University of Amsterdam

MSc Stochastics and Financial Mathematics

Master Thesis

Optimal Hedging in StochasticBlack-Box Models

Author:Wouter van Krieken

Supervisors:Dr. K.E. Bouwman (Cardano)Dr. ir. R. Lord (Cardano)Dr. P.J.C. Spreij (UvA)

Second reader:Dr. A.J. van Es (UvA)

August 23, 2015

Abstract

Derivatives are, in general, hard to include in scenario sets that are generated by a com-plex or unknown stochastic model. We investigate whether the Hedged Monte Carlo(HMC) method can be used to price and hedge derivatives in scenario sets without us-ing specific knowledge about the scenario generating model. We provide a mathematicalframework and justification for the use of the method, to determine risk-neutral pricesand min-variance hedges. Thereby there is made use of a new notion, ‘self-financing inexpectation’. HMC is analyzed in the Black-Scholes and the Heston model using differ-ent derivatives. Moreover, methodology to analyse and improve the method’s accuracyin determining prices and hedges is developed. These analyses focus on the basis func-tions the method uses for regressions. Furthermore, variations on HMC are proposed. Adirect minimization which restricts to self-financing portfolios is developed. This enablesthe use of time-dependent basis functions, which can result in more accurate optimalhedges. In addition, adjustments for the inclusion of stochastic interest rates in HMC areproposed. Features and limitations of the developed methodology are demonstrated inthe KNW model, which is used for pension regulations by the Dutch central bank (DNB).

Keywords: Hedged Monte Carlo, scenario sets, min-variance hedging, KNW model

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Preface

For the final research of my Master ‘Stochastics and Financial Mathematics’, I wantedto do an internship in the financial industry. I was curious to experience how the thingsI learned during my study would relate to practice. Cardano Rotterdam gave me thegreat opportunity of working in an organisation for which financial mathematics is dailypractice. For six months, I did research for this thesis as an intern in the QuantitativeAnalytics (QA) team.

I found the people of the QA team a very enthusiastic and competent group of ex-perts. I appreciate that all the ‘quants’ took the effort to teach me various lessons aboutquantitative analytics, pension funds and programming. The interesting discussions andlaughs I had with them contributed to the amazing time I had at Cardano. I would espe-cially like thank my supervisors at Cardano, dr. ir. Roger Lord and dr. Kees Bouwman.They where always full of ideas and able to push me in the right directions when needed.I would also like to thank my supervisor from the UvA, dr. Peter Spreij, who was a greathelp for my thesis as well. Finally, I would like to thank my fellow student Kay Bogerd,who read along during the process and gave a lot of useful feedback.

Wouter van KriekenHaarlem, August 1, 2015

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Contents

1 Introduction 6

2 Risk-neutral pricing and risk 82.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Risk-neutral measure and completeness . . . . . . . . . . . . . . . . . . . 102.3 Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Heston model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Hedging 183.1 Optimal hedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Continuous hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Static hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Discrete hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 The Hedged Monte Carlo (HMC) Method 254.1 Introduction to the method . . . . . . . . . . . . . . . . . . . . . . . . . 264.2 Discussion of the HMC method . . . . . . . . . . . . . . . . . . . . . . . 284.3 Results in the Black-Scholes model . . . . . . . . . . . . . . . . . . . . . 334.4 Results in the Heston model . . . . . . . . . . . . . . . . . . . . . . . . . 414.5 Goodness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.7 Computational time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5 Variations on the HMC method 555.1 Direct minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.2 Stochastic interest rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

6 Hedging in DNB scenario sets 636.1 Introduction of the KNW model . . . . . . . . . . . . . . . . . . . . . . . 636.2 Simulating the KNW model . . . . . . . . . . . . . . . . . . . . . . . . . 646.3 Applying HMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Conclusion 75

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A Descriptions of financial instruments 78A.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78A.2 Interest rate instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

B Three well-known risk-neutral derivative pricing methods 80B.1 Monte Carlo (MC) Method . . . . . . . . . . . . . . . . . . . . . . . . . 80B.2 Least Squares Method (LSM) . . . . . . . . . . . . . . . . . . . . . . . . 83B.3 Finite Difference (FD) Method . . . . . . . . . . . . . . . . . . . . . . . . 84

C Basis functions 86

D Option pricing in the Heston model 88

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Chapter 1

Introduction

Scenario-based studies are widely used in the financial industry. These studies consistof a (great) amount of, often stochastically generated, trajectories of relevant economicvariables. Although these scenario sets can have several purposes, for example pricing ofderivatives, they serve mainly as a financial risk management tool. The future profit andloss (P&L) of a certain portfolio, liability, balance sheet, or derivative can be tracked.In particular, asset and liability management (ALM) studies heavily rely on scenariosets. ALM is used by pension funds, banks and insurance companies, in order to studythe balance sheet’s risk. These studies are used to determine consequences of strategicdecisions or the impact of certain scenarios. In addition, if a probabilistic model is used,probabilities can be assigned to certain events.

Obviously, whatever a scenario set’s purpose is, the way scenarios are chosen is crucial.Often a stochastic model is used, which will naturally be the starting point of this thesis.In order to have realistic scenarios, these models tend to become increasingly complex. Adirect consequence of this is that it is not straightforward how to price financial deriva-tives in a scenario set. Even if pricing formulas are available, including derivatives in ascenario set can become problematic, as den Iseger and Potters pointed out [1]. This isbecause the scenario model and the model used to price the derivative are often incon-sistent, for two reasons. First, pricing must be done under a risk neutral measure, whilescenario sets are often generated under the ‘real’ measure. Second, the scenario model’seconomic assumptions can be different than the assumptions underlying the pricing for-mulas. Typically, scenario sets are based on certain views on the future developmentof the market, while these views are not necessary consistent with how instruments arepriced in the market.

Creating realistic and useful scenario sets is a whole study itself, but this will not bethe point of interest in this thesis. Different parties are specialized in developing scenariostudies, mainly for ALM purposes. They supply scenario sets for external risk managers.This introduces two challenges for users who want to perform extensive analysis on ascenario set. First, the scenario generating model is often complex as various economicviews are included. Second, the model might not be fully known to the user, for whichthe model is then virtually a ‘black box’. Therefore, pricing derivatives in a scenariocontext is often non-trivial.

In view of the no-arbitrage principle (Chapter 2), pricing of financial instruments

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goes hand in hand with hedging . A hedge of a derivative, is a portfolio that matches thefuture’s derivative’s payoff as accurate as possible. Besides that hedge portfolios can beused to determine prices, they have practical uses itself as well. Hedges are used to reducea portfolio’s risk. However, like pricing in a black-box or complex setting, constructinghedge portfolios is far from trivial. This motivates to developing a widely applicablenumerical method to determine prices and hedges of derivatives in a scenario set, whichwill be this thesis’ main objective. Potters, Bouchaud and Sestovic’s Hedged Monte Carlo(HMC ) method will be used as a starting point to tackle the above problem of includingderivatives in scenario sets [2].

Various numerical methods are developed for pricing (e.g. Monte Carlo and FiniteDifference) and for hedging (e.g. Dynamic Programming and Optimal control). Wesee great potential in the HMC method, since both prices and hedges are computed.Moreover, the method relies on scenario sets, which allows us to apply the method in thesetting described above, where the model is complex or unknown.

This thesis’ contribution to research on HMC is threefold. First, since HMC is arelatively new method, this thesis contributes by presenting results in different set-upsand proposing methods how to track and improve the method’s performance. Second, amathematical justification of the method is presented, which can not be found in litera-ture. Third, some variations on the method are introduced, which enables to apply themethod to a larger set of problems.

A possible application of the research done for this thesis lies in the research of pen-sion funds. Due the recent financial crisis, pension funds’ financial positions worsened.In addition, changes in society, such as public health, raise a demand for pension schemeswhich are sustainable in the uncertain future. Therefore, there are discussions within thepension fund sector, regulators and politics about how pension designs should be restruc-tured. One of the main topics of debate is how and whether risk should be shared betweendifferent pension participants. We think the methodology developed in this thesis canbe used to compare different pension solutions, by replicating the pension outcomes withsome relevant assets. In particular, replicating portfolios could be used to determine towhat extend an individual bank account can achieve the same results as a more complexpension scheme in which risk is shared within different groups of participants.

Chapter 2 and 3 introduce the reader to the basic principles of pricing and hedging anddevelop the mathematical framework used in the thesis. The Hedged Monte Carlo (HMC)method is introduced and analyzed in detail in Chapter 4. Finally, some variations onthe HMC method are proposed in Chapter 5 and the methodology is applied on an ALMmodel in Chapter 6.

We expect the reader to be familiar with financial derivatives. In Appendix A thepayoffs of the instruments used in this thesis are described. In Appendix B three well-known derivative pricing methods are discussed, namely the Monte Carlo (MC) method,the Least Squares Method (LSM) and the Finite Difference (FD) method. The HMCmethod has similarities with these methods so it is convenient for the reader to understandthese methods. Appendix C discusses the basis functions used in this thesis and AppendixD describes how options are priced in the Heston model in this thesis.

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Chapter 2

Risk-neutral pricing and risk

Innumerable methods have been developed in order to determine prices of financial instru-ments. Some well-known methods are discussed in Appendix B. Although the techniquesand underlying models tend to have great differences, almost all pricing methods arebased on the fundamental principles of asset pricing. Probably the most important as-sumption is the postulate of an arbitrage-free market. As we will see in Section 2.2 thisis equivalent to the existence of so-called risk-neutral measure.

In this chapter some definitions, axioms and notations are introduced in order to builda mathematical framework, which enables us to speak of prices and hedges of financialinstruments. These principles form the basis of modern portfolio theory and will be usedthroughout this thesis. We illustrate arbitrage-free pricing in the Black-Scholes model ,which is certainly the most well-known pricing model based on the discussed principle [3].That the arbitrage principles are not sufficient to determine prices in every model, wewill show by introducing another well-known model, namely the Heston model . Fordetermining prices and hedges in such models, the notion of risk becomes important,which will be the final topic in this chapter. Risk plays a fundamental role in this thesis,since it will enable us to compare hedge portfolios.

2.1 Set-up

We use a set-up based on Follmer and Schied [4], which is standard in the context ofstochastic portfolio theory. Let (Ω,F ,P) be the probability space, where we consider Pas the ‘real world’ measure. We consider a market consisting of a + 1 assets which canbe traded at times t ≥ 0. The random vector St = (S0

t , . . . , Sat ) denotes the prices of the

assets at a certain time t, from which we will assume that the time instants 0 = t0, t1, t2, . . .are discrete. We will often abbreviate Sti with Si, as it is in general clear what is meantby a time index. Although in practice it is not possible to trade at infinitely small timeinstants, a few times in this thesis we will consider a continuous time line R≥0, whichwill enable us to elaborate on certain concepts of asset pricing and hedging. At time t,all prices up to t are known. In mathematical terms this means that St is adapted to acertain filtration (Ft) ⊂ F . Opposed to the random assets S1, . . . , Sn, the zeroth assetwill be considered as risk free (a discount factor), so S0 will be assumed to be completelydeterministic or, more general, predictable with respect to the filtration. We write S and

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S when, respectively, including and excluding the zeroth risk-free asset, so S = (S0, S).Additionally, if the opposite is not stated, we will often assume that S has the Markovproperty under P. This means that the distribution of the future state only depends onthe current state.

A portfolio process ξ = (ξ0, . . . , ξn) is an adapted process that denotes weights foreach asset. The value (price) of a portfolio is given by the adapted process Wt = ξt · St,which we will refer to as the wealth process . In this thesis, ‘·’ denotes the inner product.A portfolio process is self-financing (SF ) if there are no external cash flows. In the dis-crete case this means that ξi · Si = ξi−1 · Si. In the continuous case this is expressed asd(ξt · St) = ξt · dSt.

Note that this set-up includes no bid-ask spread, so it implies a perfectly liquid andefficient market. In addition, it is assumed that any portfolio can be achieved by aninvestor and this does not influence the prices. We will make use of the notion of arbitrage.This means that it is not possible to have a portfolio that earns more than the risk-freeasset, without taking risk:

Definition 2.1. A self-financing portfolio process ξ is an arbitrage opportunity if for afixed time T we have

P(WT

S0T

>W0

S00

)> 0 and P

(WT

S0T

≥ W0

S00

)= 1.

I.e., the discounted profit is non-negative and positive with positive probability. Here, Wdenotes the portfolio’s wealth process, Wt = ξt · St

A market with no arbitrage opportunities is called an arbitrage free market .

For a fixed time T , a contingent claim CT is an FT -measurable random variable,representing an uncertain cash flow at time T . A contingent claim is called a derivativeif it only depends on the stock prices up to time T , so if it is σ((St)t≤T )-measurable.All derivatives and other financial instruments used in this thesis are described in Ap-pendix A. A contingent claim is called attainable if there exists a self-financing portfolioprocess such that CT = WT a.s. In this thesis, we call such a portfolio a perfect hedge.We define subhedge and a superhedge as a self-financing portfolio process such that, re-spectively, WT ≤ CT a.s. and WT ≥ CT a.s.

It is in general not clear what a ‘fair price’ of a contingent claim is. Clearly, a minimumrequirement of this notion would be that it is an arbitrage free price, i.e. adding the claimto the market would not result in an arbitrage opportunity. The following propositionshows that this definition is in essence a restricting of prices in terms of hedges:

Proposition 2.2. Suppose that the market is arbitrage free. Let C be a contingent claim.

1. C is attainable if and only if C has a unique arbitrage free price.

2. If C is not attainable, then its arbitrage free prices are given by the following well-defined open interval:

(maxξ0 · S0 : ξ is a subhedge,minξ0 · S0 : ξ is a superhedge)

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Proof. (Sketch) Note that the first assertion is a (almost) direct consequence of the sec-ond. Suppose that C is not attainable. It can be shown that the set of arbitrage freeprices is convex. Denote by I the open interval above, which can be proved to be well-defined. Let ξ′ be a subhedge. Since C is not attainable and by definition of a subhedge,W ′T > CT with positive probability. Therefore, adding C with price ξ′0 · S0 would in-

troduce an arbitrage opportunity. Similar reasoning holds for superhedges, so I at leastcontains all arbitrage free prices. Now let p ∈ I. Then any portfolio ξ′ with price p isneither a sub- or super hedge. Construct ξ′′ by adding a short position of C to ξ′. Notethat ξ′′ has price zero, and since ξ′ is neither a sub- or super hedge, ξ′′ is not an arbi-trage opportunity. So there can not be constructed an arbitrage opportunity by havinga portfolio holding C with price p. Therefore, the interval I contains only arbitrage-freeprices, which completes the proof.

Chapter 3 elaborates on hedging. There it becomes clear that the above concept ofarbitrage free pricing is not sufficient, since in general for unattainable contingent claimsthe interval of arbitrage free prices is large. In Section 2.5, some basics of risks areintroduced, which will enable us to further specify the concept of a fair price.

2.2 Risk-neutral measure and completeness

It is convenient to define prices as the discounted expectation of the value of a contingentclaim. The real world measure P is, however, not suitable for this purpose. This is becauserisks can be priced differently in the market. Prices can, hence, not simply be computedby considering only the assets’ expected return. Therefore, adjusted the measures areused such that prices are discounted conditional expectations:

Definition 2.3. A probability measure Q on (P,F) is a risk-neutral measure (also calledmartingale measure) if

EQs [St]

S0t

=SsS0s

, Q-a.s.,∀s ≤ t. (2.1)

Here EQs denotes the expectation with respect to Q conditional on Fs. In most literature,

the set of risk-neutral measures is restricted to the measure which are equivalent to P.This means that Q and P are absolutely continuous with respect to each other, denotedby Q ∼ P.

Intuitively, a risk-neutral measure is thus a ‘probability measure conform the marketprices’. The use of a risk-neutral measure can be motivated by the fact that it indeedexists in an arbitrage-free discrete market, which is stated in the following importanttheorem:

Theorem 2.4. The First Fundamental Theorem of Asset Pricing: A market isarbitrage free if, and only if, there exists a risk-neutral measure that is equivalent to themeasure P.

We omit the proof of this theorem as well as those of the following two, as they can befound in any introduction to modern portfolio theory (e.g. [4]). The second fundamentaltheorem characterizes a complete market . That is, a market in which every contingentclaim is attainable.

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Theorem 2.5. The Second Fundamental Theorem of Asset Pricing: An arbitrage-free market is complete if, and only if, there exists a unique risk-neutral measure that isequivalent to the measure P.

In the literature, often the above two fundamental theorems are extended by char-acterizing markets as the amount of atoms in the probability space. An atom is a setA ∈ F such that any measurable subset of A has probability zero or equal probability asA. One can show that the amount of atoms is limited in a discrete complete market:

Theorem 2.6. Let d denote the amount of atoms in which (Ω,F ,P) can be divided.Consider a market where (a + 1) stocks S take values on T + 1 discrete time instants.Then the market is complete if and only if d ≤ (a+ 1)T .

This indicates that the setting of a complete discrete market is relatively simple andunrealistic. In this case, all possible asset paths can be represented in a binomial tree.Therefore, we will typically not consider complete discrete markets in this thesis. Forcontinuous markets, however, the notion of completeness is more relevant. In the con-tinuous case, a similar result can be obtained. Loosely, we can state that a continuousmarket is complete if, and only if, the amount of tradeable assets is greater or equal thanthe amount of risk drivers [5]. We will often consider a model which describes a completecontinuous market, but restrict hedge portfolios to re-balance on discrete time instants.

2.3 Black-Scholes model

Black and Scholes introduced a method to price options, using the concepts of arbitragefree pricing [3]. Since then, this method has become commonly known as the Black-Scholes model or the Black-Scholes-Merton model, due to R.C. Merton’s contributionsto this pricing method. Although the model, due to its assumptions that are in practicetoo simplistic, is often not directly applicable, the model is still important nowadays asthe principles of the pricing method are widely used in various financial models. Further-more, as the model is relatively simple, the pricing formulas can be used as a benchmarkfor more complex models, which is exactly the way this thesis uses the Black-Scholesmodel as well.

In the Black-Scholes model, a stock is modelled which returns are normally dis-tributed. Hence, the asset price process S follows a geometric Brownian Motion:

St = S0 exp

((µ− σ2

2)t+ σWt

).

Here Wt denotes a standard Brownian Motion and the constant parameters µ and σdenote, respectively, the drift and volatility of the stock price. The interest rate r isassumed to be constant as well. In addition, it is possible to have a hedge portfolio thatcan be re-balanced continuously.

Let Vt be the time t value of an option that matures at time T . Note that VT equalsthe option’s payoff. Black and Scholes derived a formula for the price of an option for

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t ≤ T . There are various ways to derive this formula. We follow the method using aStochastic Differential Equation (SDE). A portfolio is constructed by having ∆t stocksat time t, so the value Πt of the portfolio is given by

Πt = Vt −∆tSt.

We assume a self-financing portfolio, so d(∆tSt) = ∆tdSt. Now by Ito’s lemma we have

dΠt =∂Vt∂t

dt+∂Vt∂St

dSt +σ2

2S2∂

2V

∂S2dt−∆tdSt.

The hedge weight ∆t is chosen such that the above is completely deterministic. Thus∆t = ∂Vt

∂St, such that the dSt part of the above equation vanishes. Since the above portfolio

is now risk-less and we assume no-arbitrage, the expected return of Π must equal the risk-free rate r. So we have, dΠt = rΠt, which leads to the famous Black-Scholes equation:

∂V

∂t+ rS

∂V

∂S+σ2

2S2∂

2V

∂S2− rV = 0. (2.2)

This is a partial differential equation with a boundary condition at t = T , at which thevalue of the option is known. Solving this partial differential equation for a call-optionwith strike K, yields the Black-Scholes formula:

Vt = SΦ(d1)−Ke−r(T−t)Φ(d2). (2.3)

Here Φ denotes the standard cumulative normal distribution and

d1,2 =ln(S/K)± (r + σ2/2)(T − t)

σ√T − t

. (2.4)

Remark 2.7. In the above approach, the hedge weight on the risk-free asset is notdirectly used, as only the hedge weight on the risky asset is computed. The process Π isnot as a the wealth process W introduced in Section 2.1. However, a perfect hedge canbe computed by investing in the bank account process S0

t = B(t) = ert as well.

