0. Fur-thermore, E[ξ1] ≤ E[Egt (ξ1)] ≤ Y0 and thus

0 ≤ E[Yt − Egt (ξ1)] ≤ −E[ξ1] + Y0 < +∞.

From Egt (ξ1) ∈ L1, we deduce that Yt ∈ L1. Since ξ = YT , this implies in particularthat ξ ∈ L1. The supermartingale property follows by a similar argument. Moreover,[Dellacherie and Meyer, 1982, Theorem VI.18] implies that Y is indistinguishable froma càdlàg process. Hence, Y is a càdlàg supermartingale.Theorem 1.7 provides a sequence of optimal controls (Zn) such that (Eg(ξn), Zn) ∈A(ξn, g), for all n ∈ N. Now we apply the procedure introduced in the proof of Theorem1.7 and obtain a candidate control process Z. The only notable difference in the proof,except for the fact that Y is already càdlàg, is that, here, the sequence (Eg(ξn)) is in-creasing instead of decreasing. Thus, the càdlàg supermartingales Y and Eg(ξ1) serve asupper and lower bound, respectively. Consequently, we replace Y 1 by Y and E[ξ− | F·]by Eg(ξ1) in the key Inequality (1.15). The verification follows exactly the same argu-mentation as in the proof of Theorem 1.7 for both monotonicity Assumptions (Inc) and(Dec). Finally, to get the admissibility of Z we denote with (ξn) the sequence of convexcombinations of (ξn) corresponding to (Zn). Monotonicity of the sequence (ξn) impliesξ1 ≤ ξn ≤ ξ, for all n ∈ N. We may and do switch to a subsequence such that (ξn) isincreasing as well. Now, fix an arbitrary t ∈ [0, T ]. Dominated convergence implies theL1-convergence limnE[ξn | Ft] = E[ξ | Ft]. Hence, we may select a subsequence suchthat we have P -almost sure convergence. Similar to (1.34) this implies

Y0 −t∫

0

gu(Yu, Zu)du+t∫

0

ZudWu ≥ lim supn

E[ξn∣∣∣ Ft] = E

[ξ∣∣∣ Ft] .

As before, this entails that (Y, Z) ∈ A(ξ, g). Hence, from A(ξ, g) 6= ∅ and ξ− ∈ L1

we derive by Theorem 1.7 that there exists a control process Z such that (Eg(ξ), Z) ∈A(ξ, g). In particular this yields Y0 = Eg0 (ξ), that is limn Eg0 (ξn) = Eg0 (ξ), since otherwiseEg0 (ξ) were not optimal.Fatou’s lemma: The result follows by applying monotone convergence. Indeed, denote

by ζn the random variables ζn = infk≥n ξk. Then from lim infn ξn = limn ζn, ζn ≥ η,

ζn ≤ ξn, for all n ∈ N, and monotone convergence follows

Eg0 (lim infn

ξn) = Eg0 (limnζn) = lim

nEg0 (ζn) ≤ lim inf

nEg0 (ξn).

Remark 1.12. An inspection of the proof of Theorem 1.11 shows that under the as-sumptions implying monotone convergence, if limn Eg0 (ξn) < +∞, then A (ξ, g) 6= ∅and Egt (ξn) converges P -almost surely to Egt (ξ), for all t ∈ [0, T ].Similarly, given a sequence ((Y n, Zn)) ⊂ A(ξ, g) such that (Y n) is increasing and

limn Yn

0 < ∞, then there exists a control process Z ∈ L such that (Y,Z) ∈ A(ξ, g),where Yt is the P -almost sure limit of (Y nt ), for all t ∈ [0, T ].

36


A consequence of the preceding theorem is the following result on L1-lower semiconti-nuity.

Theorem 1.13. Let g be a generator fulfilling (Pos), (Con) and either (Inc) or(Dec). Then Eg0 is L1-lower semicontinuous.

Proof. Let (ξn) be a sequence of terminal conditions, which converges in L1 to a randomvariable ξ. Suppose that there exists a subsequence (ξn) ⊂ (ξn) such that (Eg0 (ξn))converges to some real a < Eg0 (ξ). We can assume, up to another fast subsequence, that‖ξn − ξ‖L1 ≤ 2−n, for all n ∈ N. Consider now the sequence (ζn), with ζn given by

ζn = ξ −∑k≥n

(ξk − ξ)−.

Clearly, ζn ∈ L1 and ζn ≤ ζn+1 ≤ · · · ≤ ξ. Moreover, (ζn) converges in L1 to ξ, and,since it is increasing, it converges also P -almost surely. Thus, from Theorem 1.11, weget limn Eg0 (ζn) = Eg0 (ξ). Now, ζn ≤ ξ−(ξn−ξ)−+(ξn−ξ)+ ≤ ξn and monotony of thefunctional Eg0 imply a = limn Eg0 (ξn) ≥ limn Eg0 (ζn) = Eg0 (ξ), which is a contradiction.Hence, lim infn Eg0 (ξn) ≥ Eg0 (ξ).

The preceding results allows to derive a dual representation, by means of the Fenchel-Moreau theorem, of the functional Eg(·) at time zero.

Corollary 1.14. Let g be a generator fulfilling (Pos) and either (Inc) or (Dec). As-sume that g is jointly convex in y and z. Then, either Eg0 ≡ +∞ or

Eg0 (ξ) = Eg0 (ξ) = infν∈L∞+

E[νξ]−

(Eg0)∗

(ν), ξ ∈ L1, (1.37)

for the conjugate (Eg0 )∗(ν) = infξ∈L1

E[νξ]− Eg0 (ξ)

, where ν ∈ L∞.

Proof. Since Eg0 > +∞ on L1, either Eg0 ≡ +∞ or Eg0 is proper. In the latter case, inview of Proposition 1.2 and Theorem 1.13, the function Eg0 is convex and σ(L1, L∞)-lower semicontinuous on L1. Hence, the Fenchel Moreau theorem yields the dual rep-resentation (1.37). That the domain of (Eg0 )∗ is concentrated on L∞+ follows from themonotonicity of Eg0 , see Proposition 1.2.

Remark 1.15. Notice that, if the generator in Corollary 1.14 does not depend on y, thenby Item 5 of Proposition 1.2 the operator Eg0 (·) is translation invariant. Therefore, it isa lower semicontinuous, convex risk measure and the Representation (1.37) correspondsto the robust representation of lower semicontinuous, convex risk measures; see Föllmerand Schied [2004].

Under additional integrability assumptions on the terminal condition we may also for-mulate stability results for supersolutions in the set A1(ξ, g) introduced in (1.35).

Theorem 1.16. Suppose that the generator g fulfills (Pos), (Con) and either (Dec)or (Inc). Let (ξn) be a sequence in L0, such that ξn ≥ η, for all n ∈ N, where(E[η | F·])∗T ∈ L1.

37


• Suppose (ξn) is increasing P -almost surely to ξ ∈ L0, where (E[ξ− | F·])∗T ∈ L1 andA1(ξ, g) 6= ∅. Then Egt (ξ) = limn Egt (ξn), P -almost surely, for all t ∈ [0, T ].

• Suppose (E[(lim infn ξn)− | F·])∗T ∈ L1 and A1(lim infn ξn, g) 6= ∅.Then Egt (lim infn ξn) ≤ lim infn Egt (ξn), P -almost surely, for all t ∈ [0, T ].

We omit the proof of the preceding theorem, as it is a simple adaptation of the proofs ofTheorems 1.9 and 1.11. Note that Theorem 1.16 is a weaker version of Theorem 1.11.Indeed, here, given a sequence (ξn) increasing to ξ, we need to assume that A1 (ξ, g)is not empty. The underlying reason being the lack of knowledge whether the limitprocess Y , defined in the proof of Theorem 1.11, fulfills Y ∗T ∈ L1.The theorem above allows to state the following result on ‖ · ‖L1 -lower semicontinuity

of Eg. Its proof is virtually the same as the proof of Theorem 1.13.

Theorem 1.17. Suppose that the generator g fulfills (Pos), (Con) and either (Dec)or (Inc). Then ξ 7→ Eg0 (ξ) is ‖·‖L1-lower semicontinuous on its domain, that is on

ξ ∈ L0 : (E[ξ− | F·])∗T ∈ L1 and A1 (ξ, g) 6= ∅. (1.38)

We conclude this section with a theorem on monotone stability with respect to thegenerator.

Theorem 1.18. Let ξ ∈ L0 be a terminal condition, such that ξ− ∈ L1, and let (gn)be an increasing sequence of generators, which converge pointwise to a generator g.Suppose that each generator fulfills (Pos), (Con) and either (Inc) or (Dec). Thenlimn Eg

n

0 (ξ) = Eg0 (ξ). If, in addition, limn Egn

0 (ξ) < ∞, then A(ξ, g) 6= ∅ and Egn

t (ξ)converges P -almost surely to Egt (ξ), for all t ∈ [0, T ].

Proof. Note that from Proposition 1.2, we have Egn(ξ) ≤ Egn+1(ξ) ≤ · · · ≤ Eg(ξ).Hence, we may set Y0 = limn Eg

n

0 (ξ). If Y0 =∞, then also Eg0 (ξ) =∞ and we are done.Suppose that Y0 < ∞. By the same arguments as in the proof of Theorem 1.11, weconstructa a càdlàg supermartingale Y . With the same procedure as in Theorem 1.11,we construct the candidate Z. It remains to show (Y, Z) ∈ A(ξ, g). However, this canbe done similarly as in the proof of Theorem 1.7. We only show how to obtain theanalogue of (1.33). Note first that the pointwise convergence of the generators impliesthat (gk(Y, Z)) converges P ⊗ dt-almost surely to g(Y, Z). Hence, Fatou’s lemma yields

Ys −t∫s

gu(Yu, Zu)du+t∫s

ZudWu ≥ lim supk

Ys − t∫s

gku(Yu, Zu)du+t∫s

ZudWu

.

(1.39)As in the previous proof, we use the expression in the bracket on the right hand side to

38


obtain

Ys −t∫s

gku(Yu, Zu)du+t∫s

ZudWu

≥ lim supn

M(n)∑i=n

λ(n)i

Y is − t∫s

gku(Y iu, Ziu)du+t∫s

ZiudWu

.

Since on the right hand side we consider the lim sup with respect to n and k being fixedfor the moment, we may assume k ≤ n, which entails by monotonicity of the sequenceof generators

Ys −t∫s

gku(Yu, Zu)du+t∫s

ZudWu

≥ lim supn

M(n)∑i=n

λ(n)i

Y is − t∫s

giu(Y iu, Ziu)du+t∫s

ZiudWu

.

From here, we obtain as before Ys −∫ tsgku(Yu, Zu)du+

∫ tsZudWu ≥ Yt, where the right

hand side does not depend on k anymore. Combined with (1.39), this yields the analogueof (1.33).

Remark 1.19. Similar to Theorem 1.11 one may formulate extensions of Theorem 1.18,allowing not only for monotone increasing approximations of the limit generator g.Consider for example a sequence of generators (gn) and a generator g, that neitherdepend on y nor, for simplicity, on (ω, t). If in addition to limn g

n(z) → g(z) wehave limn h

n(z) → g(z), where hn is defined by hn = convg, gn, gn+1, · · ·

, then by

Proposition 1.2 and Theorem 1.18, lim infn Egn

0 (ξ) ≥ lim infn Ehn

0 (ξ) = Eg0 (ξ). To giveprecise conditions in a general setting under which the convergence of gn to g impliesconvergence of hn to g is not in line with our main objective and consequently thisquestion is left for future investigations.

1.3.3. Non positive generatorsIn this section we extend our results to generators that are not necessarily positive. Thisis important with regards to applications in mathematical finance, where the generatorsare quite often of the linear-quadratic type. It turns out that we can extend the scopeof our theorems to cover precisely some of these situations; see Section 1.3.4 for suchan example. Using some measure change, the positivity assumption on the generator gcan be relaxed to a linear bound below. This leads to optimal solutions under P , wherethe admissibility is required with respect to the related equivalent probability measure.More precisely, we say in the following that a generator g is

39


(Lb) linearly bounded from below, if there exist adapted measurable R1×d and R-valued processes a and b, respectively, such that g(y, z) ≥ az> + b, for ally, z ∈ R× R1×d. Furthermore,

∫ t0 bsds ∈ L

1(P a), for all t ∈ [0, T ] and

dP a

dP= E

(∫adW

)T

,

defines an equivalent probability measure P a.

Example 1.20. For instance, given a generator g, assume that there exists a generatorg independent of y fulfilling (Con) and such that g ≥ g. Then, there exists an R1×d-valued adapted measurable process a such that g (y, z) ≥ az> − g∗(a), for all y, z ∈R× R1×d, where g∗ denotes the convex conjugate of g. ♦

In the following, we say that Z is a-admissible, if∫ZdW a is a P a-supermartingale,

where W a = (W 1 −∫a1ds, · · · ,W d −

∫adds)> is the respective Brownian motion

under P a. We are interested in the sets

Aa (ξ, g) = (Y, Z) ∈ S × L : Z is a-admissible and (1.2) holds , (1.40)

and define the random process

Eg,at (ξ) = ess infYt ∈ L0 (Ft) : (Y,Z) ∈ Aa (ξ, g)

, t ∈ [0, T ] . (1.41)

The analogue of Theorem 1.7 is given as follows

Theorem 1.21. Let g be a generator fulfilling (Lb), (Con) and either (Inc) or (Dec)and ξ ∈ L0 be a terminal condition, such that ξ− ∈ L1(P a). If Aa(ξ, g) 6= ∅, then thereexists a unique minimal supersolution (Y , Z) ∈ Aa(ξ, g). Moreover, Eg (ξ) is the valueprocess of the minimal supersolution, that is (Eg (ξ) , Z) ∈ Aa(ξ, g).

The analogues of Theorem 1.11 and Theorem 1.13 read as follows.

Theorem 1.22. Suppose that the generator g fulfills (Lb), (Con) and either (Inc) or(Dec). Let (ξn) be a sequence in L0, such that ξn ≥ η, for all n ∈ N, where η ∈ L1(P a).

• Monotone convergence: If (ξn) is increasing P -almost surely to ξ ∈ L0, then Eg,a0 (ξ) =limn Eg,a0 (ξn).

• Fatou’s lemma: Eg,a0 (lim infn ξn) ≤ lim infn Eg,a0 (ξn).

In particular, Eg,a0 is L1 (P a)-lower semicontinuous.

We only prove the first theorem.

Proof (of Theorem 1.21). In the setting of Section 1.3.1, given a positive generator gand a random variable ζ, let us denote by A (ζ, g,W a) the set defined in (1.4) to indicate

40


the dependence of this set on the Brownian motion W a and the respective probabilitymeasure P a. Let us now define the generator g as

g (y, z) = g

y +·∫

0

bsds, z

− az> − b, for all (y, z) ∈ R× R1×d. (1.42)

By Assumption (Lb), this generator fulfills (Pos), (Con) and either (Inc) or (Dec).Since

∫ZdW a is a P a-supermartingale, a simple inspection shows that the affine trans-

formation Y = Y −∫bds and Z = Z yields a one-to-one relation between Aa(ξ, g) and

A(ξ −∫ T

0 bsds, g,Wa). Hence, the assumptions of Theorem 1.7 are fulfilled for g and

A(ξ −∫ T

0 bsds, g,Wa), and thus its application ends the proof.

Remark 1.23. Note that if (Ea[(ξ −∫ T

0 bsds)− | F·])∗T ∈ L1 (P a), then Theorem 1.9applies in the same way, that is, under the assumptions of Theorem 1.21, if

A1,a (ξ, g) :=

(Y, Z) ∈ Aa (ξ, g) : Z ∈ L1 (P a)6= ∅,

then Eg,a (ξ) is the value process of the minimal supersolution with unique controlprocess Z ∈ L1 (P a).

1.3.4. Expected exponential utility maximization

In this section, we illustrate our results, in particular our set based comparison principleand the existence theorem, by maximizing exponential expected utility. For an intro-duction to this well-known problem and for similar statements, see for instance Delbaenet al. [2002], Hu et al. [2005], Mania and Schweizer [2005], and the references therein.Consider a financial market where the discounted stock prices are modelled by a

n-dimensional process S satisfying the differential equation

dSiu = Siu(µiudu+ σiudWu

), Si0 > 0, i = 1, . . . , n, (1.43)

where n ≤ d, and the drift µ = (µ1, . . . , µn)> and the volatility row-vectors σi =(σi1, . . . , σid) are progressive processes. We suppose that σσ> is invertible, P ⊗ dt-almost surely. Hence, without loss of generality, we may assume that σij = 0, for all j =n+1, . . . , d, and that the n×n-matrix σij = σij , i, j = 1, . . . , n, is invertible P⊗dt-almostsurely. By use of the d-dimensional market price of risk vector θ = (σ−1µ, 0, . . . , 0)>,the dynamics in (1.43) are equivalent to dSiu = Siuσ

iu(θudu + dWu), for i = 1, . . . , n.

We further assume that1 ∫ θ>dW ∈ BMO(P ). Therefore, E(−∫θ>dW

)T

definesan equivalent probability measure P−θ. According to Girsanov’s change of measuretheorem, the discounted price process S is then a local martingale under P−θ. We

1For the definition and important results concerning the space BMO, we refer to Kazamaki [1994].

41


consider the problem

Uϑt = E

exp

−ξ − T∫0

ϑudSu

∣∣∣ Ft→ min, (1.44)

where ξ ∈ L0(FT ) is a random endowment and ϑ an R1×n-valued admissible strategy.Here, the optimization takes place over the set Θ of those strategies ϑ such that ϑis progressive,

∫ϑdS is a P−θ-supermartingale and exp(−

∫ τ0 ϑdS))τ∈T is P -uniformly

integrable.In principle, the following proposition is well-known, see the references mentioned

above. However, our approach relies exclusively on the theory of supersolutions de-veloped in the previous sections and extends the method of pointwise optimization ofgenerators used in Horst et al. [2010] to our setting of an incomplete market.

Proposition 1.24. Suppose that∫θ>dW ∈ BMO(P ) and ξ ∈ L∞. Then, ϑ∗ defined

by the following componentwise multiplication between ϑ∗ and S,

ϑ∗tSt = (Zt + θ>t )(σ−1t

0

)= (Z1

t , . . . , Znt )σ−1

t + (θ1t , . . . , θ

nt )σ−1

t , (1.45)

is the optimal solution in Θ of (1.44). Here, Z ∈ L1(P−θ) is the control process corre-sponding to the minimal supersolution of the BSDE

Ys −t∫s

[12

(d∑

k=n+1(Zku)2 −

n∑k=1

(θku)2

)−

n∑k=1

θkuZku

]du+

t∫s

ZudWu ≥ Yt

and YT ≥ −ξ. (1.46)

Proof. Let ϑ ∈ Θ∞, the set of those strategies in Θ such that∫ϑdS is uniformly

bounded. The general result follows by stopping any strategy ϑ as soon as |∫ϑdS| is

above n ∈ N and using the uniform integrability to show that Uϑnt converges P -almostsurely to Uϑt , for all t ∈ [0, T ], where (ϑn) is the stopped strategy. In view of Itô’slemma, the certainty equivalent Y ϑt = log(Uϑt ) +

∫ t0 ϑudSu satisfies the BSDE

Y ϑs −t∫s

gu(ϑu, Z

ϑu

)du+

t∫s

ZϑudWu = Y ϑt , Y ϑT = −ξ, (1.47)

where the control process is given by Zϑt = V ϑt /Uϑt + ϑtStσt, with V ϑ coming from the

martingale representation of Uϑ. The generator is given by

gu (ϑ, z) = 12 (z − ϑuSuσu)2 − ϑuSuµu,

for all z ∈ R1×d, and, from our assumption on θ, it follows that∫ZϑdW−θ is a P−θ-

42


supermartingale. This implies that (Y ϑ, Zϑ) ∈ A−θ (−ξ, g (ϑ, ·)). Pointwise optimiza-tion of ϑ 7→ 1

2 (z − ϑSuσu)2 − ϑSuµu, for ϑ ∈ R1×n, yields a pointwise minimum ϑ∗ (z)such that

ϑ∗u (z)Su = (z + θ>t )(σ−1t

0

).