A typical result of this arbitrage-free method of pricing, is that the option price isindependent of the drift parameter of the stock. However, we will see in Chapter 3 thatthis does not hold when the assumption of continuous hedging is dropped. Furthermore,the assumption of constant drift, volatility and interest rate in practice does not hold inthe financial markets. More complex models are, therefore, developed in order to pricederivatives. For more general stock process models and derivatives there do not existpricing formulas in closed form. Therefore, various numerical methods are developed toprice derivatives, from which some of the well-known are discussed in Appendix B.

2.4 Heston model

As we would like to consider incomplete models in this thesis as well, we now introduceanother well-known model for a stock price. In the Black-Scholes model, the stock’svolatility is constant, leading to normal distributed stock returns. Heston generalized

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this model, introducing the following PDE for a stock price S and stochastic (‘local’)volatility

√v [6]:

dS(t) = µSdt+√v(t)SdW1(t),

dv(t) = k(θ − v(t))dt+ σ√v(t)dW2(t).

This model, referred to as the Heston Model , is an example of a Stochastic VolatilityModel . In this model, the stock’s volatility is square root mean reverting, with mean θand rate of reversion k. The parameter σ denotes the ‘volatility of the volatility’ and W1

and W2 are Brownian Motions with constant correlation ρ. The latter means that theBrownian motions have quadratic covariation 〈W 1,W 2〉t = ρ.

Remark 2.8. In contrast to the Black-Scholes model, the Heston Model can not beexactly simulated by a simple Euler discretization. The latter would even result in apositive probability of a negative volatility, which is not possible in the continuous version.This problem can be overcome by taking taking v′(t) = max(v(t), 0) at each time instant,but then the dynamics still do not follow the Heston Model perfectly. Various researchis done on efficient and accurate discretization of the Heston model [7–9]. In this paperwe will use Lord et al’s full truncation scheme in order to exclude the possibility of anegative volatility without losing to much accuracy [7]. Furthermore, simulation is doneunder smaller time instants than the hedging is performed, such that the error due Eulerdiscretization is reduced.

2.4.1 Incompleteness

Typically, stochastic volatility models, like the Heston model, are not complete as thevolatility is not a trade-able asset. The stochastic volatility can be seen as an extra riskdriver, which can not be hedged. If a volatility dependent asset is in the market, however,the market can become complete. Romano and Touzi showed, for example, for a largeclass of stochastic volatility models including Heston, that the market can be ‘completed’by adding any (but just one) European option [10] on the stock.

Thus, if the market contains no volatility dependent assets, the Heston model isincomplete and prices of contingent claims in the Heston model are not always unique.In order to price options, therefore, extra assumptions need to be made how assets withvolatility risk are valued. This can be done, for example, by considering option pricesin the market. In the following, we will assume that the Heston parameters can thenbe chosen in such a way, that the Heston dynamics are risk-neutral. This implies thatthe stock’s drift equals the risk-free rate and derivatives can be priced by taking thediscounted expectation.

2.4.2 Option pricing in the Heston model

If we assume the Heston model’s parameters to be chosen such that the dynamics are ina certain risk-neutral measure Q, pricing can be done, having a constant interest rate r,by

C0 = e−rTEQCT .

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Figure 2.9: The price and implied volatility as a function of call option’s strike on astock with Heston dynamics, with initial stock price 100, long-term and initial volatilityθ, v(0) = 30%, interest rate r = 5%, revert rate k = 2, correlation ρ = −0.3. Differentvalues of the parameter σ are used, 0.0 (blue), 0.5 (red), 1.0 (yellow), 1.5 (purple) and2.0 (green).

It is, however, not trivial how to compute ECT . For the simulations in this thesis we use acommon approach which makes use of Fourier Inversion, a pricing method introduced inStein and Stein [11] and Heston [6]. This pricing method is briefly discussed in AppendixD.

The distribution of stock returns in the Heston model is well-known to have, generally,‘fatter tails’ than the normally distributed stock returns in the Black-Scholes model. Thedifference of both models can be illustrated by plotting an option’s implied volatility fordifferent strikes, like in figure 2.9. The implied volatility is the unique volatility suchthat the Black-Scholes formula is consistent with a certain option price. Clearly this is aconstant for any strike in the Black-Scholes model. In the Heston model, however, impliedvolatilities for different strikes typically show a convex shape, known as the volatilitysmile. This behavior turns out to be in line with observations in the market, whichis probably the main motivation for use of the Heston model. Viewing figure 2.9, thereader should be warned though, that a different parameter set-up can lead to qualitativedifferent results.

Remark 2.10. Although implied volatility is a widely-used notion, the computation ofthe above smile may not be straightforward to the reader. For those readers, we clarify

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what is done for computing Figure 2.9 in a chronological order:

1. Certain Heston model parameters are chosen and several option strikes are chosen.

2. The Heston option prices are calculated using Equation (D.1).

3. Using these prices, the implied volatility are computed such that they match theprices in the Black-Scholes formula.

Thus, from parameters we convert to option prices, using Heston, and than back tovolatilities, using Black-Scholes.

Although the Black-Scholes model is not accepted as a correct pricing model, it isused a lot to express market option prices in volatilities.

The Heston model is a good example of a model in which perfect hedges not alwaysexist. We are, however, interested in computing portfolios that replicate contingent claimsas well as possible. To be able to compare portfolios in their ability to replicate payoffs,we will need the notion of risk , which will be introduced in the following section.

2.5 Risk

Risk is an abstract concept as there can be various drivers and the word can be referringto different concepts. As far as this thesis is concerned, our focus will be to have aquantitative and suitable measure of the risks of contingent claims. Our main motivationfor being able to quantify risk is that we would like to rate hedge portfolios in the abilityto reduce the risk of a certain contingent claim. Artzner, Delbaen, Eber and Heathintroduced a widely accepted definition of a risk measure [12]. Artzner et al. proposedaxioms for measures on random variables in L2. The latter space consist of all the randomvariables for which the expectation and variance exist and are finite, which seems to bea natural requirement.

Definition 2.11. For a positive integer n, the space Ln consists of all random variablesX in (Ω,F ,P) with E|X|n <∞.

Since hedging is about minimizing the difference in the outcome of a contingent claimand a hedge portfolio, it is convenient to use a deviation risk measure, introduced byRockafellar, Uryasev and Zabarankin [13]. This measure is similar to Artzner et al.’s riskmeasures, but reflects more a random variable’s uncertainty. The axioms of a deviationrisk measure are given by:

Definition 2.12. A deviation risk measure is a function R : L2 → [0,∞) satisfying thefollowing properties, for any X, Y ∈ L2:

1. Shift-invariant: R(X + a) = R(X), for all a ∈ R.

2. Positive homogeneous: R(λX) = λR(X) for all λ > 0.

3. Sublinearity: R(X + Y ) ≤ R(X) +R(Y ).

4. Positivity: R(X) = 0 if and only if X is a constant.

15

The first and the fourth axioms expose that deviation risk measure focuses on un-certainty. The second axiom can be interpreted, loosely speaking, as ‘risk is probabilitytimes impact’. The third axiom makes sure that no extra risk is introduced when combin-ing multiple portfolios. A fifth axiom can be imposed, in which case we have a symmetricdeviation risk measure:

5. Symmetry: R(X) = R(−X).

A symmetric deviation risk measure can also be obtained by replacing the second axiomby R(λX) = |λ|R(X) for any λ ∈ R.

Remark 2.13. Artzner et al’s risk measure takes values in (−∞,∞] and is obtainedslightly different. In addition, they introduce the notion of a coherent risk measure,which will not be discussed in this thesis. Rockafellar et al. showed that there is a oneto one relationship for risk measures ρ and deviation risk measures R:

ρ(X) = R(X)− E(X),

R(X) = ρ (X − E(X)) .

The choice of use of a risk measure or a deviation risk measure is thus simply a matterof convenience, as they can easily be exchanged with each other. Since this thesis onlymakes uses of the above type of risk measure (R), the word ‘deviation’ will sometimes beomitted.

Example 2.14. For any a ∈ (0, 1), the well-known Value at Risk (VaR) is given by:

VaRa(X) = infλ ∈ R : P(X > λ) ≤ 1− a.

One might want to define a deviation risk measure by:

RVaRa (X) = VaRa(X − EX).

However, the third (sublinearity) and fourth (positivity) axiom of a deviation risk measureare not necessarily satisfied [13]. Since these axioms seem to be a natural requirement ofa risk measure, the VaR risk measure is not used in this thesis.

The standard deviation is the obvious example of a symmetric risk measure, and it iswidely used as well. Besides that it is easy to compute, one can justify the use with theCentral Limit Theorem, by composing the outcome of a contingent claim CT into a largesum of independent random variables [14]. It is questionable, however, whether such anapproach is justifiable. Certainly not all stock returns are exactly normally distributed.Furthermore, it is not clear whether a risk measure should indeed be symmetric, as a riskaverse trader might be more interested in the amount which can be lost than the amountthat can be won. If however, there is no information available whether a contingent claimshould be considered as a long of short position, it makes sense to use a symmetric riskmeasure (see [15, §2.5] for a discussion in more detail).

Example 2.15. Suppose that one constructs a portfolio holding w and 1− w of claimsX and Y with correlation ρ, with w ∈ [0, 1]. Then the square of the standard deviationrisk measure of the payoff is:

Var(wX + (1− w)Y ) = w2Var(X) + (1− w)2Var(Y ) + 2w(1− w)ρ√

Var(X)Var(Y ).

16

Suppose that ρ > 0 and Var(X) = Var(Y ). Then the risk is minimized if w = 1/2, sothe minimization of the standard deviation risk measure imposes diversification.

The standard deviation strongly relates to a normal distribution. In practice, stockreturn distributions are well-known to have ‘fatter tales’, since there are sometimes eventswhere there is a huge jump in the stock price. The main critique on the standard deviationas a risk measure is, therefore, that huge outliers are not properly taken into account,such that risk is underestimated.

Despite some disadvantages of the standard deviation as a risk measure it is used alot in the literature. This thesis will follow this practice as this risk measure is convenientto use, because of its tractability. It should be stressed though, that in some contextsa different risk measure should be used. When considering risk in the perspective of anindividual entity, for example, risk might be quantified differently.

In an incomplete setting like, for example, the Heston model, it is not directly clearwhat the fair price of an unattainable contingent claim is. Clearly, if a hedge portfolioexist that replicates the claim’s payoff closely, with respect to a certain risk measure, theprice of the contingent claim should be close to the price of the hedge. Various researchis done for generalizing no-arbitrage pricing using the notion of risk. For example, Cernyand Hodges provided a framework, referred to as good-deal-pricing , for ‘tigher-than-no-arbitrage price bounds’ [16].

17

Chapter 3

Hedging

In Chapter 2 we saw, considering arbitrage opportunities, that hedging plays a fundamen-tal role in pricing of financial instruments. A hedge portfolio itself can have a practicaluse as well. A hedge can be performed, for example, by a trader who sold a derivative,in order to reduce the risk of the trade. A second example is a company with uncertainfuture liabilities, which reduces the risk by trading certain correlated assets. Regulatorseven prescribes banks, pension funds and other institutions to be secured for the risk theyare exposed to, which is often done by buying derivatives. Generally, this thesis refersto a hedge as a portfolio that is constructed with the intention to reduce the risk of acertain contingent claim.

First, we will discuss the notion of an optimal hedge, as it will not always be possibleto hedge a contingent claim perfectly. A hedge strategy can be characterized by theamount of re-balancing moments on the time line, which can be done continuously, onlyonce, or finitely many times. Second, each of these cases will be discussed in more detailin this chapter.

3.1 Optimal hedge

Suppose we have a contingent claim CT , which we would like to hedge with a self-financingportfolio process ξ. The objective is to choose ξ such that the accessory wealth WT isclose to CT . Therefore, in order to compare the performance of different hedge portfolios,we need to choose a certain metric on L. Since hedging is about reducing uncertainty,it is natural to normalize the space L to the space L′ with the map X 7→ X − EX,so considering only random variables with zero expectation. Note that the problem offinding an optimal hedge can now be expressed in terms of a deviation risk measure R:

Find a self-financing portfolio process ξ that minimizes R(WT − CT ). (3.1)

In the case R equals the standard deviation risk measure, the resulting portfolio processis referred to as the variance-optimal hedge. Computing this hedge will be our mainconcern in this thesis.

18

Using the self-financing condition, in the case of discrete time instants, we can extractξ0

0 , the amount invested in the risk-free asset at time zero, from WT :

WT = W1 +T−1∑i=1

ξi · (Si+1 − Si) = ξ00S

01 + ξ0 · S1 +

T−1∑i=1

ξi · (Si+1 − Si).

Since ξ00S

01 is deterministic, we see by definition 2.12 of a deviation risk measure, that

the problem of finding an optimal hedge (Equation (3.1)) is independent of ξ00 . In other

words, the initial value of the hedge portfolio W0 can be freely chosen. The most naturalassumption is to have the portfolio’s initial value such that the portfolio’s expected valueat time T equals the derivative’s expected payoff:

EWT = ECT .

In the case of a attainable contingent claim, the above approach leads to a perfect hedge:

Proposition 3.1. Let CT be an attainable contingent claim. Let ξ? be a variance-optimalhedge with respect to a deviation risk measure R, such that EW ?

T = ECT . Then ξ? is aperfect hedge.

Proof. Since CT is attainable, there exists a portfolio process ξ′ such that W ′T = CT a.s.

By the positivity property of a deviant risk measure, we have then R(W ′T − CT ) = 0.

Since ξ? minimizes R(W ?T −CT ), we obtain R(W ?

T −CT ) = 0 as well. Therefore, we haveW ?T −CT = c a.s. for certain c ∈ R. Since EWT = ECT , we obtain c = 0. We see that ξ?

is a perfect hedge, which completes the proof.

Moreover, imposing the condition EWT = ECT , it can be proved that the problem isnow equivalent to the following minimization problem:

Find a self-financing portfolio process ξ that minimizes E(WT − CT )2. (3.2)

So now we are minimizing with respect to the so called mean squared error . That theminimum variance approach is indeed equivalent to the mean squares approach, will bethe content of proposition 3.8. There, the result is proven for the static hedge case, butthe same can easily be done for the case of discrete hedging or continuous hedging.

Remark 3.2. Commonly known concepts are delta hedging and gamma hedging . In thecase of a complete market, like we saw in the Black-Scholes case, the perfect hedge weightof the stock was given by ∆ = ∂V

∂S. Then, any movement in the stock price is compensated

by the hedge portfolio. In the case of an incomplete market, this can obviously not beachieved. In that case, a delta hedge can be defined as the hedge such that is has thesame derivative with respect to S in expectation.

Gamma hedging means that the hedge portfolio is constructed, such that the secondorder derivative ∂2V

∂S2 is neutralized.

19

3.2 Continuous hedging

In Section 2.3 we saw that an option can be hedged perfectly in the Black-Scholes model.Here, the hedge portfolio was constructed by holding ∆ := ∂C

∂Sof the stock (delta hedg-

ing). It can be shown that in the Black-Scholes Model, any simple contingent claim canbe hedged perfectly [5, chap. 7]. In this context, a simple contingent claim is one thatonly depends on a stock’s value at a certain time. The Black-Scholes model is therefore acomplete model. However, not necessarily every model that allows for continuously hedg-ing is complete. For example, the Heston model introduced in Section 2.4 is incompleteif the volatility is a non-tradable risk driver. In addition, in models for which stock pricesdo not have continuous paths, perfect hedging is often impossible. Various research isdone on variance-optimal hedging in continuous time. Refer, for example, to [17, 18].

For the remainder of this thesis, continuous hedging is not considered as it is in practicenot applicable. Some Black-Scholes results will, however, be used as a benchmark for theperformance of introduced pricing and hedging methods.

3.3 Static hedging

In the case of a static hedge, a hedge portfolio is chosen at time t = 0 and remains fixeduntil maturity T of the contingent claim. In the case of a model with one risky asset,the variance-optimal hedge is relatively simple to compute, which is well-known in thecommodity future market as the optimal hedge ratio [19]:

Proposition 3.3. Consider a one time-step model with one risky asset S. Let CT be acontingent claim with Std(CT ) > 0. The variance-optimal hedge ratio is given by:

∆∗ =Cov(ST , CT )

Var(ST )= Corr(ST , CT )

Stdv(CT)

Stdv(ST). (3.3)

Proof. The time T payoff variance of a portfolio holding ∆ risky assets and the contingentclaim −CT is given by

Var(∆ST − CT ) = (∆)2Var(ST ) + Var(CT )− 2∆Cov(ST , CT ).

The above is quadratic in ∆ with minimum at

∆∗ =Cov(ST , CT )

Var(ST ).

This can be generalized to multiple dimensions. Note that, in the case the assets areindependent, the hedge ratios equal to the hedge ratios in the one-dimensional case:

Proposition 3.4. Consider a one time-step model with one risky assets S = (S1, . . . , Sn)T .If S is linearly independent, then variance-optimal hedge ∆∗ is given by

∆∗ = Cov(S,CT )TCov(S)−1.

Here, Cov(S)−1 denotes the inverse of the covariation matrix of S.

20

Proof. By differentiating

Var (∆ · ST − CT ) =n∑i=1

(∆i)2Var(SiT ) + Var(CT )

+ 2n−1∑i=1

∑i<j≤n

∆i∆jCov(Si, Sj)− 2n∑i=1

∆iCov(Si, CT )

with respect to ∆, and setting it equal to zero, we obtain

∆T ∗ Cov(S) = Cov(S,CT ).

Since S is linearly independent, Cov(S) is invertable. The result follows.

The above results show that having the standard deviations and correlations, findinga variance-optimal static hedge is relatively straight forward.

Remark 3.5. In the case the market has only one risky asset S, the variance-optimalhedge ratio ∆ of a claim CT ∈ FT results in the hedge error variance

(1− Corr(ST , CT )2)Var(CT ).

The ‘one-period’ market is thus complete if and only if ST and X are perfectly correlatedfor any X ∈ FT with Std(X) > 0. In this case, any function f(ST ) is perfectly correlatedwith ST , which is only possible if ST can attain only one or two values. This illustratestheorem 2.6, as F can then consist of a maximum of two atoms.

Example 3.6. Suppose we would like to perform a variance-optimal static hedge on acall option CT = (ST −K)+ in the Black-Scholes model. In this case, ST is log-normaldistributed with parameters µ′ = logS0 + (µ − σ2/2)T and σ′ = σ

√T . For determining

∆∗ we need to compute:

Cov(ST , CT ) = E[STCT ]− E[ST ]E[CT ].

Note that E[ST ] = S0 exp(µT ) and E[CT ] can computed by using the Black-Scholesformula with µ as the interest rate constant:

E[ST ]E[CT ] = S0eµT eµT

(S0Φ(dµ(0))−Ke−µTΦ(dµ(−1))

)= S2

0e2µTΦ(dµ(0))−KS0e

µTΦ(dµ(−1)),

where dµ(n) is given by

dµ(n) =log(S0/K) + (µ+ σ2/2)T

σ√T

+ nσ√T .

Now we compute, using the fact that S2T is log-normal distributed with parameters 2µ′

21

and 2σ′:

E[STCT ] = E[ST (ST −K) 1ST>K ]

= E[S2T1S2

T>K2 ]−KE[ST1ST>K ]

= e2µ′+(2σ′)2/2Φ

(− logK + µ′ + 2σ′2

σ′

)−Keµ′+σ′2/2Φ

(− logK + µ′ + σ′2

σ′

)= S2

0e(2µ+σ2)TΦ

(log(S0

K) + (µ+ 3σ2/2)T

σ√T

)−KS0e

µTΦ

(log(S0

K) + (µ+ σ2/2)T

σ√T

)= S2

0e(2µ+σ2)TΦ(dµ(1))−KS0e

µTΦ(dµ(0)).