Plugged into the generator gu (ϑ, z), it yields the optimized generator

g∗u (z) = 12

(d∑

k=n+1(zk)2 −

n∑k=1

(θku)2

)−

n∑k=1

θkuzk.

Now we use our comparison result, see Proposition 1.2, to obtain

A−θ (−ξ, g (ϑ, ·)) ⊂ A−θ (−ξ, g∗) , for all ϑ ∈ Θ∞,

andEg∗,−θ (−ξ) ≤ Y ϑ,

for all ϑ ∈ Θ∞. Next we want to use existence and uniqueness of the minimal super-solution given by Theorem 1.21 to ensure that Eg∗,−θ (−ξ) has indeed a modificationEg∗,−θ (−ξ) which is the value process of the minimal supersolution and to obtain thecorresponding control process. To that end note the generator g∗ fulfills

g∗u (z) ≥ −12

n∑k=1

(θku)2 − θ>z>

and that this is a valid lower bound in the sense of Assumption (Lb) in Section 1.3.3.From our assumptions on θ and ξ follows

(E−θ[|ξ|+ 12

T∫0

n∑k=1

(θku)2du | F·])∗T ∈ L1(P−θ).

This and ξ bounded below imply that

(∥∥ξ−∥∥∞ − 1

2

∫ n∑k=1

(θk)2du, 0) ∈ A1,−θ(−ξ, g∗),

that is A1,−θ(−ξ, g∗) 6= ∅. Hence, all the assumptions of Theorem 1.21 are fulfilled,which, together with Remark 1.23, yields that Y ∗ := Eg∗,−θ (−ξ) is the value processof the minimal supersolution of (1.46) with control process Z∗ ∈ L1(P−θ). Definingϑ∗ := ϑ∗ (Z∗), yields

(Y ∗, Z∗) = (Y ϑ∗, Z∗) ∈ A1,−θ (−ξ, g (ϑ∗, ·)) .

We finally have to show that ϑ∗ ∈ Θ and that Uϑ∗0 is minimal. From∫θ>dW−θ ∈

BMO(P−θ) it follows that∫ϑ∗dS =

∑nk=1

∫(Z∗k + θk)dW−θ,k ∈ L1(P−θ). Thus,

43


∫ϑ∗dS is a P−θ-(super)martingale. Furthermore, by (1.46), for all stopping times

τ ∈ T , holds

E−θ

T∫τ

d∑k=n+1

(Z∗k)2du | Fτ

≤ 2Y ∗τ + 2E−θ[ξ | Fτ ] + E−θ

T∫τ

d∑k=n+1

(θk)2du | Fτ

.From Y ∗ ≤ ‖ξ−‖∞ and ξ ≤ ‖ξ+‖∞, it follows(∫

Z∗n+1dW−θ,n+1, · · · ,∫Z∗ddW−θ,d

)∈ BMO(P−θ).

Thus, for V ∗ := Z∗ − ϑ∗Sσ, it holds

∫V ∗dW−θ = −

n∑k=1

∫θkdW−θ,k +

d∑k=n+1

∫Z∗kdW−θ,k ∈ BMO(P−θ),

and therefore,∫V ∗dW ∈ BMO(P ). Since

∫ϑ∗dS =

∫ϑ∗SσdW−θ and ϑ∗Sσ = Z∗ +

V ∗, the admissibility of Z∗ yields that∫ϑ∗dS is a P−θ-supermartingale. The process

Cϑ∗ = Y ∗ −

∫ϑ∗dS satisfies

Cϑ∗

s −t∫s

12(V ∗u )2du+

t∫s

V ∗u dWu ≥ Cϑ∗

t , Cϑ∗

T = −ξ −T∫

0

ϑ∗udSu. (1.48)

Consequently,

exp

− t∫0

ϑ∗dS

= exp(Cϑ∗

t − Y ∗t)≤ E

(∫V ∗dW

)t

exp(Cϑ∗

0 − Y ∗t).

Hence, from∫V ∗dW ∈ BMO(P ) and Y ∗ ≥ −‖ξ‖∞, it follows that the family

(exp(−∫ τ

0 ϑ∗dS))τ∈T is P -uniformly integrable. Thus, ϑ∗ ∈ Θ. As for the optimality

of ϑ∗, note that, for ϑ ∈ Θ∞, holds

E

exp

−ξ − T∫0

ϑ∗udSu

∣∣∣ Ft = E

[exp

(Cϑ∗

T

) ∣∣∣ Ft]

≤ exp(Cϑ∗

t

)= exp

Y ∗t − t∫0

ϑ∗udSu

≤ exp

Y ϑt − t∫0

ϑ∗udSu

,

which implies

E

exp

−ξ − T∫t

ϑ∗udSu

∣∣∣ Ft ≤ E

exp

−ξ − T∫t

ϑudSu

∣∣∣ Ft .

44

1.4. Helly’s theorem

1.4. Helly’s theoremIn principle the following result is well-known, see for example [Campi and Schacher-mayer, 2006, De Vallière et al., 2009, Kupper and Schachermayer, 2009] for similarstatements. We give a proof for sake of completeness.

Lemma 1.25. Let (An) be a sequence of increasing positive processes such that thesequence (AnT ) is bounded in L1. Then, there is a subsequence (Ank) in the asymptoticconvex hull of (An) and an increasing positive integrable process A such that

limk→∞

Ankt = At, for all t ∈ [0, T ], P-almost surely. (1.49)

Proof. Let (tj) be a sequence running through I = ([0, T ]∩Q)∪ T. Since (Ant1) is anL1-bounded sequence of positive random variables, due to [Delbaen and Schachermayer,1994, Lemma A1.1], there exists a subsequence (A1,k) in the asymptotic convex hullof (An) and a random variable At1 such that (A1,k

t1 ) converges P -almost surely to At1 .Moreover, Fatou’s lemma yields At1 ∈ L1. Let (A2,k) be a subsequence in the asymptoticconvex hull of (A1,k) such that (A2,k

t2 ) converges P -almost surely to At2 ∈ L1 and soon. Then, Ak,kt → At, on a set Ω ⊂ Ω, where P (Ω) = 1. The process A is positive,increasing and integrable on I. Thus, we may define

At = limr↓t,r∈I

Ar, t ∈ [0, T ), AT = AT .

We now show that (Ak,k), henceforth named (Ak), converges P -almost surely on thecontinuity points of A. Fix ω ∈ Ω and a continuity point t ∈ [0, T ) of A(ω). We showthat (Akt (ω)) is a Cauchy sequence in R.Fix ε > 0 and set δ = ε/11. Since t is a continuity point of A(ω), we may choose

p1, p2 ∈ I such that p1 < t < p2 and Ap2(ω) − Ap1(ω) < δ. By definition of A, wemay choose r1, r2 ∈ I such that p1 < r1 < t < p2 < r2 and |Ap2(ω) − Ar2(ω)| < δ and|Ap1(ω) − Ar1(ω)| < δ. Now choose N ∈ N such that |Amr1

(ω) − Anr1(ω)| < δ, for all

m,n ∈ N with m,n ≥ N , and |Ajr2(ω) − Ar2(ω)| < δ and |Ar1(ω) − Ajr1

(ω)| < δ, forj = m,n. We estimate

|Amt (ω)− Ant (ω)| ≤ |Amt (ω)− Amr1(ω)|+ |Amr1

(ω)− Anr1(ω)|+ |Anr1

(ω)− Ant (ω)|.

For the first and the third term on the right hand side, since Am and An are increasing,we deduce that |Amt (ω)−Amr1

(ω)| ≤ |Amr2(ω)−Amr1

(ω)| and |Ant (ω)−Anr1(ω)| ≤ |Anr2

(ω)−Anr1

(ω)|. Moreover

|Ajr2(ω)− Ajr1

(ω)| ≤ |Ajr2(ω)− Ar2(ω)|+ |Ar2(ω)− Ap2(ω)|

+ |Ap2(ω)− Ap1(ω)|+ |Ap1(ω)− Ar1(ω)|+ |Ar1(ω)− Ajr1(ω)|,

for j = m,n. Combining the previous inequalities yields |Amt (ω) − Ant (ω)| ≤ ε, for alln,m ≥ N . Hence, (Ak(ω)) converges for all continuity points t ∈ [0, T ) of A(ω), for all

45


ω ∈ Ω. We denote the limit with A.It remains to see that (Ak) also converges for the discontinuity points of A. To this

end, note that A is càdlàg and adapted to our filtration which fulfills the usual conditions.By a well-known result, see for example [Karatzas and Shreve, 2004, Proposition 1.2.26],this implies that the jumps of A may be exhausted by a sequence of stopping times (ρj).Applying once more [Delbaen and Schachermayer, 1994, Lemma A1.1] iteratively on thesequences (Akρj )k∈N, j = 1, 2, 3, . . . , and diagonalizing yields the result.

46

Part II.

Cross Hedging

47

2. Futures Cross-hedging with aStationary Spread

In the introduction we have seen that the estimated correlation between the log prices ofkerosene and crude oil strongly depends on the sampling frequency. More precisely, theshort-term correlations are considerably lower than the long-term correlations, pointingtowards a long-term relationship with potential short-term deviations. Motivated bythis observation we set up a model, with mean reverting, or asymptotic stationaryspread, that allows a rigorous study of the effect of a long-term relationship on optimalcross-hedging strategies, and at the same time allows an efficient calculation of the basisrisk entailed by the optimal cross-hedges.The chapter is structured as follows. Section 2.1 introduces our model and presents

some empirical evidence, while Section 2.2 briefly reviews hedging with futures con-tracts and derives the variance optimal hedging strategy for our model. Section 2.3develops the implied hedge errors within our model for linear and non-linear positionsand Section 2.4 compares the hedge errors between different models and (suboptimal)hedging strategies emphasizing the importance of allowing for a stationary spread. Anextension of our model to account for stochastic volatility is given in Section 2.5. Section2.6 concludes.

2.1. The Continuous-time Model with a StationarySpread

As always in modeling real-world phenomena there is a trade-off between the accuracy ofa model and its tractability. We therefore illustrate the implications of an asymptoticstationary logspread between the futures price and the price of an illiquid asset byassuming a simplified and tractable model, which is presented in Section 2.1.1. Thisapproach allows us to derive not only optimal hedging strategies and their implied hedgeerrors (see Section 2.2), but also to obtain the transition density in closed form, sothat the model can straightforwardly be estimated via the efficient maximum likelihoodmethod. An empirical illustration is provided in Section 2.1.2.

2.1.1. Model SpecificationLet I = (It)t≥0 denote the price process of an illiquid asset, and suppose that aneconomic agent aims at hedging a position h(IT ), where h : R → R is a measurablepayoff function and T > 0 is a fixed time horizon. Furthermore, we assume that there

49

2. Futures Cross-hedging with a Stationary Spread

exists a liquidly traded futures contract with price process X = (Xt)t≥0, which evolvesaccording to

dXt = µXtdt+ σXXtdW(X)t , X0 = x, (2.1)

with volatility σX > 0 and constant drift rate µ ∈ R. The process W (X) =(W

(X)t

)t≥0

is a Brownian motion on a stochastic basis with probability measure P . We denote thespread of the log prices, in the following simply referred to as the logspread, by

St = log(Xt)− log(It).

Although the non-stationarity of the logspread seems to be a plausible assumption forcertain asset classes, e.g. for stock prices, there also exist relevant examples for asymp-totic stationary logspreads as shown in the introduction and the mentioned articles.We therefore propose to account for cointegration by first modeling the logspread asan asymptotic stationary process and then derive the implied dynamics of the illiquidasset.More precisely, we assume that the logspread follows an Ornstein-Uhlenbeck process,

that is the logspread solves the SDE

dSt = κ(m− St)dt+ σS

(ρdW

(X)t + ρdW⊥t

), S0 = s, (2.2)

where W⊥ =(W⊥t

)t≥0 is a Brownian motion independent of W (X), κ ≥ 0 is the mean

reversion speed, and ρ ∈ [−1, 1] the correlation. The logspread’s volatility σS is assumedto be non-negative. Moreover, we define ρ =

√1− ρ2 and use, for ease of exposition,

the following short-hand notation

W(S)t = ρW

(X)t + ρW⊥t

for the Brownian motion driving the logspread. Note that for κ ↓ 0 the Ornstein-Uhlenbeck process becomes more and more persistent and in the limit a (scaled andshifted) Brownian motion that is correlated with the Brownian motion of the futuresprice process.The dynamics of X and S determine the dynamics of the illiquid asset price, as

It = Xte−St , t ≥ 0. A straightforward calculation shows that the dynamics of I satisfy

dIt = It

(12σ

2S − κ(m− St) + µ− ρσSσX

)dt+ ItσIdW

(I)t ,

where σI =√σ2X − 2ρσSσX + σ2

S andW (I) =(W

(I)t

)t≥0

is a Brownian motion defined

by W (I)t =

((σX − ρσS)W (X)

t − ρσSW⊥t)/σI , t ≥ 0.

Note that the correlation ρIX between the Brownian motions driving I and X is givenby

ρIX = 1σI

(σX − ρσS) , (2.3)

50

2.1. The Continuous-time Model with a Stationary Spread

which is non-negative if and only if σX ≥ ρσS . For fixed parameters ρ and σX , we canwrite ρIX as a function of σS :

ρIX(σS) = σX − ρσS√σ2X − 2ρσSσX + σ2

S

. (2.4)

It can be shown that ρIX(σS) is strictly decreasing in σS , and hence invertible on R+.

Lemma 2.1. Let σX > 0 and ρ ∈ (−1, 1). Then the mapping R+ 3 σS 7→ ρIX(σS) isstrictly decreasing. Moreover, the logspread’s volatility σS satisfies

σS = σX

√1− ρ2

IX

ρ√

1− ρ2IX + ρIX

√1− ρ2

. (2.5)

Proof. By distinguishing the cases ρ ≥ 0 and ρ < 0 one can show that ∂ρIX/∂σS ≤ 0,and that the partial derivative is strictly smaller than zero if σS > 0. Thus, ρIX isstrictly decreasing in σS . From the definition of ρIX we have

ρ2IX = (σX − ρσS)2

/(σ2X − 2ρσSσX + σ2

S

),

which leads to the quadratic equation in σS

(ρ2IX − ρ2)σ2

S + 2ρσX(1− ρ2IX)σS − σ2

X(1− ρ2IX) = 0. (2.6)

If ρ 6= ρIX , then Equation (2.6) has two solutions, namely

σS = σX

√1− ρ2

IX

−ρ√

1− ρ2IX ± ρIX

√1− ρ2

ρ2IX − ρ2 .

Since ρ2IX − ρ2 = (ρIX

√1− ρ2 + ρ

√1− ρ2

IX)(ρIX√

1− ρ2 − ρ√

1− ρ2IX), this further

yieldsσS = σX

√1− ρ2

IX

1ρ√

1− ρ2IX ± ρIX

√1− ρ2

.

If ρIX > ρ, then only one of the roots guarantees that σS ≥ 0, and we obtain Equation(2.5). If ρIX = ρ, then Equation (2.6) has a unique solution, given by σS = σX/(2ρ).The inverse function of Formula (2.4) is continuous on ρ−1(R+). Therefore, Equation(2.5) must also hold true for ρIX < ρ.

Observe that (logXt, St, log It) is a 3-dimensional Gaussian process. Furthermore,it possesses a closed-form transition density and hence efficient maximum likelihoodestimation becomes feasible. Indeed, a straightforward calculation shows that the triple(logXt, St, log It) satisfies, for (X0, S0, I0) = (x, s, xe−s),

logXt

Stlog It

∼ N

log x+(µ− σ2

X

2

)t

se−κt +m (1− e−κt)log x+

(µ− σ2

X

2

)t− se−κt −m (1− e−κt)

,Σ

,

51


where, with at = 1− e−κt,

Σ =

tσ2X

ρσXσSκ at tσ2

X − ρσXσSκ at

ρσXσSκ at

σ2S

2κ a2tρσXσSκ at − σ2

S

2κ a2t

tσ2X − ρ

σXσSκ at

ρσXσSκ at − σ2

S

2κ a2t tσ2X − 2ρσXσSκ at + σ2

S

2κ a2t

and N (m,V ) denotes the normal distribution with mean vector m and covariancematrix V .As we specify first the dynamics of the futures contract as a GBM and the logspread

as an Ornstein-Uhlenbeck process the price of the futures contract leads the risk price.For example if the futures price is subject to a (demand or supply) shock the risk pricefollows and reduces the distance to the futures price. This asymmetric behavior is in linewith empirical findings as there is strong evidence that futures prices lead the spot prices,e.g. see Chan [1987], Kawaller et al. [1987] and Stoll and Whaley [1990]. Note thatmost studies investigate the relationship between a stock index and the correspondingfutures contract. However, their main argument of less frequent trading in the spotmarket and differences in transaction costs are also valid in our setup. They both leadto asymmetric access to information which in turn results in an asymmetric behavior ofthe spread. Therefore, for the applications we have in mind, it seems natural to modelthe futures and the spread first, and then to derive the spot dynamics endogenously. Ofcourse, it is also possible to specify the relation in the reverse direction, i.e. to derivethe dynamics of the futures price process based on the dynamics of the risk process andthe logspread. One can then proceed in a similiar manner.The model introduced has some similarities with Gaussian commodity spot models,

e.g. with the ones discussed in Schwartz [1997] or with the more general model providedin Casassus and Collin-Dufresne [2005]. In these models the triple of futures log price,spot log price and logspread is a 3-dimensional Gaussian process, too. These spotmodels, however, have different aims; e.g. they can be used for pricing long term forwardcommitments on the same commodity. Any forward position can be hedged by usingone interest rate derivative and two short term futures contracts. This means thatthe latter models are complete and hence the model dynamics under the risk-neutralmeasure have to be calibrated to current futures and derivative prices.The main aim of our model instead is to analyze the hedge error entailed when cross

hedging risk exposures with futures written on a correlated, but different risk source.Our model includes a non-hedgeable risk factor, the spread, leading to incompleteness.We work under the physical measure since this is the only measure under which thehedge errors characteristics are relevant for risk management. Moreover, in a crosshedging situation a calibration is not always possible, e.g. if there are no liquid kerosenefutures.

2.1.2. An Empirical Illustration

In the following we illustrate the estimation of the model by reconsidering the exampleof kerosene and crude oil. We use daily data of the spot kerosene price and the price of

52

2.1. The Continuous-time Model with a Stationary Spread

Table 2.1.: Estimates

contract nobs ADF µ σx σs κ m ρ201010 1147 0.0888∗ 0.0751

(0.1302)0.2768(0.0059)

0.2871(0.0060)

2.5548(1.0976)

−0.1661(0.0534)

0.3401(0.0260)

201009 1147 0.0750∗ 0.0742(0.1324)

0.2795(0.0059)

0.2877(0.0060)

2.7131(1.1078)

−0.1687(0.0515)

0.3475(0.0260)

201008 1145 0.0631∗ 0.0763(0.1361)

0.2823(0.0059)

0.2886(0.0061)

2.8890(1.1728)

−0.1693(0.0492)

0.3550(0.0258)

201007 1123 0.0555∗ 0.0778(0.1348)

0.2855(0.0061)

0.2908(0.0062)

3.0632(1.1704)

−0.1731(0.0473)

0.3646(0.0259)

201006 1375 0.0310∗∗ 0.1703(0.1209)

0.2758(0.0053)

0.2993(0.0058)

2.5899(1.1150)

−0.1946(0.0493)

0.3404(0.0241)

201005 1080 0.0436∗∗ 0.1114(0.1405)

0.2901(0.0064)

0.2942(0.0064)

3.5046(1.2913)

−0.1752(0.0410)

0.3771(0.0262)

201004 1058 0.0326∗∗ 0.0906(0.1453)

0.2954(0.0065)

0.2970(0.0065)

3.8040(1.3324)

−0.1808(0.0390)

0.3867(0.0262)

201003 1035 0.0250∗∗ 0.0737(0.1494)

0.3002(0.0066)

0.3006(0.0067)

4.1174(1.4241)

−0.1836(0.0371)

0.3990(0.0262)

201002 1015 0.0193∗∗ 0.0906(0.1518)

0.3035(0.0068)

0.3020(0.0069)

4.5175(1.6076)

−0.1866(0.0332)

0.4043(0.0263)

201001 994 0.0126∗∗ 0.0789(0.1551)

0.3098(0.0070)

0.3045(0.0069)

4.9939(1.3783)

−0.1916(0.0312)

0.4178(0.0262)

200912 1245 0.0097∗∗∗ 0.1852(0.1345)

0.2979(0.0060)

0.3114(0.0064)

3.8232(1.2755)

−0.2137(0.0374)

0.3892(0.0241)

200911 950 0.0046∗∗∗ 0.0860(0.1564)

0.3197(0.0073)

0.3108(0.0072)

6.3841(2.0112)

−0.1994(0.0238)

0.4415(0.0261)

200910 928 0.0029∗∗∗ 0.0603(0.1687)

0.3229(0.0075)

0.3135(0.0074)

7.0739(1.9253)

−0.2032(0.0231)

0.4532(0.0261)

200909 906 0.0014∗∗∗ 0.0819(0.1738)

0.3275(0.0077)

0.3178(0.0076)

8.0814(2.1057)

−0.2061(0.0208)

0.4677(0.0260)

200908 885 ≤ 0.0010∗∗∗ 0.0430(0.1807)

0.3321(0.0079)

0.3223(0.0078)

9.5437(2.2822)

−0.2120(0.0178)

0.4806(0.0259)

200907 862 ≤ 0.0010∗∗∗ 0.0743(0.1828)

0.3388(0.0082)

0.3275(0.0080)

11.5957(2.5298)

−0.2166(0.0153)

0.5011(0.0256)

200906 1114 ≤ 0.0010∗∗∗ 0.1416(0.1498)

0.3191(0.0068)

0.3298(0.0072)

6.6833(1.7074)

−0.2404(0.0233)

0.4579(0.0239)

200905 819 ≤ 0.0010∗∗∗ −0.0114(0.1953)

0.3516(0.0086)

0.3407(0.0086)

15.7614(2.9919)

−0.2274(0.0111)

0.5364(0.0250)

200904 797 ≤ 0.0010∗∗∗ −0.0655(0.1973)

0.3513(0.0088)

0.3429(0.0087)

15.9689(3.0523)

−0.2307(0.0103)

0.5594(0.0245)

200903 775 ≤ 0.0010∗∗∗ −0.0672(0.1953)

0.3436(0.0086)

0.3351(0.0086)

15.0723(2.8192)

−0.2362(0.0135)

0.5631(0.0244)

200902 755 ≤ 0.0010∗∗∗ −0.0694(0.2005)

0.3469(0.0089)

0.3257(0.0085)

12.1794(2.4786)

−0.2411(0.0146)

0.5815(0.0242)

200901 733 0.0021∗∗∗ −0.0802(0.1966)

0.3306(0.0086)

0.3053(0.0081)

10.9528(2.4631)

−0.2398(0.0160)

0.5742(0.0247)

The first column presents the maturity date of the contract, the second column gives the number ofobservations. The third column reports the p-value of the augmented Dickey-Fuller test for the null ofnon-stationarity. The ∗ (∗∗,∗∗∗) indicates the rejection of non-stationary at the 10% (5%,1%) level. Theremaining columns show the parameter estimates with the corresponding asymptotic standard errorsgiven in parenthesis.

53


different crude oil futures contracts. The maturities of the futures contracts range fromJanuary 2009 until October 2010, resulting in 21 (overlapping) time series.In a first step we check via the augmented Dickey-Fuller test whether the logspread of

the kerosene spot price and the corresponding crude oil futures price is stationary. Table2.1 reports the results for the different futures contracts. For most of the pairs we rejectthe null of a non-stationary logspread at any reasonable level. Of course, as one wouldexpect for a statistical test, the procedure does not suggest the existence of a long-termrelationship for every pair even if it is present. For these cases the model uncertaintyis obvious and we will check in Section 2.4.1 how the application of an optimal hedgingstrategy influences the hedging performance if the strategy is derived under our modelbut is applied to the 2GBM model, and vice versa.Table 2.1 also presents the estimation result for our data sets. To concentrate on

one asset in the remaining part of the chapter we use one representative contract withmoderate, not extreme, parameter values especially for κ and ρ. We choose the contractwith maturity in August 2009. The number of observations at which both assets, thefutures contract and the spot kerosene, are traded is 885. Figure 2.1 shows the timeevolvement of these two price series. Obviously, the price evolvements are very similar,which is also supported by the time-series plot of the logspread of the log prices (depictedin the lower panel).In the next sections we derive the variance optimal hedge, the corresponding variance

of the hedge error and derive quantitative and qualitative statements in terms of thestructural parameters.

2.2. Optimal Variance Hedging with Futures ContractsSuppose that a hedger sets up a portfolio consisting of futures contracts and cash po-sitions, in order to hedge the risk position h(IT ). In the following we denote by ξt thenumber of futures contracts held in the portfolio at time t. We assume that any futuresposition strategy ξ = (ξt)t∈[0,T ] is non-anticipating, i.e. at any time it incorporates onlyinformation publicly available. In mathematical terms, this means that ξ is progressivelymeasurable with respect to (Ft)t≥0, the filtration generated by the Brownian motions(W (X), W⊥

)T and completed with the P -null sets of the basis.If the futures price changes by ∆Xt from one trading day to the next, the hedger’s

margin account is adjusted by ∆Xt per futures contract. The cash position in the hedg-ing portfolio is changed accordingly, entailing a portfolio value change due to variationmargins of ∆V mar

t = ξt∆Xt.Denote by V = (Vt)t∈[0,T ] the total value of the hedging portfolio. Given an interest

rate r, the cash position contributes to the portfolio by rVtdt, hence the total valuesatisfies the continuous-time self-financing condition

dVt = ξtdXt + rVtdt. (2.7)

Equation (2.7) is linear. Given an initial portfolio value of V0 = v, the portfolio process

54

2.2. Optimal Variance Hedging with Futures Contracts

40

80

160

2006 2007 2008 2009 2010-0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

2006 2007 2008 2009 2010

time

time

price

spread

oflogprices(S

t=

log(X

t/I t))

logreturnskerosene


Figure 2.1.: The upper left hand panel depicts the time evolvement of the daily price of the crudeoil futures with maturity in August 2009 in US$/BBL (dashed line) and for spot jetkerosene in US$/BBL (solid line) from 2006/02/27 until 2009/07/16 (resulting in 885observations). The structure of this figure is the same as Figure 0.2 on page 10. Theupper right hand panel exhibits the scatter plot of the daily returns. The lower paneldepicts the time evolvement of the logspread of the log prices.

55


has the explicit representation

Vt = ert

v +t∫

0

e−rsξsdXs

.

Consider a self-financing hedge portfolio with futures position ξt at time t. Theconditional hedge error of the portfolio at time t ∈ [0, T ] is then given by

Ct(ξ, v) = E[e−r(T−t)h(IT )|Ft

]− Vt (2.8)

= ert

E (e−rTh(IT )|Ft)− v −

t∫0

ξse−rsdXs

.

CT (ξ, v) will also be referred to as the realized hedge error. Note that if Ct(ξ, v) isnegative, the combined value of the risk and the hedge portfolio is expected to end upwith a plus.To determine the variance optimal strategy within our model, i.e. the strategy min-

imizing the variance of the realized hedge error, we suppose that X is a martingale.This is a plausible assumption since the empirical analysis of crude oil futures pricesshows that the estimated drift parameter is close to zero for all contract months andstatistically insignificant for most assets (see Table 2.1 in Section 2.1.2). In addition, asthe estimation of the drift is notoriously challenging and very often highly speculative,it can easily distort the main aim of hedging, which is the reduction of risk. We there-fore discuss here the martingale case in depth and postpone the discussion of the moregeneral case to Section 2.5.Assuming that X is a martingale means that µ = 0 and dXt = σXXtdW

(X)t . Then

Equation (2.8) implies that the discounted conditional hedge error is also a martingale.The martingale e−rTE(h(IT )|Ft) can be written as

e−rTE (h(IT )|Ft) = e−rTE(h(IT )) +t∫

0

asdW(X)s +

t∫0

bsdW⊥s , t ∈ [0, T ], (2.9)

where a and b are progressively measurable and square-integrable processes. The firststochastic integral on the right hand side is hedgeable, since it is driven by the sameBM as the futures X. More precisely, following the strategy

ξ∗t = atert

σXXt, (2.10)

the gain from the futures position up to time t satisfies∫ t

0 ξ∗se−rsdXs =

∫ t0 asdW

(X)s .

The second integral in Equation (2.9) is orthogonal to W (X), and hence completelynon-hedgeable with X. This implies that the strategy ξ∗ minimizes the variance ofthe realized hedge error (see Theorem 1 in Föllmer and Sondermann [1986] where this

56

2.2. Optimal Variance Hedging with Futures Contracts

argument has been employed for the first time). Moreover, the conditional hedge errorsatisfies

Ct(ξ∗, v) = ertt∫

0

bsdW⊥s + E

(e−(T−t)rh(IT )

)− ertv.

Profiting from the Markov property of the processes I and S, we may express a and bin terms of sensitivities of the expected risk with respect to the futures and the logspreadvalues. More precisely, let

ψ(t, x, s) = e−r(T−t)E(h(Xt,xT e−S

t,sT

)),

where Xt,x and St,s are the solutions of Equation (2.1) resp. Equation (2.2) on [t, T ]with initial conditions Xt,x

t = x resp. St,st = s. We will refer to ψ as the value function.If h is Lipschitz continuous and its weak derivative h′ is Lebesgue-almost everywheredifferentiable, then ψ is continuously differentiable with respect to x and s, and

ψx(t, x, s) = ∂

∂xψ(t, x, s) = e−r(T−t)E

(h′(Xt,xT e−S

t,sT

)Xt,1T e−S

t,sT

). (2.11)

For details we refer the interested reader to Section 3.3.3, where a similar a statementis shown. Notice that ∂St,sT /∂s = e−κ(T−t), and hence by the same reasoning

ψs(t, x, s) = ∂

∂sψ(t, x, s) = −e−r(T−t)E

(h′(Xt,xT e−S

t,sT

)Xt,xT e−S

t,sT e−κ(T−t)

)(2.12)

= −e−κ(T−t)x∂

∂xψ(t, x, s).

The pair (X,S)T is a 2-dimensional SDE, driven by(W (X),W⊥

)T via the diffusionmatrix

Σ(x, s) =(σXx 0ρσS ρσS

).

With Ito’s formula we obtain that the processes a and b, appearing in the martingalerepresentation (2.9), are given by(

atbt

)= e−rt ΣT(Xt, St)

(ψx(t,Xt, St)ψs(t,Xt, St)

).

From Equation (2.10) and Equation (2.12) we can deduce the following result describingthe variance optimal hedge in terms of the futures-Delta, and the minimal hedge errorin terms of the logspread-Delta.

Theorem 2.2. The variance optimal futures position in the hedging portfolio is givenby

ξ∗t =[1− σS

σXρe−κ(T−t)

]ψx(t,Xt, St), (2.13)

57


and entails a realized hedge error of

CT (ξ∗, v) = E(h(IT ))− erTv − ρ T∫

0

e−rtσSψs(t,Xt, St)dW⊥t

.

Observe that the optimal hedge ξ∗ is the Delta of the position’s expectation, dampenedby the hedge ratio defined by

f(T − t) = 1− σSσX

ρe−κ(T−t).

The factor essentially equals 1 if the product of time to maturity T − t and reversionspeed κ is large. In this case the logspread is expected to return to its mean reversionlevel before maturity, and hence the optimal hedge ratio equals one. As maturity ap-proaches, the short-term fluctuations have an increasing impact on the terminal hedgeperformance, making the hedge ratio converge to

h = ρIXσIσX

. (2.14)

Indeed, due to Equation (2.3), we have lim(T−t)↓0 1−σSρe−κ(T−t)/σX = 1−σSρ/σX =h. We remark that h, defined in Equation (2.14), is sometimes referred to as theminimum variance hedge ratio (see e.g. Chapter 3.4 in Hull [2008]).Observe that if κ is equal to zero, which essentially means that there is no mean

reversion, then the logspread is not stationary and its variance increases linearly withtime. The hedge ratio is not dampened and it coincides with h. In this case the strategyξ∗ is equal to the optimal strategy in a model where both X and I are modeled as GBMs(see Section 2.4.1 for more details).In Formula (2.13) the optimal hedge is expressed in terms of the Delta with respect to

the futures price. In order to obtain a representation in terms of the Delta with respectto the illiquid asset price I, define first ϕ(t, y, s) = e−r(T−t)E(h(It,y,sT )), where It,y,sis the solution of the SDE for the illiquid asset on [t, T ], with initial values It,y,st = y

and St,st = s. Note that ψ(t, x, s) = ϕ(t, e−sx, s), and in particular, ψx(t, x, s) =e−sϕy(t, e−sx, s). Thus, the optimal hedge may be rewritten as

ξ∗t = e−St[1− σS

σXρe−κ(T−t)

]ϕy(t, It, St). (2.15)

If the logspread is positive, then the illiquid asset price is expected to rise relative to thefutures price. This explains why in Equation (2.15) the Delta is reduced by the factore−St . Conversely, if the logspread is negative, then the illiquid asset is expected to fallrelative to the futures price. In this the case the Delta is augmented by the factor e−St .Finally, we remark that the hedge ratio remains the same if the hedger uses an option

on the futures for hedging the risk exposure h(IT ). Denote by ∆(t, x) the option’s Deltaat time t, given a futures price of Xt = x. The dynamics of the option price P (t,Xt)satisfy dP (t,Xt) = rP (t,Xt)dt+∆(t,Xt)dXt, and the value of a self-financing portfolio

58

2.3. Standard Deviation of the Hedge Error

containing ξt options at time t is given by

Vt = ert

V0 +t∫

0

e−rsξs∆(s,Xs)dXs

.

The variance minimizing option position can be shown to be equal to

ξ∗t =[1− σS

σXρe−κ(T−t)

]ψx(t,Xt, St)

∆(t,Xt).

Suppose that a non-linear position of kerosene is hedged with an option, written on acrude oil futures, having a similar payoff profile. Then the ratio ψx(t,Xt, St)/∆(t,Xt)is usually stable and hence the hedging portfolio does not need to be rebalanced asstrongly as when using futures for hedging.


Having derived the variance optimal strategy and the corresponding hedge error, seeTheorem 2.2, we now aim at computing the implied standard deviation of the hedgeerror. This allows us to quantify the risk associated with the optimal strategy, whichis important for risk management and performance evaluation of the hedging strategy.We therefore derive analytic and semianalytic formulas for the standard deviation ofthe hedge error when minimizing the variance of risk exposures within our model. Asin the previous section, we assume that the hedger does not have any directional viewconcerning the futures. This means that the futures price X is a martingale withdynamics dXt = σXXtdWt.

We aim at computing the standard deviation of the hedge error when cross-hedgingthe position h(IT ) following the strategy ξ∗ of Equation (2.13). Note that the standarddeviation of CT (ξ∗, v) coincides with the standard deviation oferT ρ

∫ T0 e−rtσSψs(t,Xt, St)dW⊥t . The Itô isometry implies

std(CT (ξ∗, v)) = erT ρσS

√√√√√ T∫0

e−2rtE[ψ2s(t,Xt, St)]dt. (2.16)

In general, there is no closed-form expression for the formula for the integral in Equation(2.16). For linear positions, however, we may explicitly calculate the variance, since thefutures and the spread are lognormally distributed. Besides, for positions correspondingto Plain Vanilla options, the logspread-Delta ψs has an explicit representation, andthus allows for an efficient Monte Carlo simulation of the error (2.16). We proceed bydiscussing both cases separately.

59


2.3.1. Linear Positions

In this subsection we derive an analytic formula for the hedge error variance when cross-hedging a linear position. This is the most relevant case, since most of the risk positionsof industrial companies are linear. Think for instance of an airline being exposed to ashort position of kerosene.Suppose that the payoff function h is given by h(y) = cy, with c ∈ R. In this case

the Delta of the value function with respect to the futures price satisfies ψx(t, x, s) =e−r(T−t)E(cXt,1

T e−St,sT ) (see Equation (2.11)). Thus, with Equation (2.12), we get

∂

∂sψ(t, x, s) = −e−κ(T−t)xe−r(T−t)E(cXt,1

T e−St,sT ).

In the following we do not only need to compute the expectation of the productXt,1T e−S

t,sT , but also the expectation of the product of higher moments of the logspread

and the illiquid asset. Therefore we provide the following lemma.

Lemma 2.3. Let a ∈ R and b ∈ R+, then

E

(e−aS

0,st

(X0,xt

)b)= A(a, b, x, s, t),

where A(a, b, x, s, t) is defined by

A(a, b, x, s, t) =

xb exp[−1

2σ2Xt(b− b2)− ase−κt − a(m+ bρσXσS

1κ

)(1− e−κt

)]× exp

[12a

2σ2S

12κ(1− e−2κt)] . (2.17)

Proof. Since

S0,st = se−κt +m(1− e−κt) +

t∫0

e−κ(t−u)σS(ρdW (X)u + ρdW⊥u ),

we get

e−aS0,st (X0,x

t )b =

xb exp

− b2σ2Xt− ase−κt − am(1− e−κt) +

t∫0

(bσX − ρaσSe−κ(t−u))dW (X)u

× exp

− t∫0

(aρσSe−κ(t−u))dW⊥u

.

We calculate the variances of the independent normal variables given by the integrals

60


in the last two factors. We havet∫

0

(bσX − ρaσSe−κ(t−u))2dt = b2σ2Xt− 2abρσXσS

1κ

(1− e−κt) + a2ρ2σ2S

12κ (1− e−2κt)

andt∫

0

(aρσSe−κ(t−u))2dt = a2ρ2σ2S

12κ (1− e−2κt).

We use this to derive

E(e−aS

0,st (X0,x

t )b)

= xb exp(− b2σ

2Xt− ase−κt − am(1− e−κt)

)

× E

exp

t∫0

(bσX − ρaσSe−κ(t−u))dW (X)u

× E

exp

− t∫0

(aρσSe−κ(t−u))dW⊥u

= xb exp

(− b2σ

2Xt− ase−κt − am(1− e−κt)

)× exp

[12

(b2σ2

Xt− 2abρσXσS1κ

(1− e−κt))]

× exp[

12

(a2ρ2σ2

S

12κ (1− e−2κt) + a2ρ2σ2

S

12κ (1− e−2κt)

)],

from which the result follows.

With this at hand the standard deviation in (2.16) simplifies to

std(CT (ξ∗, v)) = |c|ρσS

√√√√√ T∫0

e−2κ(T−t)E [X2tA

2(1, 1, 1, St, T − t)] dt. (2.18)

From this we are able to derive the following explicit formula for the hedge error variance.