Using Var(ST ) =(eσ

2T − 1)S2

0e2µT , we can now conclude that the variance-optimal

hedge ratio is given by

∆∗ =1

eσ2T − 1

[eσ

2TΦ(dµ(1))− K

S0

e−µT (Φ(dµ(0))− Φ(dµ(−1)))− Φ(dµ(0))

]. (3.4)

The option can now be priced by putting C0 = W0 and ECT = EWT :

C0 = W0

= e−rTE[CT ] +W0 − e−rTE[CT ]

= e−rTE[CT ] + ξ0 · S0 − e−rTE[ξT · ST ]

= e−rTE[CT ] + ∆∗S0

(1− e(µ−r)T ) .

Note that if r = µ we obtain the Black-Scholes price e−rTE[CT ]. The Black-Scholes hedgedelta equals dr(0), and is different from the hedge obtained here, however. For example,if µ = r = 0.5, σ = 30%, K = S0 = 100, the Black-Scholes delta equals 0.56 while thestatic delta equals 0.59. Figure 3.7 shows the resulting price and hedge ratio as a functionof the drift.

If r 6= µ, the stock’s dynamics are not risk-neutral. One can argue that the price of thehedge in this case does not necessarily equal the option price. In this case, the quantityC0 can be seen as ‘production cost of replicating the payoff’, as argued by Bertsimas etal. [15, §2.5].

In this thesis, risk is often minimized using a mean-squared risk measure. The follow-ing proposition shows that this approach is equivalent to a minimization of the varianceor standard deviation. The minimization with squares, in a Monte Carlo setting, turnsout to be more straightforward than a variance minimization. Furthermore, a price iscomputed.

Proposition 3.8. The following minimization results in a variance-optimal hedge ratio:

minC0,∆

E[(C0 + ∆ST − CT )2] . (3.5)

Moreover, C0 = ∆E[ST ]−E[CT ], so the expected result of the hedge equals the derivative’sexpected result.

22

Figure 3.7: The optimal hedge portfolio price and hedge ratio as function of the drift of acall option on a Black-Scholes stock, with initial stock price 100, volatility 30%, interestrate 5%.

Proof. By definition of the variance, we have

E[(C0 + ∆ST − CT )2] = Var [∆ST − CT ] + E [C0 + ∆ST − CT ]2 .

Using Bellman’s principle of optimization, we can first minimize with respect to C0:

minC0,∆

[Var [∆ST − CT ] + E [C0 + ∆ST − CT ]2

]= min

∆

[Var [∆ST − CT ] + min

C0

E [C0 + ∆ST − CT ]2].

Now clearly C0 = ∆E[ST ]− E[CT ] is optimal, and the second term vanishes. So we haveshowed that the minimization is now equivalent to the problem of finding the minimum-variance optimal hedge:

min∆

Var [∆ST − CT ] = minC0,∆

E[(C0 + ∆ST − CT )2] .

3.4 Discrete hedging

In the case of discrete hedging (also referred to as dynamic hedging), the portfolio isre-balanced at time instants t0, t1, . . . , tT−1. Obviously a discrete hedge should be able tooutperform a static hedge, but the construction is far more complex than in the static or

23

continuous case. In view of theorem 2.6, only relatively simple models can be completeunder discrete hedging.

For small time increment sizes, the results from continuously hedging can be used, inorder to compute a nearly perfect hedge, and thus close to optimal hedge. However, iftime increments are not small, formulas obtained from continuous models can result inwrong prices and sub-optimal hedges, as we already saw in the case of static hedging.Furthermore, in many cases no perfect hedge exist and there is some risk left-over evenafter performing an optimal hedge, which is sometimes referred to as basis risk .

Wilmott used a Taylor expansion based approach in order to determine minimum-variance hedges of options in the Black-Scholes model [20]. Wilmott derived an optionprice as well, but the formula is wrong due to a mistake, as he remarked on his forum [21].As far as this thesis’s author knows, no analytic formula exists. An interesting resultfrom Wilmott is, however, that the price and hedge are dependent on the stock’s drift,in contrast to the continuous case. Moreover, this correction to the drift is proportionalto the size of the time instants. If re-balancing is done relatively frequently, ordinaryresults from the continuous case can be used. If however, the stock’s drift is not equalto the interest rate and hedging takes place on discrete time instants, the option’s priceand hedge depend on the stock’s drift parameter.

Schweizer published several papers on variance-optimal hedging in discrete time in-stants, for example [17, 22, 23]. Here, existence of the optimal hedge and expressionsfor the hedge portfolio are derived under different conditions. These expressions are,unfortunately, not directly applicable to specific models, for example the discrete versionof the Black-Scholes model. Bertsimas, Kogan and Lo derived expressions for optimalhedges in more specific cases [15], in the case of a Markov model. These results, stillrequire numerical methods in order to determine the prices and optimal hedges. In thisthesis, therefore, different methods are used to determine performance to certain hedgemethods. Often we will use the Monte Carlo (MC) method as a benchmark, introducedin Appendix B.

24

Chapter 4

The Hedged Monte Carlo (HMC)Method

The (Optimal) Hedged Monte Carlo (HMC) Method, introduced by Potters, Bouchaudand Sestovic [2], is inspired by the Least Squares Monte Carlo (LSM ) method (Sec-tion B.2). Both methods use regressions on Monte Carlo (Section B.1) simulations. Adifference is that the LSM method uses regressions to determine when an American-stylederivative needs to be exercised, while the HMC method uses regressions to determinethe value and hedge weights at each time instant.

An interesting feature of HMC is that it uses paths which are not necessarily gener-ated under the risk-neutral measure. Therefore, in contrary to many widely used pricingmethods (see Appendix B), a model can be used that generates scenarios from the ‘real’measure, which is typically the case with ALM. Moreover, since the method makes useof scenarios, no knowledge is needed about the scenario generating model. Therefore,the method can be applied to our main problem described in the introduction Chapter1, namely pricing and hedging in models which are unknown or to complex to deriveanalytic equations for.

Probably since the HMC method is relatively new, it is not as well known as otherpricing or hedging methods. Moreover, there is little research published on HMC. Petrelliet al. applied and analyzed the method for multi-asset derivatives [24]. Moreover, theylisted various advantages of the HMC approach and discussed the inclusions of transactioncosts and conditions on the hedge portfolios’ volatility [25]. Transaction costs and hedgeportfolio volatilities are not taken into account in this thesis.

This chapter contributes to the little research available on HMC by applying themethod to different models and analyzing various aspects of the method. In addition,we provide a mathematical justification of the method by introducing the new notionself-financing in expectation (SFE ).

This chapter is structured as follows. First, the HMC is introduced and discussedin detail. After that, the method is applied to both the Black-Scholes model and theHeston model. Furthermore, the method’s use of basis functions is tested and analysesare done on the regressions. By doing different analyses, we develop insights in how wellthe method works, how it can be used, and how we can track the performance of themethod in pricing and determining hedge portfolios. The chapter is closed with some

25

reports on computational time.

4.1 Introduction to the method

We return to Chapter 2’s set-up, so we have an adapted stock price process St =(S1

t , . . . , Snt ) and an adapted portfolio processes ξt = ξ1

t , . . . , ξnt to be constructed. We

write St and ξt if we include the zeroth risk-free asset in the vectors as well. Suppose wewant to price and hedge a derivative CT with payoff at time T . Recall from Chapter 3that our objective is to compute minimum variance hedges, and that this is equivalentto finding the least squares hedge. So our main objective is the following minimizationproblem:

minξ

E[(CT −HT )2] . (4.1)

Here the ξ are portfolio processes that are re-balanced at discrete time instants ti. Therandom variable CT denotes a certain contingent claim’s payoff and the processHt denotesthe result of the hedge portfolio ξ:

Ht = C0 +t−1∑i=0

ξi ·(Si+1 − Si

).

Here we have defined a new hedge process H, instead of using the wealth process W ,since we do not consider self-financing portfolios at this stage. The local risk Ri at timeti is defined by

Ri = E[(Ci+1 − ξi · Si+1

)2]1/2

. (4.2)

Here Ci denotes the estimated value of the claim at time ti, which is given by

Ci =

ξi · Si i < T

CT i = T.(4.3)

Clearly, Ci is Ft-adapted. Moreover, having chosen any ξi, equation (4.2)’s conditionalexpectation can not be calculated easily. In a MC context, however, the expectation canbe estimated as an average over all paths.

Since CT is given, backward induction can be used to construct a hedge portfolioweights ξi as a function of Si, such that at each time instant the local risk is minimized:

ξi = arg minξiRi, i = 0, . . . , T − 1. (4.4)

Recall that the portfolio process ξ has to be adapted, so the ξi are restricted to be Fi-measurable. In many cases, S is the only stochastic process that is observed, and it isMarkov under P as well. In that case, we can assume the ξi to be Si-measurable, which wewill assume in the discussion of the HMC method. In order to include this dependency ofthe stock process, a set of l basis functions Lk is used (see appendix C on basis functions).

26

At time ti, the jth asset portfolio weight, is then given by:

ξji =l∑

k=1

aji,kLk(Si). (4.5)

For our first analyses, we use polynomial basis functions and consider only one randomasset. So the basis functions are given by

Lk(s) = sk−1.

Summarizing, the HMC method uses backward induction and performs a linear leastsquares regression at each time instant, minimizing local risks.

Remark 4.1. Potters et al.’s paper uses a slightly different approach for the definitionand optimization of the local risk. In their paper the deterministic asset is not in theprice vector S, and a constant interest rate r over each time instant is chosen. Instead ofletting the derivative price be given by ξ · S, the price Ci at time ti is approximated bybasis functions. Equation (4.2) is then given by

R′i = E[(e−rCi+1(Si+1)− Ci(Si) + ξi(Si) · (Si − e−rSi+1)

)2]1/2

.

Note that the above includes ξ and not ξ, so the risk-free asset weight is not estimated.We now show that both approaches are equivalent, if the interest rate r over each timeincrement is constant. We put, for any i < T :

Ci = ξ0i S

0i + ξi · Si. (4.6)

In Potters et al.’s paper, ξi and Ci are determined by regression at each time instant,while in this thesis ξ0

i and ξi are computed. Therefore, this thesis’ ξ0i can be derived from

Potters et al.’s Ci and vice versa. Furthermore, from equation (4.6) we see ξ0i and Ci can

be expressed in each other in any system of basis functions.Using S0

i = e−rS0i+1, we can now show that that Ri is a multiple of R′i:

Ci+1 − ξi · Si+1 = Ci+1 − erξi · Si − ξi · Si+1 + erξi · Si= Ci+1 − erCi − ξi · Si+1 + erξi · Si= Ci+1 − erCi + ξi ·

(erSi − Si+1

)= er

[e−rCi+1 − Ci + ξi ·

(Si − e−rSi+1

)],

so Ri = erR′i. Hence, we have shown that both minimization problems are equivalentif interest rates are constant, by showing that R′ is basically the discounted version ofR. In the case of stochastic interest rates, both approaches can differ. The inclusion ofstochastic interest rates in HMC is discusses in more detail in Section 5.2.

The notation used in this thesis is more convenient for the adjustments made in thisthesis. Using Potters et al.’s definition of R′, however, it is more clear that in fact pricesare determined along each path.

27

4.2 Discussion of the HMC method

Potters et al. present HMC as a method for both pricing and hedging, but omitted themathematical details in their paper. In this section we justify the method mathemati-cally, by introducing the notion of ‘self-financing in expectation’. As far as this thesis’author’s knowledge reaches, this approach can not be found in the literature. In thissection’s discussions, the errors introduced by the regressions are not taken into account,as this will be the topic of Section 4.5. After having discussed the HMC minimization indetail, several ways to generalize the method to more complex problems are discussed.

In our analyses we will consider three minimization problems, which were introducedin the previous section:

1. Equation (4.1), the direct computation of the optimal hedge.

2. Equation (4.4), the backward induction version of the previous problem.

3. Equation (4.5), the HMC minimization, which is the previous problem estimatedwith basis functions.

For all these three minimization problems, it is easy to see that they have a solution. Alldomains are closed spaces, and the objective has lower limit zero.

4.2.1 Self-financing in expectation

The HMC method does not necessarily compute self-financing portfolios. In Chapter 5,a variation of the method is presented that restricts to self-financing portfolio’s. We willshow that HMC produces portfolios having the following notion of self-financing:

Definition 4.2. Let ξ be a portfolio process and CT be a contingent claim. Let (Ci)0≤i≤Tbe defined as Equation (4.3). Then we call ξ self-financing in expectation (SFE or P-SFE)if

Ei[ξi · Si+1

]= Ei [Ci+1] . (4.7)

An interpretation of a SFE hedge portfolio can be, that hedge errors are immediatelycompensated with an inflow or outflow of cash. The following results show that thebackward induction approach indeed computes a SFE portfolio process:

Lemma 4.3. Let a ∈ R \ 0 be fixed, f : X → Rm a function with domain X . Supposethat the following minimization problem is well-defined (the minimum exists):

minc∈R,X∈X

‖ac1− f(X)‖2 .

Here 1 denotes the unit vector in Rm and ‖ · ‖ the standard norm on Rm. Then for anysolution (c,X)

ac =1

m

∑f(X)

holds.

28

Proof. The minimization problem can be solved sequentially, so we can first consider

minc∈R

‖ac− f(X)‖2 = min

c∈R

m(ac)2 + ‖f(X)‖2 − 2ac

∑f(X))

.

Differentiating with respect to c, we find that for an optimal solution

2ma2c− 2a∑

f(X) = 0

must hold. Since we assumed a 6= 0, we obtain the result.

Lemma 4.4. Let (P,F ,Ω) be a probability space, and let G ⊂ F be a σ-algebra. LetY ∈ F be a random variable with E[Y 2] < ∞. Suppose that the following minimizationproblem is well-defined:

X = arg minX∈G

E[(Y −X)2] .

Then X = EG[Y ], P-a.s.

Proof. We prove by contradiction. Suppose that X 6= EG[Y ]. Assume, without lost ofgenerality, that there exists an ε > 0 and A ∈ G such that

E1AX < E1AY + ε

and P(A) > 0. We define X ′ = X + 1Aε. Note that X ′ ∈ G. What is left to show is that

E[(Y −X)2

]− E

[(Y −X ′)2

]= E

[(Y −X)2 − (Y −X ′)2

]= E

[1A (Y −X)2 − 1A (Y −X ′)2

]= E

[1A (Y −X)2 − 1A (Y −X − ε)2]

= E[1A(2ε(Y −X)− ε2

)]> P(A)ε2

> 0.

Proposition 4.5. Let CT be a contingent claim and let ξ be a portfolio process definedby Equation 4.4 (the min-variance backward induction approach). Then ξ is SFE.

Proof. Let i be fixed. Then ξi is defined such that Equation (4.2) is minimized, thus

ξ0i = arg min

ξ0i ∈Fi

E[(Ci+1 − ξi · Si+1 − ξ0

i S0i+1

)2].

Since S0i+1 > 0, the above problem can be written in the form of lemma 4.4:

ξ0i = arg min

ξ0i ∈Fi

E[Ci+1 − ξi · Si+1

S0i+1

− ξ0i

].

By the lemma we obtainEi[Ci+1 − ξi · Si+1

]= 0.

Hence, ξ is SFE.

29

Now, a similar result can be obtained for HMC. A difference is, however, that thereis not minimized over a certain expectation, but over the average in a set of scenarios.We will, therefore, write Emc for the MC average over the scenarios:

Proposition 4.6. Suppose that there is a linear combination of basis functions thatis constant. The HMC methods computes a portfolio process that is self-financing inEmc-expectation.

Proof. Assume, without lost generality, that the basis functions contain a constant func-tion Lk. Similar to Proposition 4.5, we can now apply Lemma 4.4 to obtain the result(minimizing over the weight a0

i,k).

4.2.2 HMC as a pricing method

Clearly, if the algorithm computes an (almost) perfect hedge, the estimated price is thecost of the hedge, which equals the unique arbitrage-free price. Therefore, the methodcomputes correct prices and hedges in a (almost) complete model. The following propo-sition shows that, in a risk-neutral setting where perfect hedging is not possible, pricescan be estimated as well. Here we assume, for ease of notation, a constant interest rate,but the result can easily be generalized.

Proposition 4.7. Let Q be a risk-neutral measure, hence

Si = e−rEQi

[Si+1

].

Then ξ is self-financing in Q-expectation if, and only if, C equals the risk-neutral priceprocess of CT .

Proof. The process ξ is self-financing if and only if EQi [ξi · Si+1] = EQ

i [Ci+1]. Since Q isrisk-neutral, this is equivalent to erCi = EQ

i [Ci+1], which completes the proof.

Hence, if the HMC method is applied in a risk-neutral setting, the computed accessoryprice process C is an unbiased estimator of the risk-neutral prices of the contingent claim.

So far, we have not discussed accuracy. Potters et al. showed that, in a Black-Scholesset-up, HMC computes an option price with higher accuracy than standard MC. Wereplicate this result in Section 4.3. Now, we are going to argue, heuristically, why theHMC method can compute more accurate prices than a standard MC. Thereby we ig-nore the error introduced by the use of regressions, which will be topic of discussion inSection 4.5.

Since HMC computes Emc-SFE portfolios, for each time instant i, HMC computesportfolio weights such that

Emc[ξ · Si+1] = Emc[Ci+1].

Here Emc means that we are averaging over scenario set paths. The error εi in the aboveestimation should be close to

εi = Emc[(Ci+1 − ξi · Si+1

)2]1/2

≈ Ri.

30

The accuracy can thus be estimated as εi√N

, where N denotes the number of MC pathsused. The standard deviation of the hedge error thus provides an estimation of the pricingerror E at t = 0:

E =

√∑i ε

2i

N.

So, while the MC method’s error depends on the contingent claim’s payoff variance, theHMC reduces this variance by computing a hedge which reduces the payoffs uncertainty.An extra error is, however, introduced by the use of basis functions.

Remark 4.8. The above heuristics illustrate that HMC has similarities with the MC’scontrol variate method (see Section B.1). An advantage of HMC over the control variatemethod is that portfolio weights and prices along paths are computed. An advantage ofthe control variate method in comparison with HMC is, however, that it does not rely onregressions.

In a non-risk-neutral setting it is questionable whether C0 can be viewed as a priceif some basis risk is left. In that case, the quantity C0 can better be seen as ‘productioncost of replicating the payoff’, as Bertsimas et al. pointed out [15, §2.5].

4.2.3 HMC as a hedge method

HMC computes the hedge portfolio by backward induction. The following theorem jus-tifies this approach, if the stock process S is risk-neutral.

Theorem 4.9. Let Q be a risk-neutral measure. Let all F-adapted portfolio processeswhich are Q-SFE be denoted by Ξ. Then an optimal portfolio process ξ? which is definedby

ξ? = arg minξ∈Ξ

EQ [(CT −H?T )2]

can be computed by backward induction:

ξ?i = arg minξi∈Fi

EQi

[(Ci+1 − ξiSi+1

)2], i = T − 1, T − 2 . . . , 0.

Proof. For any Q-SFE portfolio process:

CT −HT = CT − ξ0 · S0 −T−1∑i=0

ξi ·(Si+1 − Si

)= CT − ξ0 · S0 −

T−1∑i=0

ξi · Si+1 +T−1∑i=0

ξi · Si

= CT − ξ0 · S0 −T−1∑i=0

ξi · Si+1 +T−1∑i=0

Ci

=T−1∑i=1

(Ci+1 − ξi · Si+1

).

31

If we take the square and the expectation, all cross terms equal zero, since for any i < jwe have

EQ [(Ci+1 − ξi · Si+1

) (Cj+1 − ξj · Sj+1

)]=EQ [EQ

j

[(Ci+1 − ξi · Si+1

) (Cj+1 − ξj · Sj+1

)]]Tower rule

=EQ [(Ci+1 − ξi · Si+1

)EQj

[(Cj+1 − ξj · Sj+1

)]]ξi, Si+1, Ci+1 ∈ Fi+1 ⊂ Fj

=0 ξ is SFE.

Combining both results we obtain

EQ[(CT −HT )2] = EQ[T−1∑i=0

(Ci+1 − ξi · Si+1

)2] =

T−1∑i=0

EQ[(Ci+1 − ξi · Si+1

)2] = R2

i .

In addition, by Proposition 4.7 we see that the process C is independent of the choice ofSFE portfolio process ξ. Therefore, the above can be minimized by backward induction.