Theorem 2.4. The variance optimal cross-hedge of a linear position cIT entails a hedge

61


error with standard deviation

std(CT ) =

σS√

1− ρ2x exp(

(m− s)e−κT −m− ρσXσS1κ

(1− 2e−κT

)+ σ2

S

14κ(1− 2e−κT

))

× |c|

√√√√√ T∫0

exp(−2κ(T − t) + σ2

Xt− 2ρσXσS1κe−κ(T−t) + σ2

S

12κe

−2κ(T−t))dt.

(2.19)

Proof. Recall that from Equation (2.18) we have

Var (CT (ξ∗, v)) = c2ρ2σ2S

T∫0

e−2k(T−t)E(X2tA

2(1, 1, 1, St, T − t))dt. (2.20)

By the definition of A, we have

E(X2tA

2(1, 1, 1, St, T − t))

=

exp(−2(m+ ρσXσS

1κ

)(1− e−κ(T−t)) + σ2S

12κ (1− e−2κ(T−t))

)× E

(X2t exp

(−2Ste−κ(T−t)

)), (2.21)

and again using the definition of A we get

E(X2t exp

(−2Ste−κ(T−t)

))=A(2e−κ(T−t), 2, x, s, t)

=x2 exp[−2(m+ 2ρσXσS

1κ

)(e−κ(T−t) − e−κT )]

× exp[σ2Xt− 2se−κT + 2σ2

S

12κ (e−2κ(T−t) − e−2κT )

].

Combining the last equation with Equation (2.21) we further obtain

E(X2tA

2(1, 1, 1, St, T − t))

=

x2 exp(σ2Xt− 2se−κT − 2ρσXσS

1κ

(1 + e−κ(T−t) − 2e−κT ))

× exp(−2m(1− e−κT ) + σ2

S

12κ (1− 2e−κT + e−2κ(T−t))

). (2.22)

The previous calculations yield, by combination of Equation (2.20) and Equation (2.22),Expression (2.19).

The integral in Expression (2.19) can be computed in a straightforward manner using

62


standard numerical quadratures algorithms.

When analyzing the dependence of the hedge error on the different model parametersit is convenient to rewrite the hedge error formula (2.19) as follows:

std(CT ) = |c| σS√

1− ρ2x exp((m− s)e−κT −m

) T∫0

exp(−2κ(T − t) + σ2

Xt)

× exp(−2ρσX

σSκ

(1 + e−κ(T−t) − 2e−κT

)+ σ2

S

2κ

(1 + e−2κ(T−t) − 2e−κT

))dt

] 12

.

The hedge error (2.19) can be approximated with simplified formulas. For long maturi-ties, i.e. large T , the factor exp

((m− s)e−κT −m

)approximately coincides with e−m.

Moreover,T∫

0

e−2κ(T−t)+σ2Xtdt = eσ

2XT − e−2κT

2κ+ σ2X

≈ eσ2XT

2κ+ σ2X

,

and hence

for long maturities: std(CT ) ≈ |c|xe−mσS√

1− ρ2e

(−ρσSσX+

σ2S4

)/κ

√eσ

2XT

2κ+ σ2X

.

Observe that the hedge error increases exponentially with time to maturity. However,the volatility squared σ2

X is usually low (see Table 2.1 in Section 2.1.2), and thus thehedge error increases approximately linearly, with slope σ2

X/2, in the first several years(compare with the upper left panel in Figure 2.2).

For short maturities, i.e. small T , the factor exp((m− s)e−κT −m

)approximately

coincides with e−s. Besides, by linearly approximating exponentials with the Taylorexpansion up to the first order, we have

T∫0

e−2κ(T−t)+σ2Xtdt = e−σ

2XT − e−2κT

2κ+ σ2X

≈ T.

Therefore, we get the approximation

for short maturities: std(CT ) ≈ |c|σS√

1− ρ2xe−s√T .

Note that the hedge error increases with order T 1/2, for short maturities (see the upperleft panel in Figure 2.2). The kink in Figure 2.2 is determined by how fast the OrnsteinUhlenbeck process describing the logspread attains its stationary distribution. Thevariance of the logspread at time T , as a function of time to maturity T − t, is given byσ2S(1− e−2κ(T−t))/(2κ). The hedge error is roughly proportional to the variance of the

logspread.

The hedge error vanishes as the variance of the stationary logspread distribution,

63


0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.08

0.09

-10 -5 0 5 10

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0 2 4 6 8 10 12 14 16 18 200.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

-1 -0.5 0 0.5 1

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.08

0.09

0.10

0.11

0.12

0.13

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ρκ

σS σX

sT − t

std(C

T)

std(C

T)

std(C

T)

std(C

T)

std(C

T)

std(C

T)

Figure 2.2.: This figure shows the sensitivity of the standard deviation of the hedge error with respectto the time to maturity (in the upper left hand panel) and with respect to the parametersof the model (remaining panels). In each panel only the parameter indicated on theabscissa is varied while the others remain fixed at the estimates from the futures contractwith maturity in August 2009.

64


σ2S/(2κ), converges to zero. Moreover, it is straigthtforward to show

limκ↓0

std(CT ) = |c|xe−s σSσX

√1− ρ2

√eσ

2XT − 1.

Of course, not only the impact of the time to maturity to the hedge error is of interestbut also the influence of the model’s parameters. Figure 2.2 highlights the sensitivityof the standard deviation of the hedge error towards changes in the parameters. Thefigure depicts the resulting standard deviation by changing one parameter and keepingthe others constant. The fixed parameters are set to the estimates of the futures contractwith maturity in August 2009. The figure shows (in the middle left panel) the decreasingstandard deviation in the mean reversion speed κ. This comes at no surprise as a largerκ results in a faster return to the long-term relationship. The reverse U-shaped behaviorwith respect to the correlation ρ (in the right middle panel) highlights the change fromthe incomplete market setting for |ρ| < 1 to the complete market setting for |ρ| = 1.With increasing instantaneous variance of the logspread and the futures price process(σS and σX) the variance of the hedge error also increases (shown in the lower panels).

2.3.2. Non-linear Positions

In practice the case of non-linear risk positions is also relevant. For instance, considerthe very illiquid German natural gas futures markets. Due to the illiquidity gas tradersfrequently use futures of neighbouring countries for hedging purposes. So, if operatorsof German gas power plants protect themselves against changing gas prices by buyingDutch gas on the futures market (e.g. natural gas futures of the Dutch market TTF)the basis risk is due to a geographical spread in commodity prices which arises fromdifferent trading places for the same underlying. In this case TTF contracts serve asproxies that are cointegrated with the natural gas prices in the German market area.The profit margin of a gas power plant is essentially determined by the spark spread, i.e.the spread between the electricity price per MWh and the price of the amount of gasthe plant needs for producing 1 MWh of electricity. Electricity will only be producedif the profit margin exceeds the operating costs. A gas power plant can thus be seenas a call option, a highly non-linear position, on the spark spread. We therefore alsoconsider here the hedging of non-linear risk positions.When it comes to hedging non-linear risk positions, the problem is that in general

there are no explicit formulas available for the standard deviation of the hedge error.However, for Plain Vanilla options there are analytic formulas for the Deltas, allowingfor swift simulation analysis. Indeed, the Deltas resemble the Deltas for Plain Vanillaoptions in the Black-Scholes model. As an example we provide the relevant formulas fora European call option with strike K, i.e. h is given by h(y) = (y−K)+. First observethat the value function ψ for call options is given by

ψ(t, x, s) = E((It,x,sT −K

)+).

65


From the proof of Lemma 2.3 it can be seen that It,x,sT = e−St,sT Xt,x

T can be written as

It,x,sT = x exp(−1

2σ2X(T − t)− se−κ(T−t) −m(1− e−κ(T−t))

)exp(σN),

where N is a standard normal variable and σ2 is given by

σ2 = σ2X(T − t)− 2ρσXσS

1κ

(1− e−κ(T−t)

)+ σ2

S

12κ

(1− e−2κ(T−t)

).

We get

ψ(t, x, s) = 1√2π

∞∫−∞

(xe−

12σ

2X(T−t)−se−κ(T−t)−m(1−e−κ(T−t))+σy −K

)+e−y/2dy.

(2.23)In analogy to the standard Black-Scholes case we define the functions d+(t, x, s) andd−(t, x, s) as

d+(t, x, s) = 1σ

[log(K

x

)+ 1

2σ2X(T − t) + se−κ(T−t) +m

(1− e−κ(T−t)

)]and d−(t, x, s) = d+(t, x, s)− σ.

Note that the integrand in the integral in Equation (2.23) equals zero if y < d+(t, x, s),and hence

ψ(t, x, s) = x exp(−1

2σ2X(T − t)− se−κ(T−t) −m(1− e−κ(T−t)) + 1

2σ2)

Φ(d−(t, x, s))

−KΦ(d+(t, x, s)),

where Φ denotes the cumulative distribution function of the standard normal distribu-tion. The above explicit representation of ψ(t, x, s) yields

ψx(t, x, s) = exp(−1

2σ2X(T − t)− se−κ(T−t) −m(1− e−κ(T−t)) + 1

2σ2)

× [Φ(d−(t, x, s)) + ϕ(d−(t, x, s))∂xd−(t, x, s)]−Kϕ(d+(t, x, s))∂xd+(t, x, s),

where ∂xd+(t, x, s) = ∂xd−(t, x, s) = 1/(σx).

The analytic and semianalytic formulas for the hedge error allow the efficient compu-tation and the comparison of the hedge error variance for different futures contracts. Upto now, however, we assume that we hedge within the correct model (with an asymp-totic stationary logspread). Although, statistical tests may help to decide whether thelogspread is asymptotic stationary or not, there is always the risk to hedge within thewrong model and a relevant question arises: how sensitive is the hedge error with respectto the model choice? We address this question in the next Section.

66

2.4. The Performance of Suboptimal Hedging Strategies

2.4. The Performance of Suboptimal Hedging StrategiesSo far we have assumed that we know with certainty that the price of the illiquid assetand the price of the liquid futures contract are cointegrated and evolve according to ourmodel. However, it may happen that a statistical test leads to a wrong conclusion ordifferent tests lead to different implications. In other words, we face model uncertainty.In the following Section 2.4.1 we consider a 2GBM model and derive the hedge error

obtained by using the optimal strategy from our model. Furthermore, we analyse theimpact of applying the optimal hedging strategy from the 2GBM model to our model.We then proceed by comparing the optimal dynamic hedge with its optimal static coun-terpart. In practice, traders often hedge linear positions statically, holding a positionin futures that corresponds to the size of the risk. By this they intuitively reflect thatthe hedge ratio essentially equals 1 whenever time to maturity is long. In Section 2.4.2we first derive the optimal static hedging strategy and compare it with the hedgingstrategy ξ∗, which allows for portfolio regrouping.

2.4.1. The Costs of Ignoring a Long-term Relationship or FalselyAssuming a Long-term Relationship

We next introduce a simple model where both X and I are GBMs, hence are notcointegrated and the logspread does not have an asymptotic stationary distribution.We will refer to this model as the 2GBM model.In both models, the futures price is assumed to satisfy the dynamics

dXt = σXXtdW(X)t ,

but in contrast to the model with a mean reverting logspread, discussed in the previoussections, the 2GBM model assumes that the illiquid asset price process is also a GBMwith dynamics

dIt = σIIt

(ρIXdW

(X)t + ρIXdW

⊥t

).

In this model the variance minimizing hedging strategy for European options with payofffunction h are known to have the simple form

ζt = ρIXσIItσXXt

ψy(t, It), (2.24)

where ψ(t, y) = e−r(T−t)E(h(It,yT )). For a derivation of Equation (2.24) we refer toHulley and McWalter [2008]; see also Chapter 3 for a derivation in a slightly moregeneral setting using BSDEs.The optimal cross-hedge within the 2GBM model (2.24), is essentially determined

by the cross correlation. If an airline company used a 2GBM model estimated withdaily data to hedge kerosene short positions, then it would considerably underhedgeits kerosene risk, facing thus unnecessarily high variations in costs. But, by how much

67


does the hedge error increase if we use the wrong model? We next quantify the risk bycalculating the hedge error when using the optimal strategy ζ of the 2GBM model whilethe log prices are cointegrated and evolve according to the dynamics of our model.We restrict our analysis to linear positions of the form cIT . As before we denote the

realized hedge error by

CT = cIT − erTv +

T∫0

e−rsζsdXs

.

The following proposition provides the hedge error variance for the strategy (2.24)under our model with cointegration and under the 2GBM model.

Proposition 2.5. Hedging the linear position cIT with the strategy ζ entails a hedgeerror in the cointegration model with variance

Var (CT ) = c2

A(2, 2, X0, S0, T )−A2(1, 1, X0, S0, T )

−2T∫

0

ρIXσI(σX − ρσSe−κ(T−t))Btdt+T∫

0

ρ2IXσ

2IA(2, 2, X0, S0, t)dt

,

where Bt is given by

Bt = X20 exp

(σ2Xt−

1κσSσXρ

(3− 2e−κT − 2e−κt + e−κ(T−t)

)− S0

(e−κT + e−κt

))× exp

(1

4κσ2S

(2− e−2κT − e−2κt + 2e−κ(T−t) − 2e−κ(T+t)

)−m(2− e−κT − e−κt)

).

Under the correct model, the 2GBM model, the minimal variance of the realized hedgeerror is given by

Var (CT ) = c2y2 (1− ρ2IX

) (eσ

2IT − 1

).

Proof. Since I is a GBM in the 2GBM model, the value function ψ associated withthe linear position h(x) = cx is given by ψ(t, y) = e−r(T−t)cy. Therefore, ψy(t, y) =e−r(T−t)c, and the realized error variance in Model 1, following strategy

ζt = ρIXσIIte−r(T−t)c/ (σXXt) ,

satisfies

CT = cIT − erTv +

T∫0

e−rT cρIXσIItdW(X)t

,

68


and consequently we have

Var (CT ) = c2 Var (IT )− 2c2 Cov

IT , T∫0

ρIXσIItdW(X)t

+ c2E

T∫0

ρ2IXσ

2II

2t dt

.

Note that Var (IT ) = A(2, 2, X0, S0, T )−A2(1, 1, X0, S0, T ) and observe further that

E

T∫0

ρ2IXσ

2II

2t dt

=T∫

0

ρ2IXσ

2IA(2, 2, X0, S0, t)dt.

It remains to calculate the covariance above. To that effect we recall the decomposi-tion (2.30) of IT from the proof of Proposition 2.7 and borrow the respective notationto write

∫ T0 ρIXσIItdW

(X)t =

∫ T0 ρIXσIItdW

(X)t +

∫ T0 ρIXσIIt(σX − ρσSe

−κ(T−t))dt.Consequently,

Cov

IT , T∫0

ρIXσIItdW(X)t

= X0eλ(T )EQ

T∫0

ρIXσIIt(σX − ρσSe−κ(T−t))dt

= X0e

λ(T )T∫

0

ρIXσI(σX − ρσSe−κ(T−t))EQ (It) dt

=T∫

0

ρIXσI(σX − ρσSe−κ(T−t))E (IT It) dt.

In order to calculate E (IT It) we proceed similar as in Lemma 2.3. We decompose IT Itinto

IT It =X20 exp

T∫0

[σX(1 + 1u≤t)− σSρ

(e−κ(T−u) + e−κ(t−u)1u≤t

)]dWX

u

× exp

− T∫0

σS ρ(e−κ(T−u) + e−κ(t−u)1u≤t

)dW⊥u

× exp

−12

T∫0

σ2X(1 + 1u≤t)du

× exp

(−S0

(e−κT + e−κt

)−m(2− e−κT − e−κt)

).

69


The variances of the stochastic integrals are given by

T∫0

σ2X(1 + 2u≤t + 1u≤t)− 2σSσXρ

[e−κ(T−u) + 1u≤t

(2e−κ(t−u) + e−κ(T−u)

)]+ σ2

Sρ2(e−κ(T−u) + e−κ(t−u)1u≤t

)2du

andT∫

0

σ2S ρ

2(e−κ(T−u) + e−κ(t−u)1u≤t

)2du.

Hence, taking expectation yields

E (IT It) =

X20 exp

12

T∫0

[σ2X2u≤t − 2σSσXρ

[e−κ(T−u) + 1u≤t

(2e−κ(t−u) + e−κ(T−u)

)]]du

× exp

12

T∫0

σ2S

(e−2κ(T−u) + 2e−κ(T+t−2u)1u≤t + e−2κ(t−u)1u≤t

)du

× exp

(−S0

(e−κT + e−κt

)−m(2− e−κT − e−κt)

).

The result follows by a simple calculation.

Proposition 2.5 allows us to analyse the ignorance of a long-term relationship with re-spect to the variance of the hedge error. Of course the variance of the optimal strategyξ∗ given by Equation (2.13) is less than the standard error using the strategy ζ fromthe 2GBM model given by Equation (2.24). The upper panels of Figure 2.3 comparethe performance of the two strategies.1 When following the strategy ζ, the risk positionis underhedged, yielding the hedge error (dashed line) to grow continually with time tomaturity. The hedge error does not flatten as strongly as when following strategy ξ∗,whose corresponding hedge error is depicted by the solid line. For very short maturities,the mean reversion has little time to develop and hence the hedge error entailed by ζis similar to the hedge error of ξ∗. However, for long maturities ζ is considerably out-performed by ξ∗, leading for example to a more than three times higher error standarddeviation over a two year hedging period.Since by definition there is no strategy with a smaller variance than the variance

optimal strategy the proportion of a three times larger value is strongly convincing. Tofairly compare our model with the 2GBM model, we also consider the inverse case, thatis we study the hedge error of the optimal strategy from the model with the asymptotic

1Note that we used the estimated parameters obtained for the August 2009 crude oil futures contract(see Table 2.1) and that the correlation ρIX can be expressed in terms of the structural parametersof our model, see Equation (2.4).

70


stationary logspread, ξ∗, when there is no cointegration. To this end we have to derivethe resulting hedge error in the 2GBM model which is given in the next proposition.

Proposition 2.6. Hedging the linear position cIT with the strategy ξ∗, see Formula(2.13), entails a hedge error in the 2GBM model with variance

Var (CT ) = c2 B(0, 2, T )−B2(0, 1, T )

− 2T∫

0

σIρ

[1− σS

σXρe−κ(T−t)

]A(1, 1, 1, 0, T − t)σXB(1− e−κ(T−t), 1 + e−κ(T−t), t)dt

+T∫

0

[1− σS

σXρe−κ(T−t)

]2A2(1, 1, 1, 0, T − t)σ2

XB(2− 2e−κ(T−t), 2e−κ(T−t), t)dt

,

(2.25)

where B(a, b, t) is given by

B(a, b, t) = E(Xat I

bt ) = Xa

0 Ib0 exp

(12 t[σ2X(a2 − a) + 2abσXσIρ+ σ2

I (b2 − b)])

.

(2.26)

Proof. Recall from Equation (2.13) that ξ∗t =[1− σSρe−κ(T−t)/σX

]ψx(t,Xt, St), with

ψ(t, x, s) = e−r(T−t)E(h(Xt,x

T e−St,sT )). Hence, for h(y) = cy we get ψx(t, x, s) =

ce−r(T−t)E(Xt,1T e−S

t,sT

)= ce−r(T−t)A(1, 1, 1, s, T−t). Thus, the realized error variance

in the 2GBM model, following the strategy above, satisfies

CT = cIT − erTv + c

T∫0

e−rT[1− σS

σXρe−κ(T−t)

]A(1, 1, 1, St, T − t)dXt

.

Consequently, setting v = ce−rTE (IT ), we have

Var (CT ) = c2 Var (IT )

− 2c2 Cov

IT , T∫0

[1− σS

σXρe−κ(T−t)

]A(1, 1, 1, St, T − t)σXXtdW

(X)t

+ c2E

T∫0

[1− σS

σXρe−κ(T−t)

]2A2(1, 1, 1, St, T − t)σ2

XX2t dt

.

It is straightforward to see that B(a, b, t) defined as B(a, b, t) = E(Xat I

bt

), with X

and I as in the 2GBM model, fulfills Equation (2.26). Hence Var (IT ) = B(0, 2, T ) −B2(0, 1, T ). Observe that we may, via Fubini’s theorem, write the expectation in the

71


last term in the variance of CT above as

T∫0

[1− σS

σXρe−κ(T−t)

]2A2(1, 1, 1, 0, T − t)σ2

XE(

exp(−2Ste−κ(T−t)

)X2t

)dt.

In order to simplify further recall that St = log(Xt)− log(It). Thus,

E(

exp(−2Ste−κ(T−t)

)X2t

)= E

(X

2−2e−κ(T−t)t I2e−κ(T−t)

t

)= B(2− 2e−κ(T−t), 2e−κ(T−t), t),

which combined with the previous integral yields the last term in Expression (2.25). Inorder to simplify the remaining term in the variance of CT we apply the standard trickof a change of measure, here with dQ = (IT /I0)dP , Fubini’s Theorem, and reversingthe measure change to obtain

T∫0

σIρ

[1− σS

σXρe−κ(T−t)

]A(1, 1, 1, 0, T − t)σXE

(IT exp

(−Ste−κ(T−t)

)Xt

)dt.

Finally, using St = log(Xt) − log(It), and the independent increments of a Brownianmotion, we get

E(IT exp

(−Ste−κ(T−t)

)Xt

)= E

(IT I

e−κ(T−t)

t X1−e−κ(T−t)

t

)= E

I1+e−κ(T−t)

t X1−e−κ(T−t)

t exp

T∫t

σIdWIu −

12

T∫t

σ2Idu

= B(1− e−κ(T−t), 1 + e−κ(T−t), t),

which together with the previous integral yields the middle term in Expression (2.25)and thus finishes the proof.

Note that now not all parameters are identified. This is in contrast to the previousscenario, where we investigated the impact of ignoring a long-term relationship. As thelogspread is not mean reverting in the 2GBM model the parameter κ is implicitly set tozero. However, to provide a realistic comparison we estimate the implied distributionof κ in the following way: we simulate 10000 sample paths from the 2GBM model withT = 885 observations. Based on these time series we estimate our model leading to thedistribution of κ. For the graphical illustration we use the corresponding 10%, 50% and90% quantiles leading to 0.0927, 0.8365 and 2.2450 respectively. The dashed lines in thelower left panel of Figure 2.3 show the hedge error standard deviation, for these differentmean reversion speeds, when the real prices behave as in the 2GBM model, but the riskis hedged according to the cointegration model optimal strategy ξ∗. As a benchmark,the panel depicts also the genuine minimal error standard deviation (solid line) implied

72


0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.5 1 1.5 2

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2

1

1.5

2

2.5

3

3.5

0 0.5 1 1.5 2time to maturity (in years)

time to maturity (in years)time to maturity (in years)

time to maturity (in years)

ratio

ratio

std(C

T)

std(C

T)

Figure 2.3.: The upper left panel shows the standard deviation of the hedge error under cointegrationusing the optimal strategy (solid line) and the strategy from the 2GBM model (dashedline). The lower left panel shows the standard deviation under the 2GBM model usingthe optimal strategy (solid line) and the strategy from the cointegrated model (dashedlines) for κ ∈ 2.2450, 0.8365, 0.0927, from top to down, respectively. The panels on theright depict the ratio of the standard deviation of the strategies.

73


by the optimal strategy ζ. The smaller the mean reversion speed, the smaller the hedgeerror. Moreover, the hedge error converges to the minimal hedge error as the meanreversion speed converges to zero, showing that the cointegration model embeds the2GBM model.In many real world applications, it may not be obvious that there is a long-term

relationship between the hedging instrument and the risk to be hedged. Comparing theleft upper and left lower panel of Figure 2.3 we conclude that in ambiguous situations itis nevertheless better to use a model allowing for an asymptotic stationary spread ratherthan using a simpler model that does not. The error implied by mistakenly assuminga mean reverting logspread is significantly smaller than the error made by mistakenlyassuming that the hedging instrument is not cointegrated with the risk.

2.4.2. The Costs of Using a Static HedgeIn practice linear positions are often only statically hedged, even though the varianceminimizing hedge is not constant. In the numerical example below we compare thestandard deviations of static and dynamic hedges of a linear position, and address thequestion by how much the dynamic variance minimizing strategy outperforms the staticone.For that purpose we need to derive the optimal static hedge position a ∈ R that

minimizes the variance

CT (a) = cIT − erTv +

T∫0

e−rtadXt

.

With this at hand we can calculate the minimal error standard deviation that can beachieved by hedging statically with futures.

Proposition 2.7. The optimal static hedging position a in futures contracts which min-imizes the variance of the hedge error is given by

a =Cov

(cIT , e

rT∫ T

0 e−rtdXt

)Var

(erT

∫ T0 e−rtdXt

) = ce−rTE(IT∫ T

0 e−rtdXt

)σ2XE

(∫ T0 e−2rtX2

t dt) (2.27)

with corresponding variance

Var (CT (a)) = E(C2T (a)

)= c2 Var (IT )−

c2[E(IT∫ T

0 e−rtdXt

)]2

σ2XE

(∫ T0 e−2rtX2

t dt) . (2.28)

The expectation in the denominator is given by

E

T∫0

e−2rtX2t dt

=

X20

σ2X−2r (e(σ2

X−2r)T − 1), if σ2X 6= 2r,

X20T, if σ2

X = 2r.

74


For the expectation in the numerator we have, assuming σ2X > r,

E

IT T∫0

e−rtdXt

=

X20eλ(T ) σ2

X

σ2X−r

(e(σ2

X−r)T − 1), if ρ = 0,

X20eλ(T )+ ρσXσS

κ e−κT (Λ1(T )− Λ2(T )), if ρ 6= 0,(2.29)

with

Λ1(T ) = σ2Xe

(σ2X−r)T (|ρ|σXσS)−

σ2X−rκ

κ1−σ2X−rκ

(γ

(σ2X − rκ

,1κ|ρ|σXσS

)−γ(σ2X − rκ

,1κ|ρ|σXσSe−κT

)),

Λ2(T ) = e(σ2X−r)T

(|ρ|σXσS

κ

)−σ2X−rκ(γ

(σ2X − rκ

+ 1, 1κ|ρ|σXσS

)−γ(σ2X − rκ

+ 1, 1κ|ρ|σXσSe−κT

)),

λ(T ) = −S0e−κT −m(1− e−κT ) + σ2

S

4κ(1− e−2κT )− ρσSσX

κ

(1− e−κT

).

Here γ(s, x) =∫ x

0 ys−1e−ydy denotes the incomplete Gamma function. Furthermore wehave

Var (IT ) = A(2, 2, X0, S0, T )−A2(1, 1, X0, S0, T ),

where A is as in Equation (2.17).

Proof. The variance of the hedge error does not depend on the initial capital v, andhence we may assume that erT v = cE (IT ). Holding the constant position a between 0and T then entails the hedge error

CT (a) = cIT − E (cIT )− aerTT∫

0

e−rtdXt,

and since Xt is a martingale we have E (CT (a)) = 0. Hence, the variance of CT (a) is

75


given by

E(C2T (a)

)= c2E

((IT − E (IT ))2)− 2acerTE

(IT − E (IT ))T∫

0

e−rtdXt

+ a2e2rTE

T∫

0

e−rtdXt

2= c2E

((IT − E (IT ))2)− 2acerTE

IT T∫0

e−rtdXt

+ a2e2rTE

T∫0

e−2rtσ2XX

2t dt

.

The optimal a which minimizes the variance of the hedge error is given by

a = ce−rTE(IT∫ T

0 e−rtdXt

)σ2XE

(∫ T0 e−2rtX2

t dt) .

For the variance of the corresponding hedge error we have

E(C2T (a)

)= Var (cIT )−

Cov(cIT , e

rT∫ T

0 e−rtdXt

)2

Var(erT

∫ T0 e−rtdXt

)= c2 Var (IT )−

c2[E(IT∫ T

0 e−rtdXt

)]2

σ2XE

(∫ T0 e−2rtX2

t dt) .

The expectation in the denominator is given by

E

T∫0

e−2rtX2t dt

=T∫

0

e−2rt E(X2t

)︸︷︷︸=X2

0 exp(σ2Xt)

dt

=

X20

σ2X−2r (e(σ2

X−2r)T − 1), if σ2X 6= 2r,

X20T, if σ2

X = 2r.

The computations for the expectation in the numerator are somewhat more involved.Using the explicit expressions for ST and XT , we may decompose IT into

IT = XT e−ST = X0e

λ(T )DT , (2.30)

76


where DT is the value at time T of the process D defined by, for all t ∈ [0, T ],

Dt = exp

t∫0

(σX − ρσSe−κ(T−u))dW (X)u − 1

2

t∫0

(σX − ρσSe−κ(T−u))2du

× exp

− t∫0

ρσSe−κ(T−u))dW⊥u −

12

t∫0

ρ2σ2Se−2κ(T−u)du

and λ(T ) is a constant given by

λ(T ) = − S0e−κT −m(1− e−κT )− σ2

X

2 T + 12

T∫0

(σX − ρσSe−κ(T−u))2du

+ 12

T∫0

ρ2σ2Se−2κ(T−u)du

= − S0e−κT −m(1− e−κT ) + σ2

S

4κ (1− e−2κT )− ρσSσXκ

(1− e−κT ).

Note that D is a strictly positive martingale and satisfies Novikov’s condition. Thereforewe can define a probability measure Q via dQ = DT dP . Under Q the processes W (X)

and W⊥, for all t ∈ [0, T ],

W(X)t = W

(X)t −

t∫0

(σX − ρσSe−κ(T−u))du

W⊥t = W⊥t +t∫

0

ρσSe−κ(T−u)du

are independent Brownian motions. The dynamics of X, rewritten in terms of W (X),satisfy

dXt = σX(σX − ρσSe−κ(T−t))Xtdt+ σXXtdW(X)t .

Observe that the expectation of Xt with respect to Q is given by

EQ (Xt) = X0 exp

t∫0

σX(σX − ρσSe−κ(T−u))du

= X0 exp

(σ2Xt+ ρσXσS

κe−κT

)exp

(−ρσXσS

κe−κ(T−t)

).

77


Now the expectation term in the numerator can be written as

E

IT T∫0

e−rtdXt

= X0eλ(T )EQ

T∫0

e−rtdXt

(2.31)

= X0eλ(T )

T∫0

e−rtσX(σX − ρσSe−κ(T−t))EQ (Xt) dt

= X20eλ(T )+ ρσXσS

κ e−κT

×T∫

0

e(σ2X−r)t(σ2

X − ρσXσSe−κ(T−t))e−ρσXσS

κ e−κ(T−t)dt

︸︷︷︸=A

.

For ρ = 0 we are done. For ρ 6= 0 we continue by substituting u = |ρ|σXσSe−κ(T−t)/κ

which leads to an explicit expression for the above integral A in terms of the incompleteGamma function γ(s, x) =

∫ x0 ys−1e−ydy.

A = e(σ2X−r)T

(|ρ|σXσS

κ

)−σ2X−rκ

1κ |ρ|σXσS∫

1κ |ρ|σXσSe−κT

uσ2X−rκ (σ2

X − κu)e−u 1κudu

= σ2Xe


σ2X−rκ

κ1−σ2X−rκ

1κ |ρ|σXσS∫


uσ2X−rκ −1e−udu

− e(σ2X−r)T

(|ρ|σXσS

κ

)−σ2X−rκ

1κ |ρ|σXσS∫


uσ2X−rκ e−udu

= Λ1(T )− Λ2(T ),

where

Λ1(T ) = σ2Xe


σ2X−rκ

κ1−σ2X−rκ

(γ

(σ2X − rκ

,1κ|ρ|σXσS

)−γ(σ2X − rκ

,1κ|ρ|σXσSe−κT

))

78


0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 0.5 1 1.5 2 1

1.02

1.04

1.06

1.08

1.1

1.12

0 0.5 1 1.5 2time to maturity (in years)time to maturity (in years)

ratio

std(C

T)

Figure 2.4.: The panel on the left shows the standard deviation of the static (dashed line) versus thedynamic hedge error (solid line). The right hand panel depicts the ratio. As we have toassume a constant interest rate for the computation of the variance of the static hedgeerror we fix it at r = 0.02.

and

Λ2(T ) = e(σ2X−r)T

(|ρ|σXσS

κ

)−σ2X−rκ(γ

(σ2X − rκ

+ 1, |ρ|σXσSκ

)−γ(σ2X − rκ

+ 1, |ρ|σXσSe−κT

κ

)).

Plugging this into Equation (2.31) gives

E

IT T∫0

e−rtdXt

= X20eλ(T )+ ρσXσS

κ e−κT (Λ1(T )− Λ2(T )).

The expression for the Var (IT ) is straightforward.

Remark 2.8. Note that the assumption σ2X > r in the Proposition above is only needed

in order to get a closed expression with respect to the incomplete Gamma function.In case σ2

X ≤ r the defining integral of the incomplete Gamma function explodesaround 0. In this case Equation (2.29) still holds if we replace γ with the upperincomplete Gamma function γ(s, x) = −

∫∞xys−1e−ydy. In any case, regardless of

σ2X > r, Formula (2.29) holds when we replace Λ1(T ) − Λ2(T ) with

∫ T0 e(σ2

X−r)t(σ2X −

ρσXσSe−κ(T−t))e−ρσXσSe−κ(T−t)/κdt.

Proposition 2.9. The expressions for the optimal static hedge a (2.27) and for thecorresponding variance (2.28) from the previous proposition hold also in the 2GBM

79


case. For the involved expectations we have

E

T∫0

e−2rtX2t dt

=

X20

σ2X−2r (e(σ2

X−2r)T − 1), if σ2X 6= 2r,

X20T, if σ2

X = 2r,

and

E

IT T∫0

e−rtdXt

=

X0I0ρIXσXσIρIXσXσI−r (e(ρIXσXσI−r)T − 1), if ρIXσXσI 6= r,

X0I0ρIXσXσIT, if ρIXσXσI = r.

FurthermoreVar (IT ) = I2

0 (eσ2IT − 1).

Proof. The expressions for a and the corresponding variance Var (CT (a)) are modelindependent and therefore the same as in Proposition 2.7. In both models X is a GBMand hence we again have

E

T∫0

e−2rtX2t dt

=

X20

σ2X−2r (e(σ2

X−2r)T − 1), if σ2X 6= 2r,

X20T, if σ2

X = 2r.

For the expectation in the numerator we get

E

IT T∫0

e−rtdXt

= E

T∫0

e−rtd〈I,X〉t

=T∫

0

e−rtρIXσXσIE (ItXt) dt.

Straightforward computations give E (XtIt) = X0I0eρIXσXσIt, which plugged into the

above expressions gives the desired result. The expression for the Var (IT ) is straight-forward.

Using the expressions for the standard deviation of the hedge error from Theorem 2.4and Proposition 2.7 we can compare the risks entailed by both strategies. The lefthand panel of Figure 2.4 depicts the standard deviation of the static and dynamicvariance minimizing hedge against time to maturity. The right hand panel shows theincrease of the standard deviation if one confines with the static hedge. The increasein the variability by more than 10% for positions hedged over a period of one yearindicates that the hedge should be dynamically adjusted. The figure is again based onthe estimated parameter values obtained for the August 2009 crude oil futures contract(see Table 2.1 in Section 2.1.2).

2.5. Including Directional Views and Stochastic VolatilityIn this section we extend the model introduced in Section 2.1 by allowing for stochas-tic volatility of the futures and the logspread. We assume that the volatility of both

80

2.5. Including Directional Views and Stochastic Volatility

processes are proportional to a Cox-Ingersoll-Ross process. The futures dynamics thuscoincides with the dynamics of the risky asset in the Heston model.Let (W 1,W 2,W 3) be a 3-dimensional Brownian motion and suppose that the futures

price process X and its volatility ν = (νt)t≥0 evolve according to the SDE

dXt = µ(t, νt)Xtdt+√νtXtdW1t

dνt = β(ϑ− νt)dt+ σν√νt(ρ1dW

1t + ρ1dW

2t ),

where ρ1 ∈ [−1, 1], ρ1 =√

1− ρ21, β, ϑ, σν > 0, and µ : R2

+ → R is measurable. Asbefore, let St = log(Xt)− log(It). Assume that the logspread’s volatility is proportionalto ν, and that S solves the mean reverting SDE

dSt = κ(m− St)dt+ σS√νt(ρdW 1

t + ρηdW 2t + ρηdW 3

t ), S0 = s.

Since we have included a directional view in the dynamics of the futures price, we cannotdirectly invoke the method used in Section 2.2 for the derivation of the variance optimalhedge. When the trading instruments are assumed to be trended, and hence are notmartingales, then it is very difficult to determine variance optimal hedging strategy.There are, however, other quadratic optimality criteria that considerably simplify thecalculation of hedging strategies. A very intriguing type of hedging strategies are the so-called locally risk minimizing hedging strategies. These are variance optimal strategieswith respect to a particular martingale measure, usually referred to as the minimalmartingale measure. For an overview on quadratic hedging approaches we refer toSchweizer [2001] and for some more details on local risk minimization to Section 3.1.In our extended model the minimal martingale measure Q is given by

dQ

dP= exp

− T∫0

ω(t, ν)dW 1t −

12

T∫0

ω2(t, ν)dt

,

where ω(t, νt) = µ(t, νt)/√νt is the market price of risk.2

One can proceed as in Section 2.2 for the derivation of the local risk minimizinghedge, i.e. the variance optimal hedge relative to Q. The value function of h(IT ) willbe defined by

ψ(t, x, v, s) = e−r(T−t)EQ(h(Xt,x,v

T e−St,sT )).

With the same assumptions on h one can show that the local risk minimizing hedge isgiven by

ξt = ψx(t,Xt, νt, St)[1− σSρe−κ(T−t)

]+ σνρ1

Xtψv(t,Xt, νt, St).

Observe that the local risk minimizing hedge is now a weighted sum of the Delta

2A sufficient condition for this to be a proper measure change is the following growth condition on ω.For A,B ≥ 0 and δ ∈ [0, 1/2] we assume |ω(t, x)| ≤ A+Bxδ, x ≥ 0.

81


and Vega of the risk position’s expectation under the minimal martingale measure Q.Clearly, the term involving the Vega of the position appears due to the additional non-tradable risk induced by the stochastic volatility, which also needs to be cross-hedged.A similar analysis as in the previous sections, e.g. estimation of model parameters,

derivation of hedge errors and their respective standard deviations, is somewhat moreinvolved. However, one can profit of the affine model structure and express the valuefunction and its gradient in terms of generalized Ricatti equations. Fourier inversionmethods then yield semi-explicit formulas for optimal strategies, which are amenable toswift simulation analysis.