The above proof uses Proposition 4.7, which only holds in the risk-neutral measure.This raises the question whether backward induction is indeed equivalent to direct min-imization in a non-risk-neutral setting, for example the ‘real measure’. In general, theanswer is negative.

Remark 4.10. The variance risk measure is well-known to be time-inconsistent . Thatmeans that optimal hedge portfolio processes computed at t = 0 is not always optimal att > 0. This is counterintuıtive, since an investor has incentive to deviate from the optimalstrategy initially computed, although the optimal strategy at t = 0 outperforms thestrategies which are allowed to be changed. This phenomenon is introduced in literatureby Strotz [26]. For more background on time-consistent mean-variance hedging, pleaserefer to Basak and Chabakauri [27, 28]. In the latter three papers referred to, it is arguedthat in many cases only time-consistent strategies should be considered, since these arethe only ones that are actually performed.

If the model is close to complete or close to risk-neutral, the direct solution and thebackward solution are close to each-other. Moreover, it can be argued (Remark 4.10,that backward induction is the right way to compute portfolios. Therefore, we do notexpect that any problems are introduced by the use of backward induction. Section 5.1elaborates on direct minimization.

4.2.4 Extensions

As Potters et al. pointed out, the HMC method allows for some natural generalizations.First, the vectors ξt and St can be set to hold multiple assets instead of just one risk-free asset and one uncertain asset [24]. Furthermore, the basis functions can dependon more state variables than just the asset prices, letting the portfolio depend on more(nontradable) factors. This way, path-dependent derivatives can be included. For exam-ple, barrier options can be priced by adding a state variable which indicates whether thebarrier is hit. Second, the HMC method can be adjusted for American-style derivatives.

32

The latter property will not be discussed in this thesis, as this is already done by Potterset al. Fourth, Petrelli et al. discussed how transactions costs and risk premiums can beincluded [25]. In Chapter 5 some variations on the HMC method are discussed.

4.3 Results in the Black-Scholes model

In this section, HMC results are presented in a relatively simple setting, namely theBlack-Scholes model. Having shown that the method indeed estimates prices and optimalhedges, the next goal is to investigate the accuracy of the method. This section’s aimis to develop quantitative as well as qualitative tests on the method’s performance. Theinsights gained can later be used for hedging in more complex models and using differentmethods.

4.3.1 Black-Scholes option prices

First we verify the method by pricing an option, replicating Potters et al.’s results. Likein their paper, we use 8 polynomial basis functions (so order 7). We simulate a stockprice with Black-Scholes dynamics, having the drift equal to the interest rate, such thatthe price under the discrete hedging equals the usual Black-Scholes price (Section 3.4).Moreover, having the drift term equal to the risk-free rate makes the model risk-neutral,so results can be compared with Section B.1’s MC method as well. Figure 4.12 showsthe comparison of the prices of an option using both the HMC and the MC method.Over a number of 500 simulations, the HMC method results in an average price of 6.56with a standard deviation of 0.07. We obtain similar results as in Potters et al.’s paper.The MC computes an average price of 6.63, with standard deviation 0.43. Thus, bothmethods converge to the same Black-Scholes price 6.58, but the HMC method’s error isabout six times smaller than for the MC method, which is a similar result as Potters etal. obtained.

Remark 4.11. The HMC regressions take some extra computation time in comparisonwith the MC method. One could, therefore, argue that computation time should betaken into account when comparing the HMC and MC accuracy. However, since linearregressions are used, computation time of HMC is relatively small. This will be discussedin more detail in Section 4.7.

Moreover, if there is only limited data available and one is not able to simulate extrapaths, a variance reduction can be very valuable. In Chapter 6 we will consider suchsetting.

Interestingly, the HMC method does not necessarily need to be applied in a riskneutral set-up, like is needed for the MC method. In other words, the stock’s drift is notnecessarily equal to the risk-free rate. In figure 4.13, the HMC is plotted for differentdrift constants. We see that if the drift term is close to the risk-free rate, the methodproduces a price close to Black-Scholes’ price. If the drift term deviates from the risk-freerate, however, a lower price is calculated. This is in line with Section 3.4’s discussion ondiscrete hedging. As we have seen there, this ‘smile effect’ is dependent of the amountof re-balancing time instants that are chosen. In the case of a static hedge, we saw in

33

Figure 4.12: Option prices after 500 applications of the HMC method and the MCmethod. Both methods are applied on 500 simulations of 500 stock price paths. Thederivative used is an at-the-money option on a Black-Scholes modeled stock, with initialstock price 100, volatility 30%, and the drift equal to the interest rate of 5%. The HMCmethod is applied using 20 hedging time intervals and eight polynomial basis functions.The green and red histogram correspond, respectively, to the HMC and the MC method.The blue dotted line indicates the Black-Scholes price of the option, which equals 6.58.

34

Figure 4.13: HMC price for different values of the stock’s drift parameter and usingdifferent amount of re-balancing time instants, using 500 scenario sets containing 500 BSpaths. The stock price processes for different drifts are computed using the same seed.Recall that the interest rate is set to 5%, so the model is risk-neutral if the drift equals5% (dashed line). The error bars indicate the Monte Carlo errors (given by ±2 · Std/N).The price difference is relatively small. Moreover, a price difference of 0.01 translates toa implied volatility change close to 0.05%.

Figure 3.7 that the smile effect is bigger than in the case of discrete hedging, while inthe case of continuous hedging the price is the same for any drift. We are interested inthe method’s performance as a function of the amount of re-balancing time instants. Intheory, the basis risk should decrease to zero as the length of the time increments goto zero. In figure 4.14, the HMC method’s performance is compared with the a hedgestrategy using the continuous Black-Scholes delta. A couple of observations can be madefrom figure 4.14. First, as is known from the theory, the Black-Scholes hedge’s residualerror vanishes as the amount of re-balancing increases. On the other hand, the HMCmethod seems to have a small amount of residual risk that does not vanish as the timeinstants get smaller. This effect is probably due to the fact that the basis functions arenot able to fit the optimal hedge and price perfectly, which could be solved by choosinga bigger set of basis functions. This will be further discussed in Sections 4.5 and 4.6.A second observation that can be made, if few hedge time points are used, the Black-Scholes delta hedge is outperformed by the HMC hedge. This is probably because theBlack-Scholes delta is not optimal for discrete hedging, as we have seen in Chapter 3.

35

Figure 4.14: Variance of the hedge error for different amount of re-balancing instants.Again the stock has Black-Scholes dynamics and the parameters are the same as inFigure 4.12 (the tenor is fixed at 3/12). Hedging is done using the HMC method (blue)and the Black-Scholes’ delta hedge (red).

4.3.2 Hedging different derivatives

We now demonstrate the method’s capability of hedging more complex derivatives aswell. Here, we consider the financial derivatives listed in Table 4.1. HMC Hedge errorresults for a different amount of re-balancing time instants are given in Figure 4.15 andHMC prices are listed in Table 4.2. Interestingly, we see that for the Asian option thehedge error increases if the amount of time instants gets large. This is certainly notin line with the theory of optimal hedges, so this effect has to be caused by the errorintroduced by the use of regressions. The influence of basis functions in the regressionswill be further discussed in the upcoming sections.

Another, perhaps more convenient, way of comparing the hedge performances, is tocompare the hedge errors with uncertainty of the different derivatives’ payoffs. In Table4.3, the standard deviations of the hedge errors are compared with the standard deviationof the derivatives’ payoffs.

Not surprisingly, the binary option is relatively inefficient to hedge, because of thepayoff’s discontinuous nature. The barrier has a relatively high residual risk as wellbecause of the discontinuity. In addition, the hedge portfolio’s are only dependent on thecurrent stock price, while the barrier option’s payoff is path dependent. The process canbe made Markov, however, by adding an extra variable Y to the state. This variable isone if the barrier is hit and zero if the barrier is not hit, so in our case:

Y barrieri = Yi−1 · 1Si≤130.

36

id derivative1 Call option2 Binary call option3 Asian option, with average over the last month.4 Option on the stock price’s maximum over the full tenor.5 Up-and-out barrier option with barrier 130.

Table 4.1: Financial derivatives used in the analysis. All derivatives are assumed to beat the money, so have strike S0 = 100. In addition, payoffs are computed based on atenor divided in 64 discrete time increments. A more detailed explanation of the payoffsis given in Appendix A.

Figure 4.15: Hedge error for Table 4.1’s derivatives, and for different amounts of re-balancing time instants, similar as in figure 4.14. However, instead of the variance of thehedge errors, the standard deviation of the hedge divided by Table 4.2’s derivative prices.

37

derivative MC price HMC Price HMC error1. Option 6.5744 6.5657 0.0032

2. Binary option 0.4969 0.4993 0.00063. Asian option 5.8076 5.8050 0.0052

4. Maximum option 11.9262 11.9280 0.00875. Barrier option 4.3763 4.4005 0.0096

Table 4.2: MC and HMC prices of Table 4.2’s derivatives. 500 simulations of 500 pathsare used, so a MC error in each HMC price is calculated using σ/

√500. MC prices are

computed using 1, 000, 000 paths, such that they have a reliable precision (< 0.01).

derivative Std payoff Std hedge error Residual risk1 10.1 1.55 15%2 0.5 0.25 51%3 8.8 2.69 30%4 10.6 4.52 43%5 6.9 4.61 67%

Table 4.3: The in Table 4.1 introduced derivatives’ payoff compared with HMC hedgeerror variances. The payoff variances are estimated using 100, 000 scenarios with 32 timeincrements. The residual risk is expressed as the hedge variance divided by the payoffvariance.

We can now extend the set of basis functions, such that the dependency of Y is included:

(Lk)(s, y) = sn|0 ≤ n ≤ 7 ∪ ysn|0 ≤ n ≤ 7.

We tested this on approach on the barrier option with 16 re-balancing time instants. Byadding this extra dependency, the hedge error standard deviation was reduced from 4.93to 3.30 (from 71% to 48% of the payoff variance). The extension did not seem to influencethe pricing much, however.

Note that the Asian option and the maximum option are path-dependent as well. Inthese cases the process can be made Markov as well, by adding the running average andmaximum to the state vector, respectively:

Y Asiani = 1ti>T0 · averageSj|T0 ≤ tj ≤ Ti,

Y maximumi = maxSj|0 ≤ j ≤ i.

4.3.3 Comparing with numerical results

We have obtained some first results and made comparisons with analytic results in thecontinuous Black-Scholes model. However, the problem of finding a variance-optimalhedge when discrete hedging is different than in standard continuous Black-Scholes model.In particular, hedge weights obtained in the continuous model are not optimal in the dis-crete version. Therefore, the aim is to compare HMC results with accurate optimal

38

discrete hedges in the Black-Scholes model.

We are going to use a numerical method that shows similarities with Section B.3’sfinite difference method, the Lattice method, and Wilmott’s approach for hedging aBlack-Scholes option in discrete time [20]. As with FD, a grid is constructed with timeinstants and different stock values (Smin, Smin +δS, . . . , Smax). In this case, however, timeincrements are not made small, but are chosen equal to the re-balancing time instants.At each grid point, at time instant i at stock price j, a probability vector pij for all stockson the next time instant are estimated. In formula form:

pijk =

P(Si+1 ≤ Sk + δS

2

∣∣Si = Sj)

Sj = Smin

P(Si+1 ≥ Sk − δS

2

∣∣Si = Sj)

Sj = Smax

P(|Si+1 − Sk| ≤ δS

2

∣∣Si = Sj)

otherwise.

Note that random variable Si denotes the stock price at time instant i and Sj denotes thejth stock price on the grid.So the conditional densities at each grid point is discretized to the used stock prices.This can of course only been done if there is an expression for the stock’s conditionalprobabilities, which is clearly the case in the Black-Scholes model. In addition, the gridcan be computed for log stock prices as the stock prices are log-normal distributed. Inmore complex models it is less trivial how to compute the above probabilities.

The derivative prices at the last time instant are known, so using the conditionalprobabilities and backward induction, optimal hedges can be computed at each gridpoint:

ξi,j = arg minξ

∑k

pijk(Ci+1,k − Sj · ξ

)2

≈ arg min

ξEi[(Ci+1,k − Sj · ξ

)2 ∣∣Si = Sj].

We write ‘≈’, since the above is an approximation. However, the approximation error is,in contrary to the HMC method, not due the use of regressions, but due the discretizedstock prices.

We validate our numerical method by applying it to Section 4.3.1’s problem for dif-ferent parameters (Smin, Smax) and size of stock increments δS. In Table 4.4 and 4.5 theprices and hedges for the different parameters are listed, respectively. Note that the priceconverges to the standard continous Black-Scholes price, while the hedge ratio convergesquickly to a slightly different value. In general, we see that results are relatively stablefor δt ≤ 0.1 and Smax−Smin ≥ 120, which rises confidence in using the numerical methodwith dt = 0.1 and (Smin, Smax) = (40, 160) for validating the HMC results.

In order to compare the HMC method with the new numerical method, we simulateagain 500 scenario sets containing 500 BS paths. To each set we apply both methods.In Table 4.6 some results are compared. Not surprisingly, the new numerical method’shedge performs better in reducing the risk, as it is not dependent on basis functions.Though, the HMC method computes an accurate derivatives price and hedge at t = 0.

Instead of just considering t = 0, we would like to track the HMC’s results along theMC paths as well. Figure 4.16 compares the HMC hedge with the numerical hedge as

39

δt \ (Smin, Smax) (80, 120) (60, 140) (40, 160) (20, 180)

1 6.5393 6.6057 6.6097 6.62060.5 6.5242 6.5886 6.5897 6.59240.1 6.5195 6.5833 6.5833 6.58350.05 6.5193 6.5831 6.5832 6.58320.01 6.5193 6.5831 6.5831 6.5831

Table 4.4: Option prices using the numerical method for different amount of grid points.The continuous Black-Scholes price equals 6.5831.

δt \ (Smin, Smax) (80, 120) (60, 140) (40, 160) (20, 180)

1 0.5556 0.5643 0.5643 0.56430.5 0.5557 0.5644 0.5644 0.56440.1 0.5558 0.5644 0.5644 0.56440.05 0.5558 0.5644 0.5644 0.56440.01 0.5558 0.5644 0.5644 0.5644

Table 4.5: Hedge weights (i.e. amount stocks bought at t = 0, using the numericalmethod for different amount of grid points. The continuous Black-Scholes delta hedgeequals 0.5629.

Method Error variance Price HedgeHMC 2.1140± 0.0154 6.5603± 0.0065 0.5630± 0.0009

Numerical 1.2857± 0.0100 6.5833 0.5644

Table 4.6: Results of HMC method and numerical method for 500 simulations of scenariosets containing 500 paths. The average price at time zero, asset hedge weight at timezero, and the mean hedge error variance are listed. The bounds are given by MC errors,so 2 ∗ Std/

√500.

40

Figure 4.16: HMC (blue) and numerical (red) hedges halfway the hedging tenor (t = 10).The HMC hedges are averaged over the 500 scenario sets.

a function of the stock price, halfway the hedging tenor, so t = 10. We see that lowaccuracy is obtained for low and high stock pricing. This is not surprising, since less MCpaths are simulated there. Moreover, the polynomial basis functions show ‘explodingbehavior’, which motivates the use of different basis functions and will be Section 4.6’sfocus. The accuracy of the hedge can now be estimated in various ways. A naturalmeasure of the accuracy would be:

εt =

√E[(

∆hmct −∆numerical

t

)2]

We calculate this expectation numerically, by using sufficient BS MC paths, and find anerror of ε10 = 0.06 at t = 10.

4.4 Results in the Heston model

In Section 2.4 the Heston model is introduced. Recall that this model is incompleteas it includes an extra risk driver which is not a tradable asset, namely the stochasticvolatility. We are going to apply the HMC method to the Heston model, such that wecan demonstrate the HMC method in an incomplete setting. We use the following set-upfor Section 2.4’s Heston model:

• Constant interest rate and stock drift µ = r = 5%.

• Initial stock price S0 = 100.

41

• Initial volatility equals the long-term average volatility,√V0 =

√θ = 30%.

• Volatility volatility σ = 1.

• Revert rate k = 2.

• Correlation ρ = 〈W1,W2〉 = −30%.

Here we chose the parameters the same as in our previous Black-Scholes set-up andchoose the extra Heston parameters such that the stochastic volatility plays significant,but realistic, role. A correlation parameter of −30% is standard in literature. We sawthat the Heston model can be made complete by adding a volatility-dependent asset, forexample an option. We are going to investigate this property using the HMC method.Moreover, we are going test whether the hedge portfolio improves if volatility-dependencyis added to the basis functions. Therefore, we consider the following four methods:

1. A standard HMC where the assets observed and used for hedging are the risk-freeasset and the stock.

2. An extra asset, an at the money BS call option, is added to the scenario set whichcan be used for hedging and is added to the basis functions.

3. Instead of the (wrong) BS option prices, the Heston at the money call option price(Appendix D) is added.

4. Volatility as an observable is added to the basis functions, while only the risk-freeasset and the stock are used for hedging.

We hedge options of different strikes using the above four methods. In Figure 4.17, theresults are shown. Not surprisingly, the at the money option is perfectly hedged if anat the money option can be used for hedging (method 2 and 3). In- and out of themoney options are hedged better if an option can be used for hedging. Though, themethod using Heston options (method 3) performs better and is more stable than themethod with Black-Scholes option (method 2). So the idea that the hedge performanceis improved when adding a volatility-dependent asset, is supported by our observations.However, the asset added to the market needs to follow a risk-neutral price process, whichwas not the case for the Black-Scholes option.

Another observation that can be made from Figure 4.17, is that adding a volatility-dependency to the basis functions, improves the hedge of the options.

We would like to test more on the theory that adding a volatility-dependent assetmakes the model complete. This, of course, only holds for the continuous version of themodel. However, we can try to approximated the continuous model by increasing theamount of re-balancing time instants. We compute hedge portfolios for a binary optionfor different amounts of re-balancing time steps and apply three different versions of theHMC method:

1. HMC hedging with the stock and the risk-free asset (method 1 in the previousexercise).

42

Figure 4.17: Residual risks (hedge error variance divided by payoff variance) for calloptions with different strikes in the Heston model, for Section 4.4’s four different applica-tions of the HMC method. Polynomials of order 6 are used, and, if applicable, correlatedterms up to order 3. The methods are applied 50 times on Heston scenario sets containing1, 000 paths. The error bars indicate the MC errors (±2σ/

√500).

43

2. HMC hedging with the stock, risk-free asset, and a 1-year at the money call optionwith Heston prices (method 3 in the previous exercise).

3. HMC hedging with stock, risk-free asset, and the volatility as an asset. Althoughthis is an unrealistic set-up, since volatility is not a tradable asset in markets, itshould give us insights in how hedge portfolios can be improved when volatility-dependent assets are included.

Hedge results are shown in Figure 4.18. Since Heston is an incomplete model, we expectthat perfect hedging is not possible. We observe that, indeed, the residual risk for thehedge with just the stock and risk-free asset does not vanish when using a lot of timesteps. It seems that approximately 6% of the risk cannot be reduced by such hedge.The hedge which makes use of an option performs much better. Furthermore, having thevolatility as an asset reduces the residual risk even more. However, when increasing theamount of re-balancing time steps, both methods’s residual risk seem not to converge tozero either. The (small) amount of risk that can not be hedged, is possibly introducedby the use of basis functions.

Remark 4.19. While doing this section’s analyses, we encountered instability problemsin some set-ups. This happened when options where used for hedging. At the time of(close to) maturity, options have price (close to) zero under different scenarios. Therefore,multicollinearity occurs as it is unclear what weight options should be given in the hedgeportfolio in certain scenarios. This, sometimes, leads to non-logical results (extreme highasset weights). This effect seems to cause performance of the method with BS options(method 2) in Figure 4.17 to be unstable. Moreover, when hedging the binary option,we used call options of 1-year tenor instead of the binary option’s tenor equal to 3/12 ,since the latter caused a lot of instability of the hedges.

When applying the HMC method with multiple assets for hedging, we have two rec-ommendations. First, do not use assets which have the same payoff under differentscenarios. For example, a binary option is unsuitable, since the payoff is indifferent formany scenarios. Second, avoid the use of assets which are strongly correlated.