2.6. Conclusion and OutlookHedging is essential for controlling and managing risk and it is an important area ofresearch. In this chapter we show that a long-term relationship between the risk andthe hedging instrument has important implications for the optimal hedging strategyand, thus, also for the hedge error. In particular, we propose a model which explic-itly takes into account such a long-term relationship. We derive the variance optimalcross-hedge strategy und provide the variance of the hedge error in terms of the model’sparameters. We demonstrate the practical relevance of incorporating the long-termrelationship through an empirical example, where we find a long-term relationship be-tween most crude oil futures contracts and the spot kerosene price. Interestingly, themodel is also consistent with the commonly observed behavior of commodity traders,who use for cross hedges a hedge ratio of one instead of a hedge ratio dampened bythe cross correlation between the risk and the hedging instrument, which is implied bymodels with only correlated Brownian motions. Furthermore, we show that even forcases where the decision concerning the asymptotic stationarity of the logspread is notobvious, it is better to allow for a long-term relationship rather than to neglect it.The model can be extended towards several directions to provide a more realistic

dynamics for asset prices. Especially the consideration of jumps in the price processseems to be an interesting extension. However, several specifications are plausible anda careful empirical investigation is needed. On an ad hoc basis it is, for example, notclear whether the price processes jump together and how the jump sizes are related.These aspects will have significant impact on the properties of the hedge error and weplan to investigate these questions in more detail in future research.

82

3. Stochastic correlation

In this chapter we extent contemporary results on quadratic hedging with basis risk byallowing for the correlation to be random. As usual, we assume that the price of thetradable asset and the value of the non-tradable index evolve according to geometricBrownian motions. However, we will assume that the correlation between the drivingBrownian motions is not constant, but a random process with values between −1 and 1.More precisely, we will assume that the correlation process is the solution of a stochasticdifferential equation. Then, our focus is on deriving an explicit representation of thelocally risk minimizing strategy in terms of simple expectations.The chapter is organised as follows. In Section 3.1 we give a short introduction into

local risk minimization. Section 3.2 introduces our model and gives an overview on themain results we obtained. The details we use to derive our hedge formula are providedin Section 3.3. We continue in Section 3.4 by analyzing the boundary behaviour andintegrability properties of correlation processes. We conclude with Section 3.5 by givingsome explicit examples of correlation processes for which our main results hold.

3.1. A brief review of local risk minimizationIn this section we give a short introduction into the theory of local risk minimization ina quadratic sense. The material presented is a streamlined version of Schweizer [2008].We start with a filtered probability space (Ω,F , (Ft)0≤t≤T , P ), where T > 0 is a

finite time horizon and the filtration (Ft) satisfies the usual conditions, that is (Ft) isright continuous and completed by the P -null sets. We consider a financial market withone risky asset S and one non-risky asset, say a money market account with dynamicsB. We suppose that the discounted asset price X = S/B is an R-valued continuoussemimartingale, and we assume that X satisfies the so-called structure condition (SC).This means that X is a special semimartingale with canonical decomposition

X = X0 +M +A = X0 +M +∫λd〈M〉,

where M is a locally square integrable martingale with M0 = 0, and λ is an R-valued, predictable process such that the mean-variance tradeoff process K =

∫λdA =∫

λ2d〈M〉 satisfies KT < ∞, P -almost surely. It is well known that (SC) is related toan absence-of-arbitrage condition; see Schweizer [2008] for a reference.

Definition 3.1. (Schweizer [2008, Definition 1.1])The space ΘS consists of all R-valued predictable processes ξ such that the stochastic

83


integral process∫ξdX is well-defined and

E

T∫0

ξ2sd〈M〉s +

T∫0

|ξsdAs|

2 <∞.An L2-strategy is a pair ϕ = (ξ, η), where ξ ∈ ΘS and η is a real-valued adapted processsuch that the value process V (ϕ) = ξX + η is right-continuous and square-integrable.ϕ is called 0-achieving if VT (ϕ) = 0, P -almost surely.

As usual, a strategy ϕ = (ξ, η) describes the investment decisions of an agent tradingin the financial market. An investor following the strategy ϕ holds ξt shares of thediscounted asset X at time t, and keeps ηt units in a money market account. In thissection we use the money market account as numeraire so that we need not to botherabout the interest rate.We next consider a payment stream H = (Ht)0≤t≤T kept fixed throughout this in-

troduction. Mathematically, H is right-continuous, adapted, real-valued and square-integrable; the interpretation is that Ht ∈ L2(P ) represents the total payments on [0, t]arising due to some financial contract. A European contingent claim with maturity Twould have Ht = 0, for all t < T , and just an FT -measurable payoff HT ∈ L2(P ) due attime T ; in general, the process H involves both cash inflows and outlays, and can butneed not be of finite variation.

Definition 3.2. (Schweizer [2008, Definition 1.2])Fix a payment stream H. The (cumulative) cost process of an L2-strategy ϕ = (ξ, η) is

CHt (ϕ) = Ht + Vt(ϕ)−t∫

0

ξsdXs, 0 ≤ t ≤ T.

ϕ is called self-financing (for H) if CHt (ϕ) is constant, and mean-self-financing if CHt (ϕ)is a martingale (which is then square-integrable). Under the assumption that X fulfills(SC) and that the mean-variance tradeoff process K is continuous we say that an L2-strategy ϕ is locally risk minimizing if ϕ is 0-achieving and mean-self-financing, andthe cost process CH(ϕ) is strongly orthogonal to M , that is 〈M,CH(ϕ)〉t = 0, for allt ∈ [0, T ].

Thus, Ct(ϕ) comprises the hedger’s accumulated costs during [0, t] including the pay-ments Ht, and Vt(ϕ) should therefore be interpreted as the value of the portfolioϕt = (ξt, ηt) held at time t after the payments Ht. In particular, VT (ϕ) is the valueof the portfolio ϕT upon settlement of all liabilities, and a natural condition is then torestrict to 0-achieving strategies as defined in Definition 3.1.

Remark 3.3. (Schweizer [2008, Remark 1.3])Observe that if ϕt = (ξt, ηt) is a 0-achieving and mean-self-financing L2-strategy for H,then ϕ is uniquely determined from ξ (and of course H).

84

3.2. The model and the main results

It is well known that the locally risk-minimizing strategy can be obtained via theso-called Föllmer-Schweizer decomposition (FS) of the final payment HT . That is thedecomposition of HT into

HT = H(0)T +

T∫0

ξHTs dXs + LHTT , P -almost surely, (3.1)

where H(0)T ∈ L2(P ) is F0-measurable, ξHT is in ΘS , and the process LHT is a (right-

continuous) square-integrable martingale strongly orthogonal to M and satisfyingLHT0 = 0. Notice that such a decomposition can be shown to be unique. Once we have(3.1), the desired strategy ϕ = (ξ, η) is then given by

ξ = ξHT , η = V HT − ξHTX,

with

V HTt = H(0)T +

t∫0

ξHTs dXs + LHTt −Ht, 0 ≤ t ≤ T,

see Schweizer [2008, Proposition 5.2]. Furthermore, the associated cost process is givenby

CHt (ϕ) = H(0)T + LHTt , 0 ≤ t ≤ T.

3.2. The model and the main resultsLetW = (W 1,W 2,W 3) be a three-dimensional Brownian motion on a filtered probabil-ity space (Ω,F , (Ft)0≤t≤T , P ), where the filtration is generated by W and augmentedby the P -null sets. Consider two processes with dynamics

dSt = St(µXdt+ σXdW

1t

),

dUt = Ut

(µIdt+ σI

(ρtdW

1t +

√1− ρ2

tdW2t

)),

where W 1 and W 2 are independent Brownian motions. To simplify the presentationwe will assume throughout that all coefficients are constant, more precisely µX , µI ∈ Rand σX , σI ∈ R \ 0.We assume that S is the price process of a tradable asset, and U the value process of

a non-tradable index. The correlation ρ is assumed to follow

dρt = a(ρt)dt+ g(ρt)dWt, (3.2)

where W is given by W = γW 1u + δW 2

u +√

1− γ2 − δ2W 3u , with W 3 being a Brownian

motion independent of W 1 and W 2, and γ and δ real numbers such that δ2 + γ2 ≤ 1.

85


For the moment, we assume that the coefficients of the correlation dynamics, a and g,belong to C1(−1, 1), and that there exists a unique solution ρ of (3.2) with values in[−1, 1].

Throughout we suppose that the interest r > 0 is constant, and let Bt = ert. Thediscounted processes S and U will be denoted by

Xt = e−rtSt, It = e−rtUt.

Notice that

dXt = Xt

((µX − r) dt+ σXdW

1t

), (3.3)

dIt = It

((µI − r) dt+ σI

(ρtdW

1t +

√1− ρ2

tdW2t

)).

Consider a derivative d(UT ) depending on the non-tradable index. Let h(x) =e−rT d(erTx). Then d(UT ) = erTh(IT ). Our goal is to analyse how to hedge the li-ability h(IT ) by trading the asset X.

Since the market is incomplete we need to choose a criteria according to which strate-gies are chosen and the prices of contingent claims are computed. We will use theframework of local risk minimization of Section 3.1.

Our first main result is an explicit hedge formula, which can be easily implemented,for example by simple Monte Carlo simulation. We will state it right away in Theorem3.4, after a brief collection of some notations and assumptions.

We will need the conditional versions of the processes I and ρ, which are given by

It,y,vs = y +s∫t

It,y,vu

((µI − r) du+ σI

(ρt,vu dW 1

u +√

1−(ρt,vu)2dW 2

u

)), (3.4)

ρt,vs = v +s∫t

a(ρt,vu)du+

s∫t

g(ρt,vu)dWu, (3.5)

for t ∈ [0, T ), (y, v) ∈ R+ × (−1, 1).

In order to find a nice representation of the quadratic hedge, we also need the dy-namics of the derivatives of It,y,v and ρt,v with respect to the initial values y and v.Note that the derivative with respect to y of It,y,v is given by ∂

∂y It,y,v = It,y,v

y = It,1,v,and obviously ∂

∂yρt,v = 0. If the correlation process neither attains −1 nor 1 up to time

T , then the derivatives of It,y,v and ρt,v with respect to v are defined. Moreover, the

86

3.2. The model and the main results

processes ∂∂v I

t,y,v and ∂∂vρ

t,v, denoted by It,y,v and ρt,v respectively, solve the SDE

It,y,vs =s∫t

It,y,vu

((µI − r) du+ σI

(ρt,vu dW 1

u +√

1−(ρt,vu)2dW 2

u

))

+s∫t

It,y,vu σI

ρt,vu dW 1u −

ρt,vu√1−

(ρt,vu)2ρt,vu dW 2

u

, (3.6)

ρt,vs = 1 +s∫t

a′(ρt,vu)ρt,vu du+

s∫t

g′(ρt,vu)ρt,vu dWu, (3.7)

for s ∈ [t, T ], see Protter [2004, Chapter V.7, Theorem 38]. Notice that the correlationboundaries −1 and 1 are not attained if and only if the stopping times τv = τ t,v =infs ≥ t : ρt,vs ∈ −1, 1 satisfy τ t,v > T , P -almost surely. We formulate this asCondition

(H1) τ t,v > T, P -almost surely, ∀ v ∈ (−1, 1).

Notice that (H1) guarantees that∫ Tt

(g′(ρt,vu ))2du <∞, P -almost surely and

T∫t

(ρt,vu )2

1−(ρt,vu)2 du <∞, P -almost surely, (3.8)

and hence the stochastic integrals appearing in (3.6) and (3.7) are defined. Finally, forour aim of deriving an explicit representation of the quadratic hedge, we need to imposea stronger integrability condition than (3.8) on ρ and ρ. More precisely, we will assumethat the following condition is satisfied

(H2) There exists a p > 1 such that for every t ∈ [0, T ] and v0 ∈ (−1, 1) there existsan open neighbourhood U ⊂ (−1, 1) of v0 such that

supv∈U

E

T∫t

∣∣∣∣∣ (ρt,vs )2

1−(ρt,vs)2

∣∣∣∣∣p

ds <∞.

We are now ready to state our first main result which gives an explicit expression forthe locally risk minimizing strategy in terms of expectations with respect to the measureP with density dP /dP = E(−

∫µX−rσX

dW 1)T , where E(·) denotes the Doléans-Dadeexponential. Note that P corresponds to the so-called minimal martingale measure, seeSchweizer [2008].

Theorem 3.4. Suppose that the coefficients a and g in the dynamics of ρ are contin-uously differentiable on (−1, 1). Assume furthermore that (H1) and (H2) are satisfied.Let h be Lipschitz such that the weak derivative h′ is Lebesgue-almost everywhere con-tinuous. Then, the locally risk minimizing strategy ϕ = (ξ, η) exists for the derivative

87


h(IT ). ξ is given by ξt = ξ(t, It, Xt, ρt), for t ∈ [0, T ], where

ξ(t, y, x, v) = y

x

[vσIσX

E[h′(It,y,vT

)It,1,vT

]+ g(v)γ

σXE[h′(It,y,vT

)It,1,vT

]]. (3.9)

The proof of this Theorem is postponed to Section 3.3.

Remark 3.5. In terms of the original processes S and U the hedge would be given byξt = ξ(t, Ut, St, ρt) where

ξ(t, y, x, v) = y

x

[vσIσX

E[d′(U t,y,vT

)U t,1,vT

]+ g(v)γ

σXE[d′(U t,y,vT

)U t,1,vT

]]e−r(T−t),

with U obtained in the same way as I.

Before we state our second main contribution, let us apply the previous result to derivethe locally risk minimizing strategy for a European Call option.

Corollary 3.6. Suppose that the correlation is a deterministic function of time. Forstrike K > 0, let d(x) = max(x − K), 0. Then, the locally risk minimizing strategyϕ = (ξ, η) exists for the derivative d(UT ). ξ is given by

ξt = ρtUtσIStσX

∆BS(t, Ut, κt, σI),

for t ∈ [0, T ], where κt = −(µI −r)(T − t)+σI(µX−rσX)∫ Ttρsds and, with Φ the standard

normal cumulative distribution function,

∆BS(t, y, q, σ) = exp(−q)Φ(

ln(y/K) +(r + σ2

I/2)

(T − t)− qσI√T − t

)is the Black-Scholes delta for options on stocks with continuous dividend yield q.

The content of the preceding corollary is only a slight extension of a result alreadyobtained in Hulley and McWalter [2008] for the case of constant correlation. The proofis a simple straightforward calculation.

Remark 3.7. From the local risk minimizing strategy we can easily deduce the so-calledmean-variance optimal hedging strategy for the payoff h(IT ). The mean variance hedgeis defined to be the self-financing strategy minimzing the variance of the global hedgingerror, and usually differs from the local risk minimizing strategy, see Schweizer [2001]for an introduction into mean-variance hedging. In the model considered here, the meanvariance trade-off process is deterministic, and hence, by an appeal to Schweizer [2001,Theorems 4.6 and 4.7], the mean variance hedge has a representation allowing to deriveit numerically by a simply recursion. Namely, letting w = E(h(IT )), the mean-varianceoptimal strategy (ξ, η) for h(IT ) satisfies, with ξ = ξ(w),

ξ(w)t = ξt + µX − r

σ2XXt

E [h (IT ) |Ft]− w −t∫

0

ξ(w)s dXs

,

88

3.3. Derivation of the hedge formula

for all t ∈ [0, T ], and

ηt = w +t∫

0

ξ(w)s dXs − ξ(w)

t Xt,

for all t ∈ [0, T ].

Our second main result concerns conditions on the coefficients a and g of ρ such that(H1) and (H2) are fulfilled.

Theorem 3.8. Let a and g be continuously differentiable with bounded derivatives. Weassume that g(−1) = g(1) = 0, and that g does not have any roots in (−1, 1). If

lim supx↑1

2a(x)(1− x)g2(x) < 0 and lim inf

x↓−1

2a(x)(1 + x)g2(x) > 0, (3.10)

then (H1) and (H2) are satisfied, and hence, the delta hedge is given as in Theorem 3.4.

The preceding theorem can be generalized, which will enable us to give an example inSection 3.5 where the derivative of g is unbounded. This, however, requires a little morenotation, which for ease of exposition is avoided here. See Section 3.4 for a proof ofTheorem 3.8 and the more general Proposition 3.20.

3.3. Derivation of the hedge formulaIn this section we will derive and prove the hedge formula stated in Theorem 3.4. InSubsection 3.3.1 we use BSDEs to derive the Föllmer-Schweizer decomposition, whichis the key to obtain the Formula (3.9). In Sections 3.3.2 and 3.3.3 we provide detailswhich we need along the way. It is there that we need Conditions (H1) and (H2). Letus first recall the definition of a BSDE.As in Section 3.2, let W = (W 1,W 2,W 3) be a three-dimensional Brownian motion.

The filtration generated byW and completed by the P -null sets will be denoted by (Ft).Let T > 0 and ξ be an FT -measurable random variable, and let f : Ω×[0, T ]×R×R3 → Rbe a measurable function such that for all (y, z) ∈ R × R3 the mapping f(·, ·, y, z) ispredictable. A solution of the BSDE with terminal condition ξ and generator f isdefined to be a pair of predictable processes (Y,Z) such that almost surely we have∫ T

0 |Zs|2ds <∞,

∫ T0 |f(s, Ys, Zs)|ds <∞, and for all t ∈ [0, T ]

Yt = ξ −T∫t

ZsdWs +T∫t

f(s, Ys, Zs)ds.

The solution processes (Y, Z) are often shown to satisfy some integrability properties.To this end one usually verifies whether they belong to the following function spaces.Let p ≥ 1. We denote by Hp(R3) the set of all R3-valued predictable processes ζ suchthat E

∫ T0 |ζt|

pdt < ∞, and by Sp(R) the set of all R-valued predictable processes δsatisfying E(sups∈[0,T ] |δs|p) <∞.

89


3.3.1. Deriving the FS decomposition with BSDEs

As stated in Section 3.2 we use the framework of local risk minimization. Accordingly,note first that X satisfies (SC), that is X is a special semimartingale with canonicaldecomposition given by M =

∫σXXdW , λ = µX−r

σ2XX

and hence Kt =(µX−rσX

)2t, for

t ∈ [0, T ]. In order to find the FS decomposition we consider a BSDE with terminalcondition h(IT ), and driver f to be specified later,

Yt = h(IT )−T∫t

ZsdWs +T∫t

f(s, Ys, Zs)ds. (3.11)

Assume that f can be chosen such that

t∫0

ξdXs =t∫

0

Z1sdW

1s −

t∫0

f(s, Ys, Zs)ds, (3.12)

for all t ∈ [0, T ]. Also, by using (3.3), we have

t∫0

ξdXs =t∫

0

ξσXXsdW1s +

t∫0

ξXs(µX − r)ds. (3.13)

Uniqueness of semimartingale decompositions yields that the martingale parts of (3.12)and (3.13) coincide, and therefore it must hold ξ = Z1

σXX, P⊗λ-almost surely. Moreover,

the driver f has to satisfyf(s, y, z) = −z1µX − r

σX. (3.14)

Indeed, one can show that the solution of the BSDE with generator (3.14) provides theFS decomposition. We summarize this in the next result, which is in fact a special caseof El Karoui et al. [1997b, Proposition 1.1].

Lemma 3.9. The FS decomposition of h(IT ) is given by

h(IT ) = Y0 +T∫

0

Z1s

σXXsdXs +

T∫0

Z2sdW

2s +

T∫0

Z3sdW

3s ,

where (Y,Z) is the solution of the BSDE (3.11) with generator f defined as in (3.14).

In order to obtain a characterization of the solution of

Yt = h(IT )−T∫t

ZsdWs −T∫t

Z1s

µX − rσX

ds,

90


we consider the conditional forward SDE given by (3.4) and (3.5) and the associatedconditional BSDE

Y t,y,vs = h(It,y,vT

)−

T∫s

Zt,y,vu dWu −T∫s

Z1,t,y,vu

µX − rσX

du, (3.15)

for s ∈ [t, T ]. Since the BSDE (3.15) is linear we know by standard results, see forexample El Karoui et al. [1997b], that

Y t,y,vs = E[h(It,y,vT

)ΓsT |Fs

]= E

[h(It,y,vT

) Γ0T

Γ0s

∣∣∣ Fs] ,where Γts = exp

(−µX−rσX

(W 1s −W 1

t )− (µX−r)2

2σ2X

(s− t))is the solution of

dΓts = Γts[−µX − r

σXdW 1

s

], Γtt = 1.