4.5 Goodness of fit

It is clear that regressions have a fundamental role in the HMC method. Up to now,polynomial basis functions up to order 7 are used for the regressions at each time instant.Although the HMC method does not restrict to linear regressions, this thesis focuses onthe use of basis functions. This still includes a great variety of models as the varioustypes of basis functions, and the amount used can be chosen at each time instant.

Extensive research is done on regressions. This section’s aim is to apply some well-known methodologies in tracking and improving regression’s performance to the HMCmethod. We illustrate this by varying the amount of polynomial basis functions used.Section 4.6 investigates different sets of basis functions.

44

Figure 4.18: Residual risks (hedge error variance divided by payoff variance) for an at themoney binary option in the Heston model, for three different applications of the HMCmethod and different amounts of re-balancing time instants. Polynomials of order 4 areused, and, if applicable, correlated terms up to order 2. The methods are applied 50times on Heston scenario sets containing 1, 000 paths. The error bars indicate the MCerrors (±2σ/

√500).

45

4.5.1 Cross validation

Cross-validation is a well-studied method for comparing regression models. Having datasets for training and validation separately is introduced by Highleyman [29] and Stone [30].Over the years, various techniques for cross-validation are developed.

We introduce a simple form cross validation, which we are going to use in this thesis.We illustrate the method by applying it to Section 4.3’s Black-Scholes call option problemfor a different amount of polynomial basis functions. The available data set, which is inthis case a scenario set consisting of Black-Scholes paths with 20 time increments, isdivided in two sets:

• Training set : On this set the regressions are performed for all models, determiningthe models parameters. In the HMC case, the HMC method is applied on thetraining set, determining the weights of the basis functions.

• Validation set : For each model, the errors are calculated on the validation set. Inour case, this means that the different hedge strategies obtained in the previousstep are applied to the validation set.

So the trained models are tested on a new set of data which is not used when determiningthe models’ parameters. This way, the models are can be compared. In Figure 4.20performance on both the training set and the validation set is plotted for a differentamount of basis functions. As expected, the error is reduced on the training set, whenmore basis functions are used, due to the decreasing lack-of-fit . The validation set,however, has an extra error, due to over-fitting , which increases as the amount of basisfunctions increases. In this case, the optimal amount of basis functions seems to bebetween 4 or 8. If larger scenario sets are used, this number increases.

4.5.2 Formulating the Ordinary Least Squares (OLS) regression

Although cross validation provides an easy to apply and useful measure on the qualityof the goodness of fits of the basis functions, we would like to analyze the regressionsin more detail. We would like to apply some methods that are widely used in the fieldof econometrics, since cross validation provides only a limited view on the regression’sperformance. Furthermore, cross validation needs a large amount of scenarios, while theamount of data is often limited.

The fact that linear least-square regressions with basis functions are applied in theHMC method is already mentioned a few times in this thesis, but the details were upto now omitted. The regressions, often referred to as Ordinary Least Squares (OLS ),assume the underlying model

y = Xβ + u.

Suppose there are m observations (Xi, yi), where Xi consists of n variables. The vectorβ denotes the n estimated linear parameters and the vector u consists of the m errors.The parameters β are chosen such that

‖Xβ − y‖2 =m∑i=1

(Xi ∗ β − yi)2

46

Figure 4.20: Hedge error variance, in Section 4.3’s BS set-up, for different amounts ofpolynomial basis functions. The average is taken over 500 scenario sets. The locally highervariance on the validation set for 5 and 6 basis functions is significant and consequentlyobserved. However, this thesis’ author has no explanation for this behavior.

47

is minimized. We illustrate the model by specifying X and y for the first minimization ofthe HMC method. We did not do this when introducing the HMC method since, as wewill see, the computation of the regression matrix gets relatively involved. At time instant

T−1, the expectation E[(ξT−1 · ST − CT

)2]

is minimized. Suppose we have simulated m

MC paths for (ST−1, ST , CT )1≤i≤m (these three variables are thus m-vectors), consistingof (a+ 1) assets Sj. Recall that the ξjT−1 are modelled using l basis functions Lk:

L(s) = (L1(s), . . . , Ll(s)).

We choose y = CT and define X as a matrix with m rows and l(a+ 1) columns:

X =(L(ST−1)S0

T , . . . , L(ST−1)SaT).

The hedge weights and prices at time instant T − 1 are now, respectively, given by:

ξjT−1 =l∑

k=1

βlj+kLk(ST−1), CT−1 =a∑j=0

l∑k=1

βlj+kLk(ST−1)SjT−1. (4.8)

So there are l(a+ 1) parameters β which need to be estimated and:

Xβ − y = ξT−1 · ST − CT .

The OLS solution is now given by:

β = (XTX)−1XTY. (4.9)

This formula can only be applied if no multicollinearity occurs, which means there islinear dependency in the matrix X, which means that (XTX)−1 not exists. If a highamount of basis functions is used, it is likely that multicollinearity occurs. Although theminimization problem might not have a unique solution in this case, we can find oneusing numerical methods.

An advantage of the OLS method is that we can determine standard errors . Thevector β is an estimation and the standard errors indicate how precise every parameter isestimated. Moreover, standard errors of other estimators can be calculated as well. Thisis done using the following covariation matrix:

Cov(β) = s2(XTX)−1, s =

√ ∑mi=1 u

2i

m− l(n+ 1).

Here the ui are the estimated errors, given by u = Xβ − y. The calculation of thiscovariance assumes, however, that all errors ui are i.i.d. and normal distributed. Thequestion whether the errors are normal is not discussed in this thesis. In our case, theerrors are independent, since we use independent MC paths (i.e. there is no autocorre-lation. The errors do depend, however, on S. For example, at the money options arefar more uncertain than options that are far out of the money. This phenomenon is

48

called heteroscedasticity . White derived a consistent covariance estimator in the case ofheteroscedasticity [31]:

Cov(β) = (XTX)−1(XTDiag(u21, . . . , u

2m)X)(XTX)−1.

We now illustrate the use of the covariance by calculating standard errors of the equa-tion (4.8)’s price at t = T − 1. Suppose we would like to compute the option price withstandard error for a certain ST−1. The estimated derivative price can be written as alinear combination α of the estimated parameters β:

CT−1 = ξT−1 · ST−1 =

L1(ST−1)S0

T−1

L2(ST−1)S0T−1

...Ll(ST−1)S0

T−1

· β = α · β.

The standard error is now given by

Std(CT−1) =

√αTCov(β)α.

In the following analyses we use polynomial basis functions up to order 3, instead of8, such that we have no multicollinearity problems. We calculate the price’s standarderrors at t = T − 1 by applying the OLS in our standard Black-Scholes set-up. InFigure 4.21 the option’s payoffs are plotted as a function of ST−1 for the 500 scenarios.Moreover, the estimated price is plotted with the standard errors. We see that the price iscalculated relatively accurately for stock values between 95 and 120. It should be stressed,though, that standard errors are derived from parameter estimation and do not reflectany model misspecification. Thus, the standard errors reflect how much more precisioncan be gained when simulating more MC paths. Moreover, analysis can be done on theparameters’ standard errors and correlation between parameters. This can be used todetermine which parameters can be left out.

4.5.3 Lack-of-fit and over-fitting

In Figure 4.20 we saw, that the amount of basis function used needs to be balanced. Toofew basis functions cause a lack-of-fit while too many lead to over-fitting . One way offinding the right amount is using cross-validation. A second method is plotting estimatesagainst observations. For example, in view of Figure 4.21, it seems that polynomials upto order 3 suffer from lack-of-fit. A third method we would like to introduce is the useof information criteria. Two examples are the Akaike information criterion [32] and theBayesian information criterion [33] (also referred to as Schwarz criterion):

AIC = 2l′ − 2 logL, (4.10)

BIC = l′ logm− 2 logL. (4.11)

Here l′ denotes the amount of model parameters (which is l(n + 1) in our case) and Lthe likelihood function. The information criteria needs to be minimized. In Table 4.7 thecriteria values for our T − 1 minimization are listed. Since m = 500 paths are used, so

49

Figure 4.21: The blue dots indicate the option’s payoff for different stock values at T −1. The red line is the estimated option price at t = 1 and the dashed lines are ±2σ,using White’s covariance estimator. A scenario set in our standard Black-Scholes set-upconsisting of 500 MC paths is used.

Order AIC·10−3 BIC·10−3

1 1.995 2.0122 1.764 1.7893 1.486 1.5204 0.803 0.8075 1.657 1.6626 2.662 2.668

Table 4.7: Akaike and Bayesian information criteria for different amount of polynomialbasis functions.

50

logm ≈ 2, both criteria do not differ much in this case. We observe that both criteriaare minimized when using polynomials up to order 4.Various research is done on regressions in the fields of statistics and econometrics, which

can be used to analyze the HMC method’s regressions. Another well-known method formodel selection is, for example, the well-known χ2-test, which will not be discussed inthis thesis.

4.6 Basis functions

So far we have only made use of polynomial basis functions. As for the LSM method, theHMC method allows for any choice of basis functions. In this section, the use of linearand cubic basis functions is discussed. In addition, some suggestions are made how theHMC method can be improved by the choice of basis functions. Appendix C provides amore detailed introduction of basis function than this section. In [24, Appendix A] thelinear and polynomial basis functions are discussed in more detail as well.

We will see that there are set-ups in which polynomials functions are outperformed,we did not found any combination of basis functions as robust and simple as the polyno-mial functions. The linear and cubic functions, for example, turned out to be unstableif the nodes are not chosen carefully. This is because nodes with no observations in itsneighbourhood could impose extreme hedge weights. In addition, cubic and linear areharder to generalize to multiple dimensions than polynomial basis functions. For sim-plicity, except for the analyses in this section, this thesis makes use of polynomial basisfunctions.

4.6.1 Linear and cubic basis functions

Linear basis functions assume a piece-wise linear model. For a finite amount of nodes ,weights are estimated. The weights for the data points are found by linear interpolation.We will analyze this in our standard Black-Scholes set-up, so the basis functions dependon just one observable, the risky asset. Linear basis functions can be used in moredimensions as well. However, the amount of nodes grows quickly as the amount ofdimensions increase.

Cubic basis functions (a.k.a. cubic splines) are similar to linear polynomials, but as-sume piecewise polynomials. The polynomials have order 3 and are chosen such that thevalue and slope of the two accessory polynomials match at each node. In other words,for each node, both a weight and a slope is estimated. The weights for the data pointsbetween two nodes are found by smoothing.

In Figure 4.22, the difference between polynomial, linear and cubic basis functions isillustrated.

Naturally, the linear and cubic regressions depend on how the nodes are chosen. Wewill do our analyses on a different amount of nodes N . We determine the location of thenodes using an independent scenario set consisting of 10,000 paths. For each time instantwe use a different set of nodes. In our first analyses we chose the first and last node to

51

Figure 4.22: The sinus function estimated by polynomial, linear and cubic basis functions.The data points are observed are 0, 0.1, . . . , 1 and the nodes used for the linear and cubicfunctions are 0, 0.2, . . . , 1.

be equal to, respectively, the minimum and maximum value of the stock. However, thisled to unstable results, as in some data sets there where few or none data points closeto these nodes, causing unstable results. For that reason, we argue that each node onaverage should cover the same amount of data points. Therefore, we chose the first andlast node to equal, respectively, the 1/2Nth and 1− 1/2N percentile in the independentscenario set. Points not between two nodes are extrapolated using the value of the closestnode.

In our analyses we divide the nodes over the stock price intervals in two ways. Inour first set-up, the nodes are uniformly over the line. The second approach spreads thenodes evenly over the data using percentiles.

Remark 4.23. In our setup, we choose the nodes for each hedge time instant on before-hand. Instead, we could have chosen to determine the nodes separately for each scenarioset. Fore sake of simplicity, we have not done this in our analyses.

4.6.2 Results

In our standard BS set-up, we test the linear and cubis basis functions with both uniformlyand evenly chosen nodes. Therefore, we used 100 test and validation scenario sets. InFigure 4.24, for different amounts of basis functions, average hedge variances on thevalidation sets for the four described basis function set-ups are plotted. From the figurewe can observe that, at least in this set-up, the linear basis functions outperform the cubic

52

Figure 4.24: Average hedge error variance for both linear and cubic basis function inour standard BS set-up. Node sets are computed both uniformly as evenly. MC errors(±σ/

√100) of these estimations are 0.003, so the results can be considered as accurate.

basis functions. The use of uniform or evenly divided nodes does not seem to matter alot in this set-up.

In our analyses using polynomial basis functions earlier in this chapter (e.g. Figure4.14), hedge variances where mostly the range of [2.0, 2.4]. Thus, if the right set of nodesis chosen, the linear or cubic basis functions are able to outperform polynomial basisfunctions.

4.6.3 Alternative basis function models

As this section illustrated, there are various possibilities for the choice of basis functions.Clearly, the choice of basis function set should depend on the context. We would like toend this section with some suggestions how to improve the basis function set:

• Use as much possible knowledge about the optimal hedges for the choice of basisfunctions.

• When making use of nodes, make sure that every node is affected by multiple datapoints in the data set.

• In most applications it is not an advantage to have orthogonal basis functions, since

53

the optimal weights are chosen by the regressions. For example, the use of Hermitepolynomials leads to the same results as the monomials we used, as they are a linearcombination of each other.

• Often, for low and high stock prices, little data is available. One can use modelknowledge to perform clever extrapolation. In addition, the basis function can beadjusted for behavior on extreme values. For example, one could multiple poly-nomial basis function with e−x to avoid them from exploding for high values ofx.

• The choice of basis function set can depend on the time instant. For example, inthe above analyses we had different nodes for each time instant. One can use lessbasis for the first time instants as well, as the hedges are often more similar sincefor each stock price in the scenario set.

4.7 Computational time

The calculations needed for this thesis are performed using Matlab and the computerused is equipped with an Intel E5-1607 3.00 GHz processor and 8.00 GB RAM memory.In 25.9 seconds we are able to apply the HMC method 100 times with polynomial basisfunctions up to order 8 on 500-sized scenario sets. Most of this time is even used to simu-late the scenario sets (17.5 seconds) and saving and analysing results (1.2 seconds). Only7.3 seconds are needed to apply the HMC itself in this set-up, which means the runningtime of the algorithm is less than 0.07 seconds. Thus, computation time does not seemto play a huge role here. Of course computation time increases if a higher dimensionproblems are considered, different regressions or different risk measures are used. Forexample, in Chapter 5 various methods are tested on a great amount of large scenariosets, which which caused computation time to increase a lot.

In order to implement the HMC method, it is recommended to use parallel program-ming as much as possible. Since the methods relies heavily on linear optimization, mostcalculations can be performed with matrix computations instead of loops, which reducesthe computation time in Matlab enormously. However, programming clear and efficientprocedures can be challenging if one wants to implement flexibility in the scenario types,methods used, basis functions used, etc. This thesis deals with only limited set-ups andthere are a lot of variations on the problems and basis functions which can be explored.

54

Chapter 5

Variations on the HMC method

Although we have justified the HMC method with derivations and numerical results, it isnot clear whether it is necessarily the optimal method in comparison with other methods.Since this thesis focuses on the problem of not knowing the scenario generating model,we are interested in finding different methods for determining hedges with regressions.

In this chapter, some variations on the optimizing algorithm are developed and tested.First, we will investigate pros and cons of the HMC’s sequential approach of the opti-mization and propose different schemes. Second, a convenient way of including stochasticinterest rates to this scheme is discussed. As far as known to this thesis’ author, the vari-ations developed in this chapter can not be found in literature.

5.1 Direct minimization

The HMC method uses multiple regressions and backward induction. The question ariseswhether a method can be derived using just one optimization, and how such a methodcompares to the HMC method. A (naive) direct formulation of the minimization problemwould be something like as follows:

minξ

E[(CT −HT )2] = min

ξE

(CT − ξ0 · S0 −T−1∑i=0

ξi ·(Si+1 − Si

))2 . (5.1)

Where the ξ are restricted to be S-adapted processes. Recall that we minimize with basisfunctions Lk, so we are actually minimizing over the weights ajik:

ξji =l∑

k=1

ajikLk(Si).

A problem with this approach is, that the risk-free asset weights ξ0 play a somewhatambiguous role, since there is no restriction on the total portfolio values. For example, ifthe interest rate is constant, S0

i = (1 + r)iS00 , the amount invested in the risk-free asset

can be interchanged between the different ξ0i :

ξ0 · S0 +T−1∑i=0

ξi ·(Si+1 − Si

)= ξ0 · S0 +

T−1∑i=0

ξi · (Si+1 − Si) + S00

(ξ0

0 + r

T−1∑i=0

(1 + r)iξ0i

).

55

In other words, it does not matter at which time instant there is invested in the risk-freeasset. Furthermore, if the interest rate is zero, the ξi would even have no influence atall on the outcome. For that reason, multicollinearity occurs. The amount of relevantparameters can thus be reduced. This directly exposes a downside of the above method:There are no sensible prices computed along the MC paths. In a risk-neutral set-up,having determined ξ, the time zero price C0 can still be estimated using a Monte Carloaverage:

ECT ≈ Emc

[ξ0 · S0 +

T−1∑i=1

ξi · (Si+1 − Si)

]= ξ0 · S0 + r

T−1∑i=1

a∑j=1

Emcξai .

The above direct minimization thus results in portfolio processes which have no restrictionon the values at the different time instants. We will see in the results that, indeed, thisapproach leads to unrealistic portfolio processes and inaccurate price estimations. In thefollowing we will impose the self-financing condition leading to more realistic portfolioprocesses.

5.1.1 Self-financing (SF) condition

Recall that, since the HMC method uses backward induction, the computed portfolioprocesses are not necessarily self-financing, a condition often imposed when computingreplicating portfolios. For example, if one is interested in computing replicating portfolios,one is often interested in portfolios with no external in- or out cash flows. An advantageof the direct version of the HMC method over the standard version, is that we can restrictto self-financing portfolios. Equation (5.1) then reads:

minξ s.f.

E[(CT −HT )2] = min

ξ s.f.E

(CT − ξ0 · S0 −T−1∑i=0

ξi ·(Si+1 − Si

))2 . (5.2)

Besides that this method is far more intuitive than the version without the self-financingcondition, the estimated price is now just given by ξ0 · S0.

Although the problem now seems to be formulated as a constrained linear least squaresproblem, it can be written as an unconstrained OLS. However, as the formulation isstraightforward but lengthy, exact computations of the matrices, like formulated in Sec-tion 4.5.2, are omitted in this thesis.

Remark 5.1. Besides the SF condition, there are more restrictions thinkable on thehedge portfolios. For example, the self-financing in expectation condition and the time-consistency condition, introduced in Section 4.2. The latter two conditions are probablynontrivial to impose on the scheme. Whether these conditions can be incorporated in anOLS optimization is not known to this thesis’ author.

5.1.2 Time-dependent basis functions

A disadvantage of the HMC method, in contrary to for example the finite differencemethod, is that little information is interchanged between different time instants. Al-though basis functions are allowed to depend on previous time steps, there is made no

56

use of the fact that for most models and derivatives, the prices and hedge weights asfunction of the state are similar at every time instant. This motivates a method wherethe same basis function weights are used at every time instant, which is not possible inHMC. An advantage of direct minimization, is that basis functions can depend on time aswell. For example, if polynomials are used and there is only one risky asset, polynomialterms correlated with t can be used:

(Lk(x, t))1≤k≤l = xatb|0 ≤ a ≤ N, 0 ≤ b ≤M.

Note that in this case we should write Lk(St, t) instead of Lk,t(St). The hedge weightsare now given by

ξji =l∑

k=1

ajkLk(Si, ti).

Typically there are less weights ajk estimated, as the same weights are used for every timeinstant.

5.1.3 Results

In order to test the variations developed in this section, cross validation is used. Weuse the same Black-Scholes scenarios as in Chapter 4. We are going to test the pricingaccuracy and hedge performance on at the money call options for the different hedgemethods. We test on scenario sets containing 500 paths as well as sets containing 5, 000paths and we use polynomial basis functions in this analysis. In order to determine theright amount of basis functions for each method, we choose the amount of basis functionsthat produces on average the smallest hedge errors on the validation sets after applyingon 100 training and validation sets. We are going to apply the following methods:

1. HMC method. Polynomial basis functions are used up to order 3 and 4 for thescenario sets containing 500 paths, respectively.

2. Direct minimization. Basis functions of order 2 and 4.

3. Direct minimization with self-financing condition. Basis functions of order 2 and 3.

4. Direct minimization with self-financing condition and time-dependent basis func-tions. The basis functions used are polynomials of order (4, 1) and (4, 2). By order(a, b) we mean that cross terms xitj have i, j ≤ a and individual terms ti and xj

have i, j ≤ b.

5. MC method. This method does not make use of basis functions, and will computeonly the t = 0 price and no hedges.

Note that for the direct minimization generally less basis functions are used than in HMC.This is because the direct method is more vulnerable for over-fitting, as there are moreparameters estimated on one minimization than in the HMC method.

Now that we have determined which methods we are going to compare, we apply themon 500 scenario sets of size 500 and 5, 000. In Table 5.1 the computed price with accuracy

57

Method 500-set 5, 000-set1. HMC 6.558± 0.006 6.580± 0.0022. Direct 6.624± 0.108 6.671± 0.0763. Direct SF 6.549± 0.007 6.572± 0.0024. Direct SF time-dep. 6.568± 0.006 6.582± 0.0025. MC 6.591± 0.040 6.581± 0.012

Table 5.1: MC prices after applying the different methods 500 times on scenario setscontaining 500 and 5, 000 paths. Recall that the analytic price equals 6.583.

Method 500-set 5, 000-set1. HMC 2.839 2.7202. Direct 2.980 1.4753. Direct SF 3.578 2.9934. Direct SF time-dep. 2.241 2.061

Table 5.2: MC variance of the hedge errors after applying the different methods 500 timeson scenario sets containing 500 and 5, 000 paths.

are listed. Clearly, the direct method is not suitable for pricing, as even standard MCis far more accurate. This is probably because unrealistic portfolios are computed, andthe ξi can be interchanged between different time instants, as described before. However,if the SF condition is imposed, the direct method computes accurate prices, which arecomparable with the HMC method. In Table 5.2 the hedge performances are summa-rized. The direct method computes small variance hedges. The self-financing condition,however, needs to be imposed to compute hedges which have a direct interpretation. In-terestingly, in this set-up the method with time-dependent basis functions outperforms allothers. Apparently, using information from different time instants indeed helps improvingthe regressions.

5.2 Stochastic interest rates

The HMC method, as introduced by Potters et al. and in this thesis, assumes a constantinterest rate. We are now going to discuss the inclusion of an adapted stochastic interestrate process rt ∈ Ft that is constant over each time increment. The risk-free asset S0

thus follows the following predictable process:

S0i+1 = ertS0

i ∈ Ft.

We propose five ways to include stochastic interest rates in HMC. We motivate thevariations, heuristically. Improved performances can not be proved, since performancewill depend on the specific problem the variations are applied to. After that, we comparethe hedge performances for some of the adjustments with numerical tests.

58

5.2.1 Extending basis functions

Recall from Section 4.2 that portfolios constructed by HMC are self-financing in expec-tation (SFE):

Ei [ξi · Si+1] = Ei [Ci] .This was true, however, with respect to the filtration generated by S, assuming S wasMarkov under the used measure. The Markov property makes sure that only informationfrom earlier time instants is not relevant when determining the hedge weights. If theinterest rate is deterministic, indeed, the process S represents all the relevant informationpresent at each time instant. In order to follow Section 4.2’s justification for HMC asa pricing and hedging method in the case of stochastic interest rates, we must includethe risk-free asset and the interest rate in the filtration. Assuming the model is modelMarkov, we thus need to consider hedges which are adapted with respect to (S, r). Thisbrings us to the most straightforward way of including stochastic interest rates in HMC,namely by having basis functions that depend on the interest rate and bank account aswell.

5.2.2 Adjusting ξ0

As we have seen in the various examples considered, derivative prices often depend onthe interest rate. The risk-free asset price (bank account), in contrary, does not influencederivative prices. For that reason, it seems inconvenient to let the regressions depend onthe bank account. We propose a scheme that excludes the value of the bank account. Wecan adjust the minimizations

minξi

Ei[(Ci+1 − ξi · Si+1

)2]

by adjusting (scaling) the regression for the amount invested in the risk-free asset ξ0:

ξ0i =

1

S0i+1

l∑k=1

a0i,kLk. (5.3)

In other words, instead of regressing the amount invested in the risk-free asset, the amountof cash needed on the next time instant is estimated. This way the dependency of therisk-free asset process S0 can be excluded from the basis functions. Moreover, as in thecase of a deterministic interest rate, there is a constant factor in the scheme on which aweights can be determined. Recall that we needed this property in order to prove SFE(Lemma 4.3 and 4.4).

5.2.3 Adjusting all hedge weights

If a risk-neutral model is used, the stock’s drift equal the risk-free rate:

EQt St+1 = ertSt.

It could, therefore, be beneficial to scale the risky asset hedge weights as well:

ξji = e−ril∑

k=1

aji,kLk, 1 ≤ j ≤ a.

59

This way, instead of regressing on St, we regress on the predictors ESt, which couldprovide a more accurate prediction of the optimal hedge weights.

5.2.4 Discounting local risk

As stated in Remark 4.1, Potters et al. used a discounted definition of local risk. Althoughwe do not have any views whether risk should be considered discounted or not, it is clearthat in the case of stochastic interest rates the discounted version of the problem isdifferent from our original scheme:

minξi

Ei[(e−riCi+1 − ξi · e−riSi+1

)2]

We see that, when using MC paths, trajectories with with higher interest rate get a lowerweight in the minimized quantity. We chose not to test the discounted HMC scheme inthis thesis, as the different minimization objective makes it difficult to compare resultswith the other adjustments.

5.2.5 Estimating the derivative’s expectation

Finally, we would like to propose a completely different approach for the case the deriva-tive’s payoff is independent of the interest rates. For example, if equity options onlydepend on the underlying stock(s), they, in some cases, can priced under a risk-neutralmeasure by

Ct = Et[S0t

S0T

CT

]= Et

[S0t

S0T

]Et [CT ] .

The above is only valid, however, if all stock returns are completely independent of theinterest rate process. The conditional expectations Et [CT ] can be estimated with a HMC

application with zero interest rate. The expectations Et[S0t

S0T

]can be estimated separately

or determined with model knowledge.Although this approach seems promising for pricing derivatives which are completely

independent of the interest rate, we keep this approach out of the scope of this thesis, asin practice the interest rates always influences the derivative prices, as stock prices arecorrelated with interest rates.

5.2.6 Results

In a stochastic interest rate setting, we are going to compare the standard HMC methodwith some of the HMC adjustments presented above. We test the different adjustmentsfor stochastic in the Black-Scholes model with normal distributed interest rates. First, weuse our standard risk-neutral Black-Scholes model, with the adjustment that the interestrate follows a Brownian Motion:

dSt+1 = µStdt+ σSStdW1t ,

drt = σrdW 2t .

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Method \ basis functions stock stock,risk-free asset stock,interest rate1. HMC 4.03 3.84 3.822. HMC adjusted ξ0 (5.2.2) 3.86 3.84 3.79

Table 5.3: Average hedge error variance after applying two methods 500 times on scenariosets containing 500 scenarios where the stocks have constant drift. The methods areapplied using three basis function dependencies. The differences are significant as theMC error of the estimates are all close to 0.01.

Here W 1 and W 2 are independent Brownian Motions. All parameters are as in theprevious cases and σr is chosen to equal 5, so that interest rate diffusion has a notableimpact (it is even likely to become negative). Note that this model is not risk-neutral,as the stock’s drift does not change according to the interest rate. Later we will use therisk-neutral version of this model.

We compare the following two methods:

1. Standard HMC

2. HMC with adjusted ξ0 (5.2.2)

Furthermore, we compare three different basis functions sets. One that depends only onthe stock price, one that depends on both the stock and the risk-free asset, and one thatdepends on both the stock and the interest rate. We use polynomial basis functions up toorder 3 and correlated terms up to order 1, as we know from practice that this results ingood fits. Hedge variances are listed in Table 5.3. We observe that the basis function setwith interest rate dependency outperforms the others. Moreover, the HMC with adjustedξ0 has a slightly lower hedge error variance than the standard HMC.

We are now going to consider the risk-neutral version of the model introduced above:

dSt+1 = rtStdt+ σSStdW1t ,

drt = σrdW 2t .

We test whether hedge errors can be reduced by adjusting all hedge weights with thestochastic interest rate (5.2.3), as described above. We compare this with only adjustingthe risk-free hedge weight, and again use the three different sets of basis functions. Av-erage hedge error variances on validation sets after 500 tests are listed in Table 5.4. Weobserve the HMC with only the risk-free hedge weight adjusted performs slightly better,but acknowledge that different set-ups might lead to different results.

Another observation that can be made from both this table as well as the previousone, that the inclusion of interest rate dependency in the basis functions only slightlyimproves the hedge when the HMC scheme is adjusted for the risk-free hedge weight.By adjusting the scheme we were thus able to improve the hedge performance, withoutthe need of extra basis functions. We therefore think that adjusting the optimization forstochastic interest rate can be beneficial for more complex models as well, since the useof smaller basis function sets, generally, leads to less problems due over-fitting.

61

Method stock stock,risk-free asset stock,interest rate1. Adjusted ξ0 (5.2.2) 4.09 4.08 4.012. All assets adjusted (5.2.3) 6.21 4.05 4.49

Table 5.4: Average hedge error variance after applying the two methods 500 times onscenario sets containing 500 risk-neutral scenarios. The methods are applied using threebasis function dependencies.

62

Chapter 6

Hedging in DNB scenario sets

As a part of laws for Dutch pension funds, ‘het Financieel Toetsingskader (FTK)’, theDutch central bank (DNB, ‘De Nederlandsche Bank’), prescribes the use of certain sce-nario sets for feasibility tests [34]. Pension funds are obliged to use these scenario sets,which are published on a quarterly basis by DNB. With this data, pension funds performan ALM study, to assess their future results and risks. These sets contain 2, 000 pathsof economic variables with a 60-year horizon, which are generated by a stochastic model,known as the KNW model [35]. Interest rates, inflation, and a stock index are modeled.For the calibration there is made use of the advice of the ‘Commissie Parameters’ [36]and a paper published by Draper [37].

Besides that these scenario sets are topical and relevant for pension funds, they seemsuitable to test the methodology developed in this thesis. The scenario sets publishedcan be used to determine prices and hedges of derivatives and the results can be com-pared with knowledge of the used model. Moreover, we can investigate to what extendthe limited data that is made available can be used to model derivatives. By doing this,we would like to illustrate to the reader how the method can perform in a real ALM study.

In this chapter, the KNW model is briefly introduced. After that, some HMC method-ologies developed in this thesis are tested on the data published by DNB and data createdfrom our own version of the model.

6.1 Introduction of the KNW model

We provide a high-level description of the KNW model [35], that should be sufficient forunderstanding this chapter’s discussions. The model can be written as a multivariateOrnstein-Uhlenbeck process :

dYt = (Θ0 + Θ1Yt) dt+ ΣY dZt. (6.1)

Here Zt is a vector consisting of 4 independent Brownian Motions and Θ0,Θ1 and ΣY arematrices which depend on the model’s parameters. The vector

Y ′ = (X1, X2, log Π, logS, logB)

consists of:

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1. A state vector X = (X1, X2). The time t values of bonds maturing at time Tprices P (t, T ) can be computed with the state vector Xt and the model parameters.Although both the nominal and real interest rates are modeled, this thesis onlymakes use of nominal interest rates.

2. A price index Π, used for modeling inflation.

3. A stock index S.

4. The cash wealth index B (bank account), which follows a predictable process. Thisprocess can be computed using the process X and the model parameters as well,but is added to the vector Y for convenience.

Equation (6.1) describes a stochastic differential equation (SDE). Exact simulation canbe achieved by discretizing the Ornstein-Uhlenbeck process [35].

The model prescribes a complete market, thus prices of contingent claims can becomputed by constructing self-financing portfolios. A much more convenient way ofcomputing prices is, however, by simulating the model under the unique risk-neutralmeasure Q. Adjusted matrices Θ∗0 and Θ∗1 can be calculated, such that the model becomesrisk-neutral. Under the risk neutral measure, all assets prices have the same drift as thebank account, and prices can be computed using B as the discount factor. So for any0 ≤ s ≤ t ≤ T and any contingent claim C:

P (s, T ) = EQs

[Bs

Bt

P (t, T )

], (6.2)

St = EQt

[Bt

BT

ST

], (6.3)

Ct = EQt

[Bt

BT

CT

]. (6.4)

6.2 Simulating the KNW model

Since the start of 2015, DNB publishes the scenario sets quarterly. For our analysis weare going to use the very first set [38, ‘HBT scenarioset 2015 Q1’]. The sets publishedcontain 2, 000 paths with the following variables for years t = 0, 1, . . . , 60:

• The state vector X = (X1, X2)

• The stock index

• The inflation index

The parameters needed to compute bond prices P (t, T ) with tenors T = 1, . . . , 75 asfunction of Xt are included as well. The initial state X0 is equal for every path. Forexample, (2.18, 2.23) in the set we consider. The set does not contain the short rate, sothe bank account process needs to be estimated from the interest rates. We will do thisby constructing a process that has each year the return of a 1-year bond.

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Figure 6.1: Nominal interest rate curve at t = 0. The blue curve is obtained from theDNB scenario set parameters and the red curve is computed by the parameters obtainedby the model.

Besides the data published by DNB, we are going to simulate extra scenarios as well,by implementing the KNW model. We do this for two different reasons. First, extra datacan give use more insight how well different methods perform in computing prices andhedges when applying on DNB’s data. Second, more information can be achieved aboutthe model, since implementing the model enables us to simulate risk-neutral scenariosand generate paths with smaller time increments than the DNB scenarios, which haveyearly time increments.

Although DNB has published the model used and how some parameters are chosen,not all parameters used are known to this thesis’ writer. For the interested reader, theparameters used for the simulations are taken from [37, table 1, first column] and adjustedas described in [39]. The initial state X0 = [2.177, 2.232] is taken from the scenario set. Totest our implemented model, we simulated 2, 000 scenarios and compared them with theDNB scenarios. In Figure 6.1 and 6.2, the interest rate curve and modeled variables arecompared, respectively. From these figures we can conclude that the scenarios of bothsets are similar, so we rely on the simulated data as a decent qualitative benchmark.Figure 6.1 and 6.2 should give the reader some insight in the model as well. For example,the stock’s drift and volatility is about 7% and 12%, respectively.

65

Figure 6.2: The blue sequences are obtained from the DNB scenario set and the redsequences are obtained by a simulation of the model. The error bars indicate the standarddeviation (±σ) of the variables.

66

6.3 Applying HMC

6.3.1 Three derivatives

For our analyses, we are going to use three financial derivatives CiT that mature at a

certain time T . An equity derivative, an interest rate derivative, and a combination ofboth:

1. An at the money stock put option. We choose the stock index S of our model tohave initial value S0 = 100, so the time T payoff is given by

C1T = (ST − 100)+ = 100 ·

(STS0

− 1

)+

.

2. A Swaption on a receiver interest rate swap, which starts at T0 > 0 and ends atTn > T0, and pays yearly (δ = Ti+1 − Ti = 1). The swap is chosen to be at themoney at t = 0. In other words, the fixed rate K equals the swap rate rswap

0 att = 0, such that the underlying has zero value at t = 0. We choose the notional toequal 100, thus the payoff at time T ≤ T0 is given by

C2T = 100 · (SwapT )+ = 100 ·

(P (T, Tn)− P (T, T0) + δK

n∑i=1

P (T, Ti)

)+

.

3. An exotic derivative which payoff depends on both the interest rate curve and thestock index. Since the instrument is derived from two different asset classes, werefer to it as the hybrid . The derivative used, is an equity linked receiver swaption.Its payoff is given by

C3T = 100 · (SwapT (K(ST )))+ .

Here, the underlying swap is the same as for the previous derivative C2, with insteadthe fixed rate replaced by a strike that depends on the stock price. The strike K(ST )is given by (

rswap0 − gearing · (ST

S0

− 1)

)+

.

In this thesis, the gearing is chosen to equal 2%, such that both the interest rateand the stock index have a comparable impact on the payoff in the DNB model.Figure 6.3 and Table 6.1 illustrate the hybrids payoff.

For an overview and description of financial derivatives, please refer to Appendix A.

Remark 6.4. The reader might wonder why a complex instrument, like the hybrid de-scribed above, would be traded in the market. There is large payoff if both the stockindex and the interest rate fall (See Table 6.1 for a schematic overview of the deriva-tive’s payoff structure.). This is typically a security a pension fund could be interestedin. Pension funds invest in stocks, which makes their investments’ performance heavilydependent on the equity market. In addition, the value of their future liabilities rises ifinterest decrease. Therefore, a pension fund can buy security for the event of low stockprices and low rates. A hybrid is likely to be relatively cheaper than a combination ofput options and swaps, securing for both low stock markets and low interest rates.

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Figure 6.3: Hybrid payoffs against the time T stock index and swaprate, in a simulated2, 000-path scenario set generated under the real measure.

rate rate→ rateequity 0 0 0equity→ 0 0 +equity 0 + ++

Table 6.1: Schematic overview of the payoff of Section 6.3.1’s hybrid. The derivativebenefits from low interest rates and a low equity market.

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Figure 6.5: Estimated prices (left) and residual risks (right) of the three derivatives,after applying risk neutral model (top) and the real model (bottom) on 100 scenariosets containing 2, 000 paths. The put option (blue), the swaption (red) and the hybrid(yellow) are hedges using different length of time increments dt, using the same seeds.For the risk-neutral scenarios and real scenarios, different seeds are used. The errors barsindicate the MC errors ±2σ/

√100.

6.3.2 First results

We test the HMC method on the three derivatives described above, for different amountsof re-balancing time instants. We simulate 100 scenario sets containing 2, 000 paths, usingboth the risk neutral version and the real version of the model. The derivatives’ tenorsare chosen to equal 6 years, and the swap parameters are set (T0 = 6, Tn = 16, δ = 1). Forour first analyses we use a standard HMC with Section 5.2’s adjustment for stochasticinterest rates (adjusting the risk-free hedge weight ξ0). Polynomial basis function areused with orders up to 4 and correlated order up to 2. The bank account process is usedas an asset and as an observable in the basis functions. In addition, the option is hedgedusing the stock index, the swaption is hedged using the underlying swap, and the hybridis hedged using both the stock and the swap. In Figure 6.5 the average prices and residualrisks are plotted. In Table 6.2 the estimated prices are listed and compared with the MCprices computed with the risk neutral scenario sets.

We observe that the put option and swaption are hedged and priced relatively well.The hybrid, on the other hand, has a higher residual risk that does not vanish as theamount of re-balancing time instants increases. Moreover, the hybrid prices are signifi-cantly different in the two different measures. In order to improve pricing precision, we

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MC HMC RN HMC Real

1. Put option 17.645± 0.151 17.603± 0.029 17.911± 0.0292. Swaption 7.563± 0.037 7.545± 0.008 7.504± 0.0133. Hybrid 3.460± 0.042 3.422± 0.015 3.949± 0.027

Table 6.2: Estimated prices after applying the HMC method 100 times on both the risk-neutral version and the real version of the model. The values are taken out of figure6.5, using the smallest time increments, dt = 0.125 (48 re-balancing time instants). Inaddition, standard MC prices are computed using the risk-neutral model. The errors barsindicate the MC errors ±2σ/

√100.

should probably aim on improving the hedge portfolios by using different variations ofHMC and different basis function sets.

Another observation that can be made in the risk-neutral model, that the estimatedprices seems independent of dt. The accuracy, however, increases if more re-balancingtime instants are used, since pricing precision depends on the hedge performance. Theseobservations are in line with Section 4.2’s mathematical discussion of the HMC method.In the model generated under the real measure, the prices decrease slightly if dt increases.

6.3.3 Applying on yearly data

Our next aim is to find the right HMC variation and basis function sets when havingscenario sets with only yearly data, which is the case for DNB scenario sets. In ourprevious exercise, we did not use cross validation and neglected errors introduced byover-fitting. We are now going to use validation sets to compare different approachesmore precisely.