Let P be the probability measure with density dPdP = Γ0

T , and denote with ψ : [0, T ] ×R+ × (−1, 1)→ R the function defined by

ψ(t, y, v) = E[h(It,y,vT

)]. (3.16)

That the function ψ is well defined and that is has first derivatives with respect to yand v follows from Sections 3.3.2 and 3.3.3. The value process of the solution of theBSDE (3.15) satisfies Y t,y,vs = ψ(s, It,y,vs , ρt,vs ). Our main goal is to derive the explicithedge formula (3.9). With the help of Lemma 3.17 we get the following representationfor Zt,y,vs .

Zt,y,vs = σ(It,y,vs , ρt,vs )∗(∂yψ(s, It,y,vs , ρt,vs )∂vψ(s, It,y,vs , ρt,vs )

),

where the volatility matrix σ(y, v) is given by

σ(y, v) =(yσIv yσI

√1− v2 0

g(v)γ g(v)δ g(v)√

1− γ2 − δ2

), y ∈ R+, v ∈ (−1, 1).

Hence, we have

Z1,t,y,vs = It,y,vs σIρ

t,vs ∂yψ(s, It,y,vs , ρt,vs ) + g(ρt,vs )γ∂vψ(s, It,y,vs , ρt,vs ), (3.17)

that is the hedge formula is given by

ξ(t, y, x, v) = yσIv∂yψ(t, y, v) + g(v)γ∂vψ(t, y, v)σXx

. (3.18)

Thus by plugging in the explicit representations of ∂yψ(s, y, v) and ∂vψ(s, y, v), givenin Section 3.3.3, we obtain (3.9), and hence have proven Theorem 3.4.

91


Remark 3.10. Note, that the approach we take by characterizing the Föllmer-Schweizerdecomposition via the solution of a linear BSDE is the same as in El Karoui et al.[1997b, Example 1.3]. In our model, however, the inverse of the volatility matrix of theasset processes X and I is unbounded and hence does not fall within the specificationsof El Karoui et al. [1997b, Hypothesis 1.1]. Moreover, the coefficients of the volatilitymatrix of the forward processes I and ρ associated with the BSDE do not satisfy theprerequisites of El Karoui et al. [1997b, Proposition 5.9], that is we do not have uniformlybounded derivatives. In order to recover our hedge formula in spite of these extensionswe apply the results of Sections 3.3.2 to 3.4.

3.3.2. Differentiability with respect to the initial conditions

In this section we want to make some remarks on the system of SDEs, given by Equations(3.4) to (3.7), concerning existence, uniqueness, continuity and differentiablity withrespect to the initial values y and v. We also observe the following.

Lemma 3.11. Suppose that (H1) holds. Then the SDE for It,y,v in (3.6) has a uniquesolution which is given by

It,y,vs = yE(Gt,v

)s

s∫t

E(Gt,v

)−1udHt,1,v

u , (3.19)

for s ∈ [t, T ], where

Ht,y,vs =

s∫t

It,y,vu σI

ρt,vu dW 1u −

ρt,vu√1−

(ρt,vu)2ρt,vu dW 2

u

,

for s ∈ [t, T ], and

Gt,vs =s∫t

(µI − r) du+s∫t

σI

(ρt,vu dW 1

u +√

1−(ρt,vu)2dW 2

u

),

for s ∈ [t, T ].

Proof. Due to (H1) we can define the semimartingales (Ht,y,vs )t≤s≤T and (Gt,vs )t≤s≤T

as above. The dynamics in (3.6) immediately imply that It,y,vs is the solution of thelinear stochastic equation

It,y,vs = Ht,y,vs +

s∫t

It,y,vu dGt,vu ,

92


the solution of which is given by

It,y,vs = E(Gt,v

)s

Ht,y,vt +

s∫t

E(Gt,v

)−1u

(dHt,y,v

u − d⟨Ht,y,v, Gt,v

⟩u

) ,

see for example Revuz and Yor [1999, Chapter IX]. Notice that

d⟨Ht,y,v, Gt,v

⟩u

= It,y,vu σ2I ρt,vu ρt,vu du− It,y,vu σ2

I

√1− (ρt,vu )2 ρt,vu√

1− (ρt,vu )2ρt,vu du = 0.

Since Ht,y,vt = 0 and Ht,y,v = yHt,1,v we obtain (3.19).

Remark 3.12. The process It,y,v is given by

It,y,vs = y exp

s∫t

σI

(ρt,vu dW 1

u +√

1−(ρt,vu)2dW 2

u

)+

s∫t

(−1

2σ2I + µI − r

)du

,for s ∈ [t, T ]. Moreover suppose that

∫ stg′(ρt,vu )dWu is well defined, then the SDE in

(3.7) has a unique solution ρt,v given by

ρt,vs = E

s∫t

g′(ρt,vu)dWu +

s∫t

a′(ρt,vu)du

,

for s ∈ [t, T ].

Before we end this section we want to give an auxiliary result which will be used in thesequel.

Lemma 3.13. Consider two predictable processes cv and dv, depending on a parameterv ∈ (−1, 1). Suppose that there exists a continuous function D : (−1, 1)→ R+ such thatcv + |dv| ≤ D(v), for all v ∈ (−1, 1). Then the process

bt,vs = E

s∫t

dvudWu +s∫t

cvudu

,

is defined, and for all p ≥ 1 and v0 ∈ (−1, 1) there exists an open neighbourhoodU ⊂ (−1, 1) of v0, such that

supv∈U

E

(sup

t≤u≤T|bt,vu |p

)<∞.

Proof. Let p ≥ 1. The Burkholder-Davis-Gundy Inequality implies that for a constant

93


Cp, depending only on p, we have,

E

(sup

t≤u≤T|bt,vu |p

)≤ ep(T−t)D(v)E

supt≤u≤T

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣p

≤ Cpep(T−t)D(v)E

∣∣∣∣∣∣∣T∫t

∣∣∣∣∣∣(dvu)2 E

u∫t

dvwdWw

∣∣∣∣∣∣2

du

∣∣∣∣∣∣∣p/2

≤ Cpep(T−t)D(v)Dp(v)E

∣∣∣∣∣∣∣T∫t

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣2

du

∣∣∣∣∣∣∣p/2

.

In the rest of the proof we have to distinguish between p ≥ 2 and p ∈ [1, 2). We firstconsider p ≥ 2. By Jensen’s inequality and Fubini’s theorem we get

E

∣∣∣∣∣∣∣T∫t

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣2

du

∣∣∣∣∣∣∣p/2

≤ ET∫t

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣p

du

=T∫t

E

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣p

du. (3.20)

Notice that |E(∫ utdvwdWw)|p = exp(

∫ utpdvwdWw +

∫ utp2(dvw)2dw −

∫ utp2(dvw)2dw −

p2∫ ut

(dvw)2dw), and thus Hölder’s inequality implies that the left hand side of Inequality(3.20) can be further estimated against

E

∣∣∣∣∣∣∣T∫t

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣2

du

∣∣∣∣∣∣∣p/2

≤T∫t

Eexp

u∫t

2pdvwdWw −12

u∫t

4p2 (dvw)2dw

1/2

×

Eexp

u∫t

2p2 (dvw)2dw − p

u∫t

(dvw)2dw

1/2

du

94


Consequently,

E

∣∣∣∣∣∣∣T∫t

∣∣∣∣∣∣E u∫t

dvwdWw

∣∣∣∣∣∣2

du

∣∣∣∣∣∣∣p/2

≤T∫t

ep2(T−t)D(v)

EE

u∫t

2pdvwdWw

1/2

du.

= (T − t)ep2(T−t)D(v),

which yieldsE

(sup

t≤u≤T

∣∣bt,vu ∣∣p) ≤ CpDp(v)(T − t)e(p+p2)(T−t)D(v),

from which we deduce the result for p ≥ 2. For 1 ≤ p < 2 we use Jensen’s inequality toobtain

E

(sup

t≤u≤T

∣∣bt,vu ∣∣p) ≤ CE∣∣∣∣∣∣∣T∫t

∣∣∣∣∣∣E u∫t

dt,vw dWw

∣∣∣∣∣∣2

du

∣∣∣∣∣∣∣p/2

,

and continue with the same arguments as for p > 2. Hence the result.

Remark 3.14. Since It,y,v is lognormally distributed, independent of v, we have

supv∈(−1,1)

E(∣∣It,y,vs

∣∣p) <∞,for all p ≥ 1. Let K be a compact subset of (−1, 1) and suppose supv∈K g′(ρt,vs ) isbounded and supv∈K a′(ρt,vs ) is bounded above, uniformly for all t ≤ s ≤ T . Then wehave supv∈K E(

∫ Tt|ρt,vs |pds) <∞, for all p ≥ 1, by Lemma 3.13.

3.3.3. Differentiability of ψIn order to derive the hedge formula (3.18) we need to ensure that ψ defined in (3.16) iscontinuously differentiable with respect to v and y. We only consider the differentiablityin v, since for y it is comparatively simpler. Since we want to use uniform integrabilitythis is where (H1) and (H2) come into play.

Lemma 3.15. Suppose (H1) and (H2) hold. Then, for all v0 ∈ (−1, 1), there exists anopen neighbourhood U ⊂ (−1, 1), such that

supv∈U

E(∣∣It,y,vT

∣∣p′) < C, (3.21)

for all p′ ∈ [1, p), with p of Assumption (H2). Moreover C is a constant that dependsonly on p from Condition (H2), the model’s parameters and U .

Proof. Let p′ ≥ 1, such that p > p′ with p of Assumption (H2). Let Gt,v and Ht,y,v

be defined as in Lemma 3.11. Notice that Gt,v is lognormally distributed. Since thedistribution does not depend on the correlation, there exists a constant C ∈ R+ such

95


that we have E(|E(Gt,v)|2p′) ≤ C, for all v ∈ (−1, 1). The Cauchy-Schwarz Inequality,the Burkholder-Davis-Gundy Inequality and Jensen’s inequality imply

E(∣∣It,y,vT

∣∣p′) = E

∣∣∣∣∣∣E (Gt,v)T

T∫t

E(Gt,v

)−1udHt,y,v

u

∣∣∣∣∣∣p′

≤√C

E supt≤s≤T

∣∣∣∣∣∣s∫t

E(Gt,v

)−1udHt,y,v

u

∣∣∣∣∣∣2p′

12

≤√CCp′

E∣∣∣∣∣∣T∫t

E(Gt,v

)−2u

(It,y,vu

)2σ2I

[(ρt,vu )2

1−(ρt,vu)2

]du

∣∣∣∣∣∣p′

12

≤√CCp′

E∣∣∣∣∣∣

T∫t

E(Gt,v

)−2p′

u

(It,y,vu

)2p′σ2p′I

[(ρt,vu )2

1−(ρt,vu)2

]p′du

∣∣∣∣∣∣

12

.

Now choose p > 1, such that pp′ < p. An application of the Hölder Inequality yields,with q = p

p−1 ,

(E(∣∣It,y,vT

∣∣p′))2≤ CC2

p′

E∣∣∣∣∣∣

T∫t

E(Gt,v

)−2p′qu

(It,y,vu

)2p′qdu

∣∣∣∣∣∣1/q

×

E∣∣∣∣∣∣

T∫t

[(ρt,vu )2

1−(ρt,vu)2

]p′pdu

∣∣∣∣∣∣1/p

.

For any U ⊂ (−1, 1), the supremum supv∈U E(|∫ TtE(Gt,v)−2p′q

u (It,y,vu )2p′qdu|) is finitedue to the lognormal distribution of It,y,v and the normal distribution of Gt,v, thedistributions of which do not depend on the correlation process ρ. Therefore, with Ufrom Assumption (H2), we get supv∈U E(|It,y,vT |p′) < C.

The following lemma states conditions under which ψ is differentiable with respect tov.

Lemma 3.16. Let h be Lipschitz such that the weak derivative h′ is Lebesgue-almosteverywhere continuous. Under (H1) and (H2) ψ(t, y, v) is continuously differentiablewith respect to v and the partial derivative ∂vψ(t, y, v) is given by

∂vψ(t, y, v) = E[h′(It,y,vT

)It,y,vT

].

96


Proof. Let v0 be an element of (−1, 1), and p > 1 as in (H2). According to Lemma 3.15we can choose a δ > 0 with (v0 − δ, v0 + δ) ⊂ (−1, 1), such that, for all p′ ∈ [1, p), wehave supv∈(v0−δ,v0+δ) E(|It,y,vT |p′) <∞. For all v ∈ (v0 − δ, v0 + δ), we will show

1. ψ(t, y, v) is well defined,

2. h(It,y,vT

)is absolutely continuous in v,

3. E[h′(It,y,vT

)It,y,vT

]is continuous at v = v0, and

4. E[∫ δ−δ

∣∣h′ (It,y,v0+uT

)It,y,v0+uT

∣∣ du] <∞.

By standard arguments these four statements imply the result, see for instance Durrett[1996].The properties of h imply that there exists a constant C ∈ R+ such that |h(x)| ≤

C(1 + |x|), and hence with Remark 3.14 we have, with q = pp−1 ,

E[∣∣h (It,y,vT

)∣∣] ≤ E (∣∣Γ0T

∣∣q)1/qE(∣∣h (It,y,vT

)∣∣p)1/p

≤ E(∣∣Γ0

T

∣∣q)1/q2C(

1 + E∣∣It,y,vT

∣∣p)1/p

<∞,

and therefore ψ is well defined.Since h is Lipschitz, it is absolutely continuous. Besides, It,y,vT is differentiable and

continuous in v (see Section 3.2), and consequently, the composition h(It,y,vT ) is abso-lutely continuous in v. With Hölder’s inequality we have, for p′ ∈ [1, p) and p > 1, suchthat p > p′p > 1,

E[∣∣h′ (It,y,vT

)It,y,vT

∣∣p′] ≤ C (E [∣∣It,y,vT

∣∣p′p])1/p.

Thus, by Lemma 3.15 we get

supv∈(v0−δ,v0+δ)

E[∣∣h′ (It,y,vT

)It,y,vT

∣∣p′] ≤ C <∞.

Hence, the family of random variables (h′(It,y,vT )It,y,vT )v∈(v0−δ,v0+δ) is uniformly inte-grable with respect to P . Now, let (vn)n∈N be any sequence in (v0− δ, v0 + δ) with limitv0. Then by continuity of h′(It,y,vT )It,y,vT and the properties of uniform integrability weget

limn→∞

∣∣E [h′ (It,y,vnT

)It,y,vnT

]− E

[h′(It,y,v0T

)It,y,v0T

]∣∣≤ limn→∞

E[∣∣h′ (It,y,vnT

)It,y,vnT − h′

(It,y,v0T

)It,y,v0T

∣∣] = 0,

97


that is the continuity of E[h′(It,y,vT )It,y,vT ] at v = v0. We use boundedness of h′ andFubini’s Theorem to get

E

δ∫−δ

∣∣h′ (It,y,v0+uT

)It,y,v0+uT

∣∣ du ≤ C δ∫

−δ

E∣∣It,y,v0+uT

∣∣ du ≤ C supv∈(v0−δ,v0+δ)

E∣∣It,y,vT

∣∣ ,which is finite by Lemma 3.15. Since we verified 1.-4. the proof of Lemma 3.16 iscomplete.

3.3.4. The hedge as variational derivativeThe control process Zt,y,v of the linear BSDE (3.15) has a representation in terms ofthe gradient of ψ and the matrix-valued function defined by

σ(y, v) =(yσIv yσI

√1− v2 0

g(v)γ g(v)δ g(v)√

1− γ2 − δ2

), y ∈ R+, v ∈ (−1, 1).

Lemma 3.17. Assume that (H1) and (H2) hold, that a and g are continuously dif-ferentiable and let h be Lipschitz such that the weak derivative h′ is Lebesgue-almosteverywhere continuous. Then, for s ∈ [t, T ],

Zt,y,vs = σ(It,y,vs , ρt,vs )∗(∂yψ(s, It,y,vs , ρt,vs )∂vψ(s, It,y,vs , ρt,vs )

). (3.22)

Proof. By Lemma 3.16, ψ(s, y, v) is continuously differentiable in y and v. As is shownin Imkeller et al. [2012, Theorem 5.2 and Remark 5.3.i] this is sufficient for Equation(3.22) to hold.

Note that in Imkeller et al. [2012] the authors establish representations as in (3.22) byusing only elementary methods. However, up to now the standard method of derivingthese relationships was to interpret Zt,y,v as the Malliavin derivative, or more preciselythe Malliavin trace, of Y t,y,v. Compared to the approach given in Imkeller et al. [2012]this has the disadvantage that additional regularity assumptions which originate inthe usage of the Malliavin calculus are needed. Nevertheless we want to outline howMalliavin calculus can be used to derive (3.22), thus giving a proof of (3.22) in this thesis(though not in full generality). Since this approach entails variational derivatives of theforward processes I and ρ, see Equation (3.23), we need the additional assumption thatthe coefficients a and g of the dynamics of ρ have bounded derivatives.

Proof (under the additional assumptions that a and g have bounded derivatives). LetIn be the solution of the SDE

dInt = Int (µI − r)dt+ σIInt

(1− 1n

)ρtdW

1t +

√1−

(1− 1

n

)2ρ2tdW

2t

.

98


It is straigtforward to show that InT converges to IT in L2. By taking a subsequence,we may assume that InT converges to IT almost surely.Next we approximate the payoff function h by a sequence of everywhere differentiable

and globally Lipschitz continuous functions. More precisely, let ϕ(x) = 1√2π e− x2

2 , x ∈ R,the density of a standard normal distibution and let ϕn(x) = nϕ(nx), x ∈ R, for alln ≥ 1. We define hn as the convolution with ϕ, that is hn = h ∗ ϕn. Observe thathn is Lipschitz continuous with respect to the same Lipschitz constant as h. Note thatLipschitz continuity of h implies uniform convergence of hn to h, hence hn(InT ) convergesP -almost surely to h(IT ). Moreover, hn is differentiable.As before, we denote by In,t,y,v the process In conditioned on Int = y and ρt = v.

We further define ψn(t, y, v) = E[hn(In,t,y,vT )], for all n ≥ 1, where E denotes theexpectation with respect to the measure P defined in Section 3.2. Note that by thesame methods as in Section 3.3.3 it can be shown that the ψn are differentiable; indeed,due to the factor (1 − 1

n ) the integrability condition in (3.21) is trivial. Moreover, itsderivatives are bounded, that is the ψn are Lipschitz continuous.We proceed by showing that ψn converges pointwise to ψ. Indeed, with L ∈ R+ being

the Lipschitz constant of h, we have

limn|ψn(t, y, v)− ψ(t, y, v)| ≤ lim

n

√E∣∣hn (In,t,y,vT

)− h

(It,y,vT

)∣∣2≤ lim

n

√4(E∣∣hn (In,t,y,vT

)− h

(In,t,y,vT

)∣∣2 + E∣∣h (In,t,y,vT

)− h

(It,y,vT

)∣∣2)≤ lim

n

√4(‖hn − h‖∞ + L2E |InT − IT |

2)

= 0.