We are going to compare three methods to compute the hedges: A standard HMC,a direct SF scheme, and a direct SF scheme with time-dependent basis functions (seeSection 5.1 for a discussion of these methods). For all three methods, stochastic interestrates are taken in account as described in Section 5.2, adjusting the risk-free hedge weightξ0. Moreover, we restrict to polynomials basis functions of certain order with correlatedterms up to another certain order. In order to compare the different methods and amountof basis functions used, we used cross-validation on 100 scenario sets containing 2, 000paths, generated under real measure. Having correlated terms up to order 1 turns outto produce optimal results for all methods, well the optimal order of the uncorrelatedpolynomial terms difference for each method, as can be seen in Table 6.3. For all threederivatives (the latter table only shows results for the hybrid), we observed that themethod with time-dependent basis functions performs best in reducing the risk, followedby the standard HMC method.

We can conclude, in the above set-ups, that approximately 75% of the hybrid’s riskcan be hedged, when there is re-balanced yearly. Now we have determined suitable basisfunction sets for each method (Table 6.3), we are going to compare this result with theDNB data. We hedge the three derivatives in our DNB data set (2015 Q1), consistingof 2, 000 paths. There is one adjustment we have to make, since the data set does not

70

order HMC HMC SF HMC SF time dep.

1 25.6 25.2 24.42 24.8 25.7 24.13 25.4 25.9 23.24 27.5 33.6 24.4

Table 6.3: For different orders of polynomial basis function sets, residual risks (hedgeerror variance divided by derivative payoff, in %) for the hybrid, after applying the threemethods on 100 scenario sets consisting of 2, 000 yearly paths which are generated underthe neutral measure. Only the order 1 correlated terms of the basis functions are used.For each method, the smallest residual is made bold. These basis function sets are usedin the following analyses.

Derivative HMC HMC SF HMC SF time dep.

1. Option 2.6 3.2 2.22. Swaption 1.9 2.0 1.83. Hybrid 29.8 28.2 30.3

Table 6.4: Residual risks (hedge error variance divided by derivative payoff, in %) for thethree derivatives in the DNB data set, using the three difference methods. Basis functionsets are selected according Table 6.3.

contain the bank account process. We estimate this process by using the 1-year bondprices. With use of our own replication of the model, we validated that this adjustmentdoes not influence the results much.

In Table 6.4, residual risks for the different methods and different derivatives, usingthe DNB data. The hybrid is hedged with a residual risk of approximately 30%, whichis slightly higher than our previous results. The option and swaption, however, seemto be hedged slightly better than the average in our previous results. We can think oftwo possible explanations for this small difference with our previous results. First, thescenarios are randomly generated, so, like we have already seen in our generated scenariosets, hedge performance is not the same for every set. The difference can be thus bedriven by the inaccuracy due to limited data. Second, the model used for the DNB setcan be slightly different from the model we implemented, as we have no full knowledgehow the DNB scenarios are generated.

Remark 6.6. Besides that we have tested only a limited set of basis functions, one canargue that the amount of basis functions should be selected differently. For example, wekept pricing accuracy out of the scope of basis function selection.

Another remark that must be made, is that, since we made a prototype of the model,we used knowledge of the model. If we would consider the data strictly as ‘Black Box’, thiswould be impossible. This could be overcome by applying Section 4.5’s techniques to thedata set. For example, cross-validation can be applied by partitioning the scenario setsin two sets (this can be done multiple times using multiple partitions). Since the amount

71

Figure 6.7: Residual risks of the three derivatives for different tenors, using the standardHMC method. Table 6.4 coincides with a 6-years tenor.

of data available is already very limited, we have chosen not to follow that approach.

In our previous analyses we used a tenor of 6 years for all the derivatives. Recall thatthe underlying swap started at t = 6 as well, and ended at t = 6 + 10. In Figure 6.7the residual risks are plotted for different derivative tenor’s T . We fixed T0 = T andTn = T + 10.

The option and swaption turn out to be hedged relatively well for short and longtenors. The Hybrid’s residual risk, increases if shorter or longer tenors are chosen thenthe 6-year tenor we used before. We tested whether this result occurred due the choiceof a sub-optimal amount of basis functions, which turned out not to be the case.

6.3.4 Improving the Hybrid hedge

In order to improve the hybrid’s hedge, we vary the assets used for hedging and for thebasis functions. In our previous analyses, the hybrid was hedged using the bank account,the stock index and the interest rate swap. The hybrid’s payoff is, however, not fullydetermined by these factors. The derivative’s payoff functions depends on bond of mul-tiple tenors, P (T, T0), P (T, T1), . . . , P (T, Tn). These assets are, however, only dependentof the two risk drivers X1 and X2. We test different the use of different sets of interestrate assets for hedging, using the swap, the T0-bond, the Tn/2-bond and the Tn-bond.Results are listed in Table 6.5. Here we used our time dependent HMC method withbasis functions of order 3 and correlated order 1, as this choice gave us the best results.From the table we can conclude that the best results where obtained when hedging whenmaking use of three bonds, which resulted in an average residual risk of 17.6% on thevalidation sets.

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Interest rate assets Validation set residual risk (%)

Swap (previous set-up) 23.8T0-bond, Tn-bond 22.8T0-bond, Tn-bond, swap 18.2T0-bond, Tn/2-bond, Tn-bond 17.6T0-bond, Tn/2-bond, Tn-bond, swap 30.2

Table 6.5: Hedge residual risks when using different interest rate assets in the hedgeportfolio, besides the bank account and the stock index.

We see that the choice in types of basis functions and assets for the hedge portfolio isendless. We have also tested different basis functions by changing the maximum order foreach observable individually. After testing different set-ups, we obtained best results whenusing time-dependent polynomials, with the time variable up to order 5 and correlated upto 1, and all order variables up to order 3 and correlated up to order 1. After 100 tests,this led to an average residual risk of 9.1% and 16.2% on the training set and validationset, respectively. This seems sufficiently small to consider the hybrid in an ALM study.

Remark 6.8. On the DNB data set, however, only 22.4% of the risk is reduced. Thisslightly higher number is probably due the fact that in the DNB set the short-rate is notobserved. The bank account is estimated with 1-year bonds. Since, both the stock andbond dynamics are dependent on the short rate, some information is lost by this approx-imation. For example, the correlation between the bank account and the stock index is30% our simulated model, while it is only 18% when the bank account is approximatedby 1-year bonds.

Summarizing, we have significantly improved the hybrid’s hedge by choosing differentbasis functions and hedge assets. However, we were not able to reduce the risk fully. Bydoing more analyses on the regressions and trying different basis functions it is probablypossible to reduce the residual risk slightly more. However, the results indicate that atleast 10% of the Hybrid’s risk can not be reduces with yearly hedging.

As we have seen multiple times in this thesis, the hybrid’s residual risk results in adiscrepancy in the price under the real world measure and under the risk-neutral measure(see Table 6.2). The reason for this lies in the residual risk that is assumed to have zeroprice in HMC. Under the risk-neutral measure, this assumption is, generally, correct.If scenarios are generated under the real measure, however, this can result in differentprices. HMC prices can then, therefore, better be seen as costs of the variance-optimalhedge. In case one wants to obtain risk-neutral prices of a not fully hedgeable derivativein scenarios generated under the real measure, one need to assign risk premiums to theresidual risks. This problem is discussed in more detail, but not solved, in [25, Sec. 3].Having only scenarios generated under the real measure, one needs more informationabout risk premiums in the model. These risk-premiums can, for example, be estimatedfrom market prices or derived from the model specification. Further research can be donehow HMC can be adjusted to incorporate risk-premiums estimated from a certain source.This can be useful since it could enable including complex derivatives more accurately inscenario studies with limited data, as is the case with the DNB scenario sets.

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Although the DNB data set provides only limited information, we were able to hedgethe put option and the swaption efficiently. The majority of the hybrid’s risk could bereduced as well, which could enable to, for example, investigate the use of this derivativein an ALM study. However, in order to be able to fully replicate and price a complexinstrument like the hybrid, scenarios with shorter time increments are needed.

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Chapter 7

Conclusion

Thesis outline

In this thesis, we investigated the HMC method for determining prices and hedge portfo-lios in scenario sets. In particular, we were interested in how to construct hedge portfoliosif only little information about the scenario generating model is known, viewing the modelas a ‘black box’. After discussing fundamentals of pricing and hedging, we introduced theHMC method and analysed the method both mathematically and practically, contribut-ing to the little literature available on the method. Thereby we introduced the notionof ‘self-financing in expectation’. We tested the method in the Black-Scholes model andthe Heston model hedging different financial derivatives, and proposed methods how toanalyze and improve the HMC’s performance. After that, we proposed variations on theHMC method, which involved putting different restrictions on the hedge portfolios andthe inclusion of stochastic interest rates. Finally, we illustrated the developed method-ology by applying HMC on a topical model meant for Asset and Liability Management(ALM), namely the KNW model used for regulations of the Dutch central bank (DNB).

Main findings

We validated Potters et al.’s result that HMC can be used to compute accurate risk-neutral derivative prices. We were able to use the method to compute min-variancehedges as well. We showed that, if the scenarios are computed under a risk-neutralmeasure and errors introduced by basis functions are ignored, the backward inductionHMC uses, indeed computes risk-neutral prices. In addition, min-variance hedges arecomputed. We observed that, if the scenarios are computed under the real measure andsmall time increments are used, relatively accurate prices and hedges can be computedas well.

In our analyses, we used little knowledge about the models tested. The only as-sumption made for application of the HMC method, is that the asset price processes areMarkov under the probability measure used. We were able to price and hedge variousderivatives in both complete and incomplete models. This raises confidence that HMC isa reliable and widely applicable method that can be applied in a black box setting.

We introduced variations on HMC using only a single optimization, instead of back-ward induction. Thereby we were able to impose the self-financing condition on the

75

scheme. Moreover, we showed that, at least in the set-ups we used, more accurate pricesand better hedges can be computed than when using a standard HMC, using this directself-financing approach with basis functions that depend on the time.

We hedged derivatives in a ALM model used in practice. We were able to priceand hedge certain derivatives in some cases, but limitations were exposed as well whenhedging a complex derivative if only limited data is available.

Future research

We provided a mathematical background for the HMC method, and justified the methodfor computing prices and min-variance hedges. We showed that min-variance hedges andunbiased prices are computed under the risk-neutral measure. We observed that accurateprices and hedges can be computed in some incomplete and non-risk-neutral settings aswell. We wonder whether analytic bounds or adjustments can be derived for the biaswhen the HMC is applied under the real measure. Also, further research can be done onhow the residual risk can be taken into account for pricing in the HMC method.

We illustrated briefly with some statistical methods how the HMC regressions per-form. We observed that the choice of basis functions has a great influence in the method’sperformance. Therefore, we think that further research on the selection of basis functionsets can be very valuable since it could result in more accurate HMC results.

The variations on the HMC method seems promising. Probably there are more wayshow the minimization problem can be restricted or specific model knowledge can be incor-porated in the scheme. This can useful when the model is applied to a specific problem.For example, different risk measures can be considered, as the standard deviation riskmeasure is not always the objective.

Possible applications

In finance, both prices and hedge portfolios are relevant for traders, risk managers, in-vestors, etc. A hedge portfolio, for example, provides insights in risk drivers of a certainclaim. Moreover, replicating portfolios are used to determine the so-called ‘Greeks ’, whichare used to characterize trading positions. We showed that HMC can be applied to diverseproblems. We would like to close this thesis’ by proposing two more specific applicationsof the research done for this thesis.

First, we think the methodology can be beneficial in ALM studies. The scenario setsconsidered in ALM are generated under the real measure, and the underlying model isoften complex or applied by an external party. Derivatives are, therefore, difficult to con-sider in these studies, while derivatives are often fundamental when analyzing a balancesheet. Even if the HMC method is not able to compute the market price of a contingentclaim when it is not efficiently hedgeable, the calculated portfolio provides insights in theclaim’s risk drivers and what assets should be traded to reduce risk. In addition, usingthe estimated prices along paths, derivatives can be added to a scenario set, such that

76

that they can be considered in an ALM study.

A second application can be found in the research of pension fund designs. At themoment of writing this thesis, there is a lot of debate about the Dutch pension system.The discussions involve, for example, whether or not risk should be shared between groupsof pension participants. We think the HMC methodology can help providing insight inthe different pension schemes which are considered. It would be interesting to be able toshow that, for example, certain pension schemes are replicable by using certain hedgesor derivatives.

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Appendix A

Descriptions of financial instruments

We expect the reader to be familiar with financial derivatives and other financial instru-ments. This chapter provides a brief overview of the instruments used in this thesis.Thereby we focus on expressing the payoff mathematically. Common practices, intu-itions, use of derivatives and other details are omitted since these are not relevant for thisthesis.

We assume that all the instruments presented here have a notional of one 1, as thevalues and payoffs for different notionals can easily be obtained by scaling.

In Section 6.3 a certain hybrid is introduced. A hybrid is a derivative that thatdepends on more than one asset class.

A.1 Options

A call option and put option have an underlying stock S, strike K and date of maturityT . The time T payoffs are, respectively given by

Ccall optionT = (ST −K)+ , Cput option

T = (K − ST )+ .

Here (·)+ means max(0, ·). The above describes European options , or (plain) vanilla op-tions. In finance, the term vanilla means ‘standard’. American options have the samepayoff, but can be exercised at any time t ≤ T . For all options described below, both theEuropean and the American version can be considered.

Binary options have the following payoff:

Cbinary call optionT =

1 ST ≥ K

0 ST < K, Cbinary put option

T =

0 ST ≥ K

1 ST < K.

An Asian option uses the average price over certain time instants T1, . . . , Tn:

CAsian call optionT =

(1

n

n∑i=1

Si −K

)+

.

78

Basket options have multiple stocks S1, . . . , Sa as underlying, with weights w1, . . . wa

(∑

j wj = 1). The value process of the basket if given by St = S0

∑j

SjT

Sj0

. So the time T

payoff is given by

CBasket call optionT =

(ST −K

)+

.

In this thesis, an option on the maximum of a stock SmaxT = max0≤t≤T St is used:

CBasket call optionT =

(SmaxT −K

)+

.

A barrier option becomes activated or worthless if a certain barrier B is hit at any timet ≤ T . There are four forms:

• Down-and-out: Option becomes worthless if the stock price comes below B.

• Up-and-out: Option becomes worthless if the stock price comes above B.

• Down-and-in: Option becomes activated if the stock price comes below B.

• Up-and-in: Option becomes activated if the stock price comes above B.

A.2 Interest rate instruments

Let P (t, T ) denote the time t value of a cash flow of 1 unit at time T . The interest rateF (t, T ) over the period [t, T ] is thus given by

F (t, T ) =1

T − t

(1

P (t, T )− 1

).

The short rate r(t) is given byr(t) = lim

T↓tF (t, T )

and the bank account B(t) evolves with the short rate:

B(t) = exp

(∫ t

0

r(t)dt

).

A.2.1 Swaps and swaptions

Interest rate swaps exchange a fixed rate K and a stochastic rate F . Cash flows are atfixed dates T1, . . . , Tn. We assume intervals of equal length, δ = Ti+1 − Ti. The timet ≤ T0 value of payer swaps and receiver swaps are given by

Cpayer swapt = P (t, T0)− P (t, Tn)−Kδ

n∑i=1

P (t, Ti), Creceiver swapt = −Cpayer swap

t .

A swaption provides the right to enter a swap Cswap, so its time T0 value is given by

CswaptionT = (Cswap

T )+ .

For a more detailed discussing of interest rate instruments, please refer to [40].

79

Appendix B

Three well-known risk-neutralderivative pricing methods

This chapter discusses three well-known methods to determine prices in the case of a risk-neutral model. The methods presented are numerical and only applicable when havinga certain model of the asset prices. However, the discussed techniques turn out to havesimilarities with the HMC method, introduced in Chapter 4.

B.1 Monte Carlo (MC) Method

B.1.1 Introduction to MC

Using Monte Carlo simulations for pricing derivatives was proposed by Boyle [41]. TheMonte Carlo method uses a risk-neutral model for asset prices (St), which are simulatedon discrete time instants. Taking m samples si from the the distribution of S, now leadsto discounted payoffs vi of a certain derivative V . In view of Section 2.2, by the Lawof Large Numbers, we obtain that the sample mean v is a consistent estimator for thederivative price V0:

V0 =S0

0

S0T

EQVT ≈S0

0

S0T

v =1

S0T

S00

m

m∑i=1

vi.

Moreover, the Central Limit Theorem provides a confidence interval, using the samplestandard deviation s. The MC error ε is given by

ε ≈ s√m.

Example B.1. Path dependent derivatives, for example Asian options (Appendix A),are hard to price due their complex structure. However, having simulated paths, thederivative’s payoffs are easily computed. Computing the MC price of this derivative hasthus become relatively simple by taking the average payoff.

The MC method is powerful, since it enables to price any path-dependent derivative inany model. In addition, when using sufficient samples, an accurate estimate of the errorcan be given. A drawback is that in many cases is it hard to make the error small, as the

80

error only decreases with factor√m as the amount of samples increases. Fortunately,

there exists methods to reduce the error by influencing s, known as variance reductiontechniques, which will discussed in B.1.2.

A second drawback of the MC method, is that the method is not appropriate forpricing American-style derivatives. This problem as well as a solution, developed byLongstaff and Schwartz, are discussed in Section B.2.

A third disadvantage, which in fact all methods in this chapter have, is that the sampleneeds to be taken from a risk-neutral distribution. This distribution can be non-trivialto compute, and the model needs to be complete, in order to have a unique risk-neutralmeasure, as discussed in Section 2.2.

B.1.2 Variance-reduction

Several variance-reduction methods are developed in order to improve the MC method’saccuracy, by reducing the sample standard deviation s. This thesis briefly introduces anti-thetic variates, control variates and importance sampling. For a more extensive discussionof variance-reduction techniques, please refer to Glasserman [8, ch. 29] or Willmot [42,ch. 4].

Antithetic variates

The antithetic variates method reduces variance by reflecting the sample variables usingthe stock’s cumulative distribution F :

s′i = F−1(1− F (si)).

This way, we obtain m extra samples s′i that follow the distribution of S as well. Clearly,F must be invertible in order to apply this method. To simplify notation, we assumethat the interest rate is zero, such that the prices are just expectations. The price cannow be estimated by

V0 ≈1

2m

(m∑i=1

vi +m∑i=1

v′i

)=

1

m

(m∑i=1

vi + v′i

2

).

Now we calculate our sample variance:

Var

(1

m

m∑i=1

vi + v′i

2

)≈ 1

m

m∑i=1

Var(V + V ′)

2= Var(V ) (1 + Corr(V, V ′)) .

So we indeed have a variance reduction if V and V ′ have negative correlation. Note thatthere MC error has reduced a factor

√2 in addition, due the fact that twice the amount

of samples are used. However, one should take in account that the computations of thes′i will cost some additional computation time as well.

Example B.2. Suppose we value a call option in the Black-Scholes model using MC.Clearly, V and V ′ have negative correlation, as a higher stock leads to a higher payoff.The samples si can easily be reversed, using

si = S0 exp((r − σ2/2)T + σZi),

s′i = S0 exp((r − σ2/2)T − σZi).

81

Where the Zi are i.i.d. standard normal distributed.

Control variates

The control variate method uses a derivative U from which the price is known, and has asimilar payoff as V . For any fixed b ∈ R, the derivative’s price can then be estimated by

V0 ≈1

m

m∑i=1

vi − b

(1

m

m∑i=1

ui − E[U ]

).

Clearly, this is a consistent estimator. Moreover, its variance is given by:

Var (V − b(U − EU)) = Var(V ) + b2Var(U)− 2bStd(V )Std(U)Corr(V, U).

It can now be shown that, choosing the optimal b∗ = Cov(V,U)Var(U)

, that the variance is

reduced with factor 1−Corr(V, U)2. Thus, the amount of reduction depends on how wellthe payoffs of U and V are correlated.

Example B.3. Suppose one would like to price an Asian option V in the Black-Scholesmodel, which has payoff:

VT =

(1

l

l∑i=1

Si −K

)+

.

One can use a set of call options as a control variate, since the Black-Scholes formula canbe used for the price and the payoff is highly correlated with the Asian option price:

VT ≈ UT =1

l

l∑i=1

(Si −K)+ .

Importance sampling

Importance sampling changes the probability measure form which the paths are simulated.Suppose that the futures stock price S has a density f and one wishes to calculated theprice of a derivative:

Ef [V (S)] =

∫V (s)f(s)ds.

Here we denote Ef since we are taking expectation with respect to the density f . Supposewe have another density g which is positive wherever f is positive. We can now expressthe derivative’s as a expectation with respect to g:

Eg[V (S)f(S)

g(S)] =

∫V (s)

f(s)

g(s)g(s)ds = Ef [V (S)].

So we can perform a MC simulation under the measure of g. The density g should bechosen in such way that the MC error is reduced. This is generally the case when g hashigher density than f at places where the derivative’s payoff is high. Thus, the methodgreatly relies on a clever change of measure.

82

B.2 Least Squares Method (LSM)

In this section we will briefly introduce the Least Squares Method (LSM) for two reasons.First, the method illustrates how more complex derivatives can be prices numerically.Second, the HMC method is inspired by LSM and both methods have similarities. BothLSM and HMC make use of backward induction and regressions using basis functions.

The MC method is not directly applicable to American-style derivatives. These arederivatives, which have the optionality to be exercised earlier than the maturity date(explained in more detail in appendix A). The difficulty when pricing American-stylederivatives, is that it is not directly clear what the optimal exercise strategy is. A (naive)solution is to do extra simulations at the different time points of the simulations paths,in order to estimate the value of the derivative if it is not exercised. In practice this is,however, not realizable, as a enormous amount of simulations are needed to have suffi-cient accuracy. The Least Squares Method is developed to overcome this problem.

We consider a derivative that can be exercised at times 0 < t1, . . . , tk = T . LetVi denote the exercise value of the derivative at time ti, such that Vi is σ(St1 , . . . , Sti)-measurable with 1 ≤ i ≤ k. The value of the derivative can now be written, usingstopping times, as

V0 = sup

EQ Vτ

S0τ

∣∣∣∣ τ is a stopping time with values in t1, . . . , tk

.

It is, however, not clear what the optimal stopping time τ is. Therefore, a differentapproach without the use of stopping times is desirable. We are going to use the followingbackward induction:

Vi = max(Vi,S0i

S0i+1

EQi [Vi+1]),

VT = VT .

Hence, at each time ti the exercise value Vi is compared with the discounted continuationvalue EQ

i [Vi+1]. Note that both values are a function of Si, so we can write Vi(Si) forthe continuation value at time ti. The continuation value depends on the distribution ofthe future values Vi+1 as well, which makes it generally difficult to compute. The pricingmethod we are going to describe, is based on an approximation of the continuation value.Having continuation values for any stock price at any time instant, it becomes trivialwhether the option should be exercised or not in each state.

We are going to follow Longstaff and Schwartz’s approach, known as the Least SquaresMethod , which is used to price an American-style derivative [43]. As Longstaff et al’spaper, this section focuses on pricing an option, but the method can easily be general-ized to more complex derivatives. Basis functions are used in order to approximate thecontinuation values:

Vi(Si) ≈l∑

i=1

aiLi(Si).

83

Here a set of l basis functions Li is used, for example certain polynomials:

Li(s) = si−1.

Appendix C discusses different types of basis functions in more detail. The weights aiare computed by a linear least squares optimization, using sufficiently large amount ofMonte Carlo simulation of the stock paths. Having determined the continuation valuesVi at time ti for any stock price, one can compute the option values Vi. This can be usedfor calculating the continuation values Vi−1 at time ti−1, and so on. Having computedcontinuation values for any stock price at each time instant, the stopping time on anyMC path can be computed. A standard MC simulation can now be used to determinethe option’s value.

B.3 Finite Difference (FD) Method

The finite difference (FD) method is used for numerically solving partial differential equa-tions (PDE). Although we do not use PDEs in this thesis, since we consider discrete timeinstants, we use a scheme similar to FD to compare HMC results with a numerical schemewithout basis functions in Section 4.3.3 (please refer to literature on the Lattice Method).

A FD application for pricing derivatives is firstly introduced by Schwartz [44]. Themethod is illustrated in this section using the Black-Scholes model on a call option. Wesaw in Section 2.3, that in the Black-Scholes case there exists a PDE for the option price:

∂V

∂t+ rS

∂V

∂S+σ2

2S2∂

2V

∂S2− rV = 0.

The methods computes option prices Vi,j on a grid, consisting of time instants 0 =t0, t1, . . . , tN = T and stock price increments Smin = S0, S1, . . . , SM = Smax. For nota-tional convenience, we take constant time instants ti − ti−1 = δt and stock price valuesSj −Sj−1 = δS, although non-constant increments could improve the performance of themethod, since certain stock prices are more likely than others. At time T , the value ofthe derivative is known, so we have:

VN,j = max(Sj −K, 0), j = 0, 1, . . . ,M. (B.1)

If Smin and Smax are, respectively, sufficiently small and large, two different borders ofthe grid can be estimated as well:

Vi,0 ≈ 0 i = 0, 1, . . . , N, (B.2)

Vi,M ≈ Smax −Ke−r(T−ti) i = 0, 1, . . . , N. (B.3)

In order to compute option prices on the other points of the grid, a discrete approximationof the PDE used. The commonly used schemes are Explicit Euler, Implicit Euler andCrank-Nicolson, each comprises its own advantages and disadvantages. We illustrate theFD method with the Explicit Euler scheme, which is given by:

Vi,j − Vi−1,j

δt+ rSj

Vi,j+1 − Vi,j−1

2δS+

1

2σ2S2

j

Vi,j+1 − 2Vi,j + Vi,j−1

(δS)2− rVi,j = 0.

84

Using this equation, option prices Vi−1,j at time ti−1 can be expressed in terms of optionprices at time ti, which allows for backward induction. Using the above three boundaryconditions (Equations (B.1),(B.2) and (B.3)), option prices can be computed on the wholegrid. An advantage of the explicit Euler scheme is that there is an explicit expressionfor the option price at each grid point. Both the Implicit Euler- and the Crank-Nicolsonscheme, in contrast, are solved by matrix inversions, which make them computationallyless efficient. A disadvantage of the Explicit Euler scheme is that wrong choices of incre-ment sizes δt and δS can result in an unstable scheme, which causes a chaotic surface ofoption prices, instead of a smooth surface.

The main benefit of the FD method is that a full grid of option prices is generated. Fordetermining prices at points which are not necessarily on the grid, linear interpolation canbe used. Another feature is that pricing American derivatives can easily be implemented,by adjusting for the early exercise value at each computed grid point. A drawback isthat, in order to be able to apply the method, a PDE needs to be present. This is oftennot the case, for example if the model is not Markov. Moreover, the method gets harderto apply and computationally inefficient for higher order state vectors.

85

Appendix C

Basis functions

The HMC method (Chapter 4) and the LSM method (Section B.2) both make use ofbasis functions . In this thesis there is made use of polynomial, linear and cubic basisfunctions. We briefly discuss how these basis functions are implemented. More detaileddiscussions of linear and cubic functions can be found in [25, Appendix B].

In this thesis, basis functions are used as follows. Suppose there are m observations(Xi, yi) ∈ X × R for a certain set X . One can model this by having a set of l basisfunctions Lk : X → R. An OLS regression can then be performed to determine weightsβ ∈ Rl for each basis function:

β = arg minβ∈Rl

(X ′β − y)2.

Here, X ′ is given by:

X ′ =

L1(X1) . . . Ll(X1)...

...L1(Xm) . . . Ll(Xm)

.

In Section 4.5.2, the formulation of the OLS with basis functions in the case of HMC isdiscussed.

We now discuss the three different types of basis functions used in this thesis. In thisthesis we consider the cases X = R or X = Rn. In Figure 4.22 the basis functions areillustrated.

Polynomial basis function

Suppose there is only one observable, so Xi ∈ R. Then we can define a set of polynomialbasis functions up to a certain order a:

(Lk(x)) =xk−1|1 ≤ k ≤ a+ 1

.

There is no use in considering certain polynomial sequences, like for example Hermitepolynomials, since these can be written in terms of the basis functions we use here.Orthogonality plays no role in our application.

86

If there are multiple variables observed, we can define a polynomial set of basis func-tion by specifying the order a and the correlated order b:

(Lk(x)) = xb11 xb22 . . . xbnn |0 ≤ bi ≤ b,∀1 ≤ i ≤ n ∪ xij|b < i ≤ a, 1 ≤ j ≤ n.

Linear Basis functions

We only discuss the 1-dimensional case where there is only 1 variable observed, butthe approach can be generalized for multiple dimensions. The linear model uses nodesN1, . . . , Nl and interpolates between these nodes. The basis basis functions Lk for 1 ≤k ≤ l are then given by:

Lk(x) =

x−Nk−1

Nk−Nk−1Nk−1 ≤ x ≤ Nk

Nk+1−xNk+1−Nk

Nk−1 ≤ x ≤ Nk

0 else.

Cubic spline basis functions

Cubic basis function are based on a cubic spline model. This assumes piecewise polyno-mials of which values and first order derivatives coincide at nodes N1, . . . , Nl. Again, weonly discuss the 1-dimensional case. Generalizing to higher dimensions is difficult. Foreach node Nk there are two basis functions Lk and Lk defined:

Lk(x) =

(x−Nk−1)2

(Nk−Nk−1)3(2(Nk − x)(Nk −Nk−1) Nk−1 ≤ x ≤ Nk

(x−Nk+1)2

(Nk+1−Nk)3(2(x−Nk)(Nk+1 −Nk) Nk ≤ x ≤ Nk+1

0 else.

Lk(x) =

(x−Nk−1)2(x−Nk)

(Nk−Nk−1)2Nk−1 ≤ x ≤ Nk

(x−Nk+1)2(x−Nk)

(Nk+1−Nk)2Nk ≤ x ≤ Nk+1

0 else.

87

Appendix D

Option pricing in the Heston model

In Section 2.4, the Heston Model is introduced. The way option prices are computedin Section 4.4 were, however omitted. For our simulations we used Fourier Inversion,a common pricing method introduced in Stein and Stein [11] and Heston [6]. In thisAppendix, the pricing of a call option is rigorously discussed. Here we assume that theused Heston model has parameters such that the model describes a risk-neutral measureand that the interest rate r is constant, so

C0 = e−rTEQCT .

A central element of the approach is the log-stock’s characteristic function:

φ(u) = E [exp(iu logST )] .

For an expression and derivation of the stock’s characteristic function in the HestonModel, please refer to Heston [6] or Bakshi, Cao and Chen [45]. Clearly the characteristicfunction depends on the chosen model. Furthermore, the time T stock value distributioncompletely determines the characteristic function and vice versa. Therefore, derivativeprices can be expressed in terms of the characteristic function. The (undiscounted) calloption price is given by:

E[(ST −K)+

]= R(F ) +

1

πRe

∫ ∞0

e−i(v−iα)k φ(v − i(α + 1))

−(v − i(α + 1))(v − iα)dv. (D.1)

Here F denotes the stock’s forward price erTS0 and R(F ) is given by:

R(F ) = F · 1α≤0 −K · 1α≤−1 −1

2(F · 1α=0 −K · 1α=−1) .

Theoretically, at least in the European option case, equation (D.1) holds for any α ∈ R.However, the behavior of the integral, which needs to be evaluated numerically, is greatlydependent of the choice of α. For a detailed discussion on pricing derivatives using FourierInversion, please refer to Lord [46]. In this thesis, for the calculation of option prices inthe Heston model (Section 4.4), we use α = −1

2, as suggested by Lewis [47].

88

References

[1] P.d. Iseger and J. Potters. Bridging the gap - arbitrage free valuation of derivativesin ALM. Journal of Financial Transformation, 25:91–94, 2009.

[2] M. Potters, J.P. Bouchaud, and D. Sestovic. Hedge Your Monte Carlo. Risk Maga-zine, pages 133–136, 2001.

[3] F. Black and M. Scholes. The Pricing of Options and Corporate Liabilities. Journalof Political Economy, 81(3):637–654, 1973.

[4] H. Follmer and A. Schied. Stochastic Finance: An Introduction in Discrete Time.Walter de Gruyter, 3rd edition, 2011.

[5] T. Bjork. Arbitrage Theory in Continuous Time. Oxford University Press, 1998.

[6] S.L. Heston. A Closed-Form Solution for Options with Stochastic Volatility withApplication to Bond and Currency Options. The Review of Financial Studies, 6(2):327–343, 1993.

[7] R. Lord, R. Koekkoek, and D. Dijk. A Comparison of Biased Simulation Schemesfor Stochastic Volatility Models. Quantitative Finance, 10(2), 2010.

[8] P. Glasserman. Monte Carlo Methods in Financial Engeneering. Springer, 2003.

[9] A.v. Haastrecht and A. Pelsser. Efficient, almost exact simulation of the Hestonstochastic volatility model. International Journal of Theoretical and Applied Fi-nance, 31(1), 2010.

[10] M. Romano and N. Touzi. Contingent Claims and Market Completeness in a Stochas-tic Volatility Model. Mathematical Finance, 7(4):399–410, 1997.

[11] E. Stein and J. Stein. Stock price distributions with stochastic volatility - an analyticapproach. Review of Financial Studies, 4, 1991.

[12] P. Artzner, F. Delbaen, J.M. Eber, and D. Heath. Coherent Measures of Risk.Mathematical Finance, 9(3):203–228, 1999.

[13] R.T. Rockafellar, S.P. Uryasev, and M. Zabarankin. Deviation Measures in RiskAnalysis and Optimization. Working paper, 2002.

[14] J.P. Bouchaud and M. Potters. Theory of Financial Risk and Derivative Pricing:From Statistical Physics to Risk Management. Cambridge University Press, 2000.

89

[15] D. Bertsimas, L. Kogan, and A.W. Lo. Hedging Derivative Securities and IncompleteMarkets: an Epsilon-Arbitrage Approach. Operations Research, 49(372-397), 1999.

[16] A. Cerny and S.D. Hodges. The Theory of Good-Deal Pricing in Financial Markets.Mathematical Financie - Bachelier Congress, 2000.

[17] M. Schweizer. Mean-variance hedging for general claims. The Annals of Probability,2:171–179, 1992.

[18] D. Duffie and H.R. Richardson. Mean-Variance Hedging in Continuous-Time. Annalsof Applied Probability, 1:1–15, 1991.

[19] L.H. Ederington. The Hedging Performance of the New Futures Markets. TheJournal of Finance, 34(1), 1979.

[20] P. Wilmott. Discrete Charms. Risk Magazine, 7:48–51, 1994.

[21] P. Wilmott. Wilmott forums. http://www.wilmott.com/messageview.cfm?catid=4&threadid=58839, Jul 2001. Accessed: 2015-05.

[22] M. Schweizer. Variance-Optimal Hedging in Discrete Time. Mathematics of Opera-tions Research, 20(1):1–32, 1995.

[23] M. Schweizer. Approximation pricing and the variance-optimal martingale measure.The Annals of Probability, 24(1):206–236, 1996.

[24] Petrelli, A. and Balachandran, R. and Zhang, J. and Siu, O. and Chatterjee, R. andKapoor, V. Optimal Dynamic Hedging of Multi-Asset Options. 2009.

[25] A.R. Petrelli, R. Balachandran, O. Siu, R. Chatterjee, Z. Jun, and V. Kapoor.Optimal Dynamic Hedging of Equity Options: Residual-Risks, Transaction-Costs,& Conditioning. , 2010.

[26] R.H. Strotz. Myopia and inconsistency in dynamic utility maximization. The Reviewof Economic Studies, 23(3):165–180, 1956.

[27] S. Basak and G. Chabakauri. Dynamic mean-variance asset allocation. Review ofFinancial Studies, 23(8):2970–3016, 2010.

[28] S. Basak and G. Chabakauri. Dynamic hedging in incomplete markets: A simplesolution. Review of Financial Studies, 25(6):1845–1896, 2012.

[29] W.H. Highleyman. The design and analysis of pattern recognition experiments. TheBell Systems Technical Journal, pages 723–744, 1962.

[30] M. Stone. Cross-validatory choice and assessment of statistical predictions. Journalof the Royal Statistical Society, 36:111–147, 1974.

[31] H. White. A Heteroscedasticity Consistant Covariance Matrix Estimator and aDirect Test for Heteroscedasticity. Econometrica, 48:817–838, 1980.

90

[32] H. Akaike. Information theory and an extension of the maximum likelihood principle.Second international symposium on information theory, pages 267–281, 1973.

[33] G. Schwarz. Estimating the dimension of a model. Annals of Statistics, pages 461–464, 1978.

[34] De Nederlandsche Bank. Pension fund feasibility test. http://www.toezicht.dnb.nl/en/2/6/51-233035.jsp, june 2015. Accessed: 2015-07.

[35] R.S. Koijen, T.E. Nijman, and B.J. Werker. When can life cycle investors benefitfrom time-varying bond risk premia? Revieuw of Financial Studies, 23(2):741–780,2010.

[36] Langejan, T.W. and Gelauff, G.M.M. and Nijman T.E. and Sleijpen O.C.H.M. andSteenbeek, O.W. Advies Commissie Parameters. febr 2014.

[37] N. Draper. A Financial Market Model for the US and the Netherlands. http://www.cpb.nl/publicatie/financial-market-model-us-and-netherlands, may 2012.

[38] De Nederlandsche Bank. Scenarioset haalbaarheidstoets pensioenfondsen. http:

//www.toezicht.dnb.nl/2/50-233246.jsp, july 2015. Accessed: 2015-07.

[39] De Nederlandsche Bank. Model en modelparameters onderliggend aan scenarioset,june 2015. Accessed: 2015-07.

[40] D. Filipovic. Term-Structure Models. Springer, 2009.

[41] P.P. Boyle. Options: A Monte Carlo Approach. Journal of Financial Economics, 4:323–338, 1977.

[42] Wilmott, P. Paul Wilmott Introduces Quantitative Finance. John Wiley & Sons,2nd edition, 2007.

[43] F.A. Longstaff and S.S. Schwartz. Valuing American Options by Simulation: ASimple Least-Squares Approach. The Review of Financial Studies, 14:113–147, 2001.

[44] E.S. Schwartz. The Valuation of Warrants: Implementing A New Approach. Journalof Financial Economics, 4:79–93, 1977.

[45] G. Bakshi, C. Cao, and Z. Chen. Emperical performance of alternative option pricingmodels. Journal of Finance, 52:2003–2049, 1997.

[46] R. Lord. Efficient pricing algorithms for exotic derivatives. PhD thesis, TinbergenInstitute, 2008.

[47] A. Lewis. A simple option formula for general jump-diffusion and other exponentialLevi processes. OptionCity.net, 2001.

91

Index

Akaike information criterion, 49American options, 78American-style, 83antithetic variates, 81arbitrage, 9arbitrage free market, 9arbitrage free price, 9arbitrage opportunity, 9Asian option, 78atom, 11attainable, 9autocorrelation, 48

bank account, 79bank account process, 12barrier option, 79Basis functions, 83basis functions, 86basis risk, 24Basket options, 79Bayesian information criterion, 49Binary options, 78Black-Scholes, 11Black-Scholes equation, 12Black-Scholes formula, 12Black-Scholes model, 8

call option, 78cash wealth index, 64characteristic function, 88coherent risk measure, 16complete market, 10contingent claim, 9continuation value, 83control variate, 31, 82Cross-validation, 46Cubic basis functions, 51cubic spline, 87cubic splines, 51

delta hedging, 19, 20derivative, 9deviation risk measure, 15discrete hedging, 23diversification, 17dynamic hedging, 23Dynamic Programming, 7

European options, 78

fair price, 9Finite Difference, 7

gamma hedging, 19good-deal-pricing, 17Greeks, 76

hedge, 18Hedged Monte Carlo, 7, 25hedging, 7Heston Model, 13Heston model, 8het Financieel Toetsingskader (FTK), 63heteroscedasticity, 49HMC, 7hybrid, 67, 78

implied volatility, 14Importance sampling, 82information criteria, 49Interest rate swaps, 79

KNW model, 63

lack-of-fit, 46, 49Least Squares