Let (Y n,t,y,v, Zn,t,y,v) be the solution of the BSDE

Y n,t,y,vs = hn(In,t,y,vT

)−

T∫s

Zn,t,y,vu dWu −T∫s

Zn,1,t,y,vu

µX − rσX

du,

for s ∈ [t, T ]. Since hn(In,t,y,vT ) converges to h(It,y,vT ) in L2(P ), standard a prioriestimates for Lipschitz BSDEs, or simply the Ito isometry under the measure P , implythat (Y n, Zn) converges to (Y,Z) in S(R)⊗H2

T (R3).Notice that, due to the Markov property, we have Y n,t,y,vs = E[hn(In,t,y,vT )|Fs] =

ψn(s, In,t,y,vs , ρt,vs ). Since the approximations ψn are Lipschitz continuous, we may applythe chain rule, which yields

DuYn,t,y,vs = ∂yψ

n(s, In,t,y,vs , ρt,vs

)DuI

n,t,y,vs + ∂vψ

n(s, In,t,y,vs , ρt,vs

)Duρ

t,vs , (3.23)

for u ∈ [t, T ], where Du denotes the Malliavin derivative of Y n,t,y,v, In,t,y,v and ρt,v

respectively. DuIn,t,y,v and Duρ

t,v are solutions of linear SDEs, see Nualart [2006,Theorem 2.2.1]. In particular this guarantees right continuity of DuY

n,t,y,vs in s.

By the Clark-Ocone formula, the process Zn,t,y,v is the predictable projection of

99


hn(In,t,y,vT ) under the measure P . More precisely, for all s ∈ [t, T ], we have Y n,t,y,vs =∫ stE[DuY

n,t,y,vs |Fu]dWu, where Wt = (W 1

t + µX−rσX

t,W 2t ,W

3t ) and E[·|Fu] stands for

the predictable projection operator with respect to P . Due to the right continuityof DuY

n,t,y,vs in s, we may interchange the Malliavin and the predictable projection

operator, which yields

Zn,t,y,vu = E[DuY

n,t,y,vs |Fu

]= lim

s↓uDuY

n,t,y,vs

= σn(u, In,t,y,vu , ρt,vu

)∗(∂yψn(u, In,t,y,vu , ρt,vu )∂vψ

n(u, In,t,y,vu , ρt,vu )

), (3.24)

where

σn(y, v) =(yσI(1− 1

n )v yσI

√1− (1− 1

n )2v2 0g(v)γ g(v)δ g(v)

√1− γ2 − δ2

).

We next show that the partial derivatives ∂yψn and ∂vψn converge pointwise to ∂yψand ∂vψ, respectively. To this end denote again the derivatives of In,t,y,v with respectto v by In,t,y,v. Lemma 3.15 yields that supnE|InT |p < ∞, which further implies thatthe sequence (InT ) is uniformly integrable. Moreover,

|∂vψn − ∂vψ| ≤ E∣∣(hn)′

(In,t,y,vT

)In,t,y,vT − h′

(It,y,vT

)It,y,vT

∣∣≤ E

∣∣(hn)′(In,t,y,vT

)∣∣ ∣∣In,t,y,vT − It,y,vT

∣∣+ E∣∣It,y,vT

∣∣ ∣∣(hn)′(In,t,y,vT

)− h′

(It,y,vT

)∣∣ . (3.25)

We show separately that both summands in (3.25) converge to 0 as n → ∞. Sincethe approximating functions hn have one common Lipschitz constant L ∈ R+, thederivatives satisfy |(hn)′| ≤ L, for all n ≥ 1. Consequently,limn E|(hn)′(In,t,y,vT )||In,t,y,vT − It,y,vT | ≤ L limn E|In,t,y,vT − It,y,vT |. Next, let τk = T ∧inft ≥ 0 : |ρt| = (1 − 1

k ), for all k ≥ 1. Then the stopped processes In·∧τk convergeto I·∧τk in L2 as n → ∞, see for example Protter [2004, Chapter V.4]. Therefore, bydominated convergence, for every k ≥ 1, we have

limnE∣∣In,t,y,vT − It,y,vT

∣∣≤ lim

n

(E∣∣1τk<T (In,t,y,vT − It,y,vT

)∣∣+ E∣∣1τk≥T (In,t,y,vT − It,y,vT

)∣∣)≤ E

(1τk<T

(∣∣In,t,y,vT

∣∣+ E∣∣It,y,vT

∣∣)) .Recall that (InT ) is uniformly integrable, and that limk P (τk = T ) = 1. Hence, by lettingk →∞ we get that limn E|In,t,y,vT − It,y,vT | = 0, and hence the first summand in (3.25)converges to 0.

In order to show that the second summand in (3.25) vanishes, we first show that,for xn → x, (hn)′(xn) → h′(x). If x is a point of continuity of h′, then we have the

100

3.4. A class of correlation dynamics which fulfill the main assumptions

following estimate

|(hn)′(xn)− h′(x)| =∣∣∣∣∫ ϕn(xn − y)h′(y)dy − h′(x)

∣∣∣∣=∣∣∣∣∫ ϕn(xn − x+ x− y)h′(y)dy − h′(x)

∣∣∣∣=∣∣∣∣∫ ϕn(x− z)h′(z + xn − x)dz − h′(x)

∣∣∣∣ (z := y − xn + x)

≤∣∣∣∣∫ ϕn(x− z)h′(z)dz − h′(x)

∣∣∣∣+∣∣∣∣∫ ϕn(x− z)[h′(z + xn − x)− h′(z)]dz

∣∣∣∣ .Applying the transformation y := n(x − z) in each term on the right hand side of theinequality, together with dominated convergence and the continuity of h′ in x, yieldsthat limn h

′(xn) = h′(x).Since IT has a density, (hn)′(In,t,y,vT ) converges to h′(It,y,vT ) almost everywhere. Con-

sequently, by dominated convergence, we obtain

limnE|It,y,vT ||(hn)′(In,t,y,vT )− h′(It,y,vT )| = 0.

Thus we have shown that limn ∂vψn(t, y, v) = ∂vψ(t, y, v), for all t ∈ [0, T ], y ∈ R and

v ∈ (−1, 1).Notice that In,t,y,vs = yIn,t,1,vs and In,t,y,vs = yIn,t,1,vs , which implies that, for all t ≥ 0

and v ∈ (−1, 1), the sequence (∂vψn(t, ·, v)) converges to ∂vψ(t, ·, v) uniformly in y onall compact sets of R. Similarly, one can show locally uniform convergence in y of thepartial derivatives ∂yψn to ∂yψ.This finally yields that (∂vψn(s, In,t,y,vs , ρt,vs )) converges to ∂vψ(s, It,y,vs , ρt,vs ), and

(∂yψn(s, In,t,y,vs , ρt,vs )) to ∂yψ(s, It,y,vs , ρt,vs ), almost surely. Moreover, by combining thiswith Equation (3.24), we get that Zn converges almost surely to

σ(s, It,y,vs , ρt,vs )∗(∂yψ(s, It,y,vs , ρt,vs )∂vψ(s, It,y,vs , ρt,vs )

). Since Zn converges also to Z in H2, we obtain

the result.

3.4. A class of correlation dynamics which fulfill the mainassumptions

In this part of the work we characterize a class of dynamics which fulfill Conditions(H1) and (H2). One result has already been mentioned as Theorem 3.8 in Section 3.2above. We will give its proof below.Moreover we will give an extension of Theorem 3.8 in Proposition 3.20. We will see

in the Section 3.5, that this extension enables us to show that the so-called Jacobiprocesses also fit into our framework. In contrast to the dynamics given in Theorem 3.8

101


the diffusion coefficient of a Jacobi processes has unbounded derivatives in −1 and 1.We first collect some notation and facts on attainability of boundaries for diffusions.

The material is taken from Karlin and Taylor [1981]. Suppose we are given a generaldiffusion

dXt = µ(Xt)dt+ σ(Xt)dWt, l ≤ X0 ≤ r,

where l (resp. r) denote the left boundary l (resp. the right boundary). In the followingwe only consider the analysis for the left boundary. We define, for x ∈ (l, r),

s(v) = exp

− v∫v0

2µ(w)σ2(w)dw

, v0 ∈ (l, x),

S(x) =x∫

x0

s(v)dv, x0 ∈ (l, x),

S[c, d] = S(d)− S(c), (c, d) ∈ (l, r),S(l, x] = lim

c→lS[c, x].

We already indicate that x0 and v0 will be of no relevance in the following. S is calledthe scale measure whereas M is the speed measure:

m(x) = 1σ2(x)s(x) ,

M [c, d] =d∫c

m(x)dx.

We also need

Σ(l) = limc→l

x∫c

M [v, x]dS(v).

According to Karlin and Taylor [1981] the boundary l is attracting if S(l, x] < ∞ andthis criterion applies independently of x ∈ (l, r). Moreover, the boundary l is said to be

1. attainable if Σ(l) <∞,

2. unattainable if Σ(l) =∞.

For a proof of the following Lemma see Karlin and Taylor [1981, Chapter 15.6].

Lemma 3.18. S(l, x] =∞ implies Σ(l) =∞.

With this at hand, we can find sufficient conditions on the coefficients of the correlationdynamics so that (H1) is satisfied. Let again ρt,v and ρt,v be defined as in (3.5) and(3.7), respectively. To simplify notation, from now on we will suppress the dependenceon t and v and only write ρ resp. ρ.

102

3.4. A class of correlation dynamics which fulfill the main assumptions

Lemma 3.19. Let a and g be continuously differentiable. We assume that g does nothave any roots in (−1, 1). If

lim supx↑1

2a(x)(1− x)g2(x) < 0 and lim inf

x↓−1

2a(x)(1 + x)g2(x) > 0, (3.26)

then Condition (H1) is satisfied.

Proof. We show that ρ does not reach −1. By (3.26), there exist ε > 0, δ > 0 andv0 ∈ (−1, 1), such that 2a(w)

g2(w) ≥ε

1+w , for all −1 < w < v0 < −1 + δ. Hence,

s(v) = exp

v0∫v

2a(w)g2(w)dw

≥ exp

v0∫v

ε

1 + wdw

= C exp(− log(1 + v)) = C1

1 + v.

Consequently,

S[c, x] ≥ Cx∫c

11 + v

dv −→∞,

for c→ −1, and thus by Lemma 3.18 we obtain that ρ does not reach −1. We treat theboundary 1 similar and hence (H1) holds.

The next proposition provides conditions under which Condition (H2) is satisfied. Wewill need two auxiliary processes a and g defined by, for u ∈ [0, T ],

au = 2ρu(1− ρ2

u)g(ρu) + 2g′(ρu), and

gu = 2ρu(1− ρ2

u)a(ρu) + 2a′(ρu) + g2(ρu)1− ρ2

u

+ 4ρ2ug

2(ρu)(1− ρ2

u)2 + (g′(ρu))2 + 4ρu1− ρ2

u

g(ρu)g′(ρu).

Proposition 3.20. Assume the conditions of Lemma 3.19 are satisfied. Then Assump-tion (H1) holds, and therefore, a and g are well defined. Suppose a is bounded and g isbounded from above. Then Assumption (H2) is satisfied, and hence, the delta hedge isgiven as in Theorem 3.4.

Proof. We start by an application of Ito’s formula on the process Φ defined by Φs =f(ρs, ρs), where f is given by f(x, y) = y2

1−x2 . Note that fx(x, y) = 2xy2

(1−x2)2 , fxx =

103


2y2

(1−x2)2 + 8x2y2

(1−x2)3 , fy(x, y) = 2y1−x2 , fyy = 2

1−x2 and fxy = 4xy(1−x2)2 . We have

Φs = Φt +s∫t

2ρuρ2u

(1− ρ2u)2

[a(ρu)du+ g(ρu)dWu

]

+s∫t

2ρu1− ρ2

u

[a′(ρu)ρudu+ g′(ρu)ρudWu

]

+ 12

s∫t

[2ρ2u

(1− ρ2u)2 + 8ρ2

uρ2u

(1− ρ2)3

]g2(ρu)du+ 1

2

s∫t

21− ρ2

u

(g′(ρu))2ρ2udu

+s∫t

4ρuρu(1− ρ2)2 g(ρu)g′(ρu)ρudu

= Φt +s∫t

Φu[

2ρu(1− ρ2

u)g(ρu) + 2g′(ρu)]dWu

+s∫t

Φu[

2ρu(1− ρ2

u)a(ρu) + 2a′(ρu) + g2(ρu)1− ρ2

u

+ 4ρ2ug

2(ρu)(1− ρ2

u)2

+(g′(ρu))2 + 4ρu1− ρ2

u

g(ρu)g′(ρu)]du.

Thus, Φ is the solution of a linear stochastic equation and given by

Φs = ΦtE

s∫t

audWu +s∫t

gudu

.

Hence by our assumptions on a and g, and by Lemma 3.13, all moments of supt≤u≤T Φuare finite, which further yields (H2).

We use the two preceding statements to prove Theorem 3.8.

Proof (of Theorem 3.8). Condition (H1) follows from Lemma 3.19. Since 1 and −1 areroots of g we can write g(x)

1−x2 = 11+x

g(x)−g(1)1−x = −1

1+xg(x)−g(1)x−1 and hence g(x)

1−x2 is boundedfor x 1 by the derivative of g at x = 1. Similarly for x −1, and consequentlythe fraction g(x)

1−x2 is bounded on [−1, 1]. Moreover, Condition (3.26) implies that thereexists an ε ∈ (0, 1), such that all x ∈ (−1, 1) with |x| ≥ 1− ε satisfy xa(x) < 0. Hencea (resp. g) is bounded (resp. bounded from above) and therefore we obtain the resultby Proposition 3.20.

Remark 3.21. 1. Note that the conditions on the coefficients a and g of the correlationdynamics in Theorem 3.8 are more restrictive than in Proposition 3.20. This is mainlyfor ease of exposition in Section 3.2. In Section 3.5.1 an example is given where thecoefficient g of the correlation dynamics does not have a bounded derivative.

104

3.5. Examples

2. It is possible to prove Theorem 3.8 without considering the auxiliary processes a andg and using Proposition 3.20. In the following we give a rough sketch of a more intuitiveproof of Theorem 3.8. That alternative proof consists in showing that all moments ofthe process Yt = 1

1−ρ2tare finite, from which one easily deduces Condition (H2) to be

satisfied. From Ito’s formula we obtain

dYt = 2ρt(1− ρ2

t )g(ρt)YtdWt + 2ρta(ρt)Y 2

t dt+(1 + 3ρ2

t

) g2(ρt)(1− ρ2

t )2Ytdt,

showing that Y is a linear SDE with an additional drift term growing quadratically inY . Condition (3.10) implies that there exists an ε ∈ (0, 1), such that all x ∈ (−1, 1)with |x| ≥ 1 − ε satisfy xa(x) < 0. Moreover, |ρs| ≤ 1 − ε = Ys ≤ 1

2ε−ε2 , andconsequently, the quadratic drift term in the dynamics of Y has a shrinking effect assoon as Y exceeds Cε = 1

2ε−ε2 . In other words, Y can be shown to be dominated bythe SDE

dYt = 2ρt(1− ρ2

t )g(ρt)YtdWt +

(1 + 3ρ2

t

) g2(ρt)(1− ρ2

t )2 Ytdt+ Cεdt,

that, by standard arguments, can be shown to possess finite moments.

3.5. ExamplesThe aim of this final section is to give some explicit correlation dynamics which fallwithin the framework above. We start by modelling correlation processes directly assolutions of various SDEs with values in [−1, 1] in Subsection 3.5.1. Another approachis used in Subsection 3.5.2 where we use mappings of an Ornstein-Uhlenbeck processonto the open intervall (−1, 1).

3.5.1. Modelling correlation directlyExample 3.22. Of course all processes that are bounded away from −1 and 1 fulfillthe Conditions (H1) and (H2). ♦

Example 3.23. For a(x) = κ(θ − x), with κ > 0, θ ∈ (−1, 1), and g(x) = α(1− x2) inthe dynamics of ρ, the prerequisites of Theorem 3.8 are fulfilled. ♦

Example 3.24. Let a and g be polynomials. Assume that g(−1) = g(1) = 0, and thatg does not have any roots in (−1, 1). If

limx↑1

a(x)g2(x) = −∞ and lim

x↓−1

a(x)g2(x) = +∞,

then the prerequisites of Theorem 3.8 are satisfied. ♦

The common denominator of the precceding two examples is that the coefficientsin the dynamics of ρ fulfill the prerequisites of Theorem 3.8, which includes bounded

105


derivatives. We now want to give an example where g does not have bounded derivativesin −1 and 1. We consider so-called Jacobi processes, which are given by the solution of

dρt = κ(θ − ρt)dt+ α√

1− ρ2t Wt. (3.27)

Jacobi processes might be of interest for modelling stochastic correlation, because theirstationary and transitional densities are well known and can be obtained quite explicit,see for example Boortz [2008].By exploiting the boundary theory at the beginning of Section 3.4 or by checking

when Condition (3.10) holds one can easily show that for κ, α > 0 and θ such that

κ ≥ α2

1± θ , (3.28)

the boundaries −1 and 1 of the process defined in (3.27) are unattainable. Hence, wehave that Assumption (H1) is fulfilled. We want to apply Proposition 3.20 and thereforehave to check the boundedness of a and upper boundedness of g. Note that g turns into

gu = 2ρu(1− ρ2

u)a(ρu) + 2a′(ρu) + α2

1− ρ2u

= 2ρu(1− ρ2

u) (κ(θ − ρu))− 2κ+ α2

1− ρ2u

,

and au = 0. In order to ensure upper boundedness of gu it is sufficient to show theexistence of an ε > 0 such that 2ρu(κ(θ − ρu)) + α2 < 0, for all |ρu| > 1− ε, P -almostsurely. This is guaranteed by choosing κ, α > 0 and θ such that the constants ρ(1) andρ(2) defined by

ρ(1),(2) = θ

2 ±√θ2

4 + α2

2κ,

fulfill−1 < ρ(1) ≤ ρ(2) < 1. (3.29)

Note, for example, that for α = 1 and θ = 0.9 Condition (3.28) is satisfied by κ = 10and that this choice of parameters also fulfills (3.29).

3.5.2. Modelling correlation with Ornstein-Uhlenbeck processes

In the previous section we assumed that the stochastic correlation process is describedin terms of the SDE (3.2). The correlation dynamics need not to be modelled directly.Alternatively, one can use a continuous bijection b : (−1, 1) → R, and model at firstplace the transformed process b(ρt) as an SDE. This has the advantage that b(ρt) canbe modelled as a diffusion on R with Lipschitz coefficients. The correlation may bemodelled as a standard mean reverting process, for example an Ornstein-Uhlenbeckprocess, the dynamics of which can be calibrated via standard methods.In this section we will discuss this alternative approach of modelling correlation. As a

paradigma example we will choose as bijection b(x) = x√1−x2 , and we will assume that

106

3.5. Examples

Ut = b(ρt) is an Ornstein-Uhlenbeck process with dynamics

dUt = a(θ − Ut)dt+ σUd(γdW 1t + δdW 2

t +√

1− γ2 − δ2dW 3t ),

where a > 0, θ ∈ R, σU > 0 and γ, δ ∈ (−1, 1) are such that γ2 + δ2 ≤ 1. Noticethat ρt = Ut√

1+U2t

. We will prove that the prerequisites of Theorem 3.4 are satisfied andhence that the local risk minimization strategy is defined as in (3.9).

Lemma 3.25. The correlation process ρt satisfies Conditions (H1) and (H2) and henceTheorem 3.4 holds in this setting.

Proof. The proof is a simple application of Ito’s formula. The first and the secondderivative of b−1 : R →] − 1, 1[, x 7→ x√

1+x2 are given by (b−1(x))′ = (1 + x2)− 32 and

(b−1(x))′′ = −3x(1 + x2)− 52 . We set Wt = γW 1

t + δW 2t +

√1− γ2 − δ2W 3

t . We obtain

dρt = (1− ρ2t )

32σUdWt + (1− ρ2

t )(aθ(1− ρ2

t )12 − aρt −

32ρt(1− ρ

2t )σ2

U

)dt.

It is straightforward to show that the coefficients of this SDE satisfy the conditions ofTheorem 3.8. Hence the result.

107

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List of Figures0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 732.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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List of Tables

2.1. Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

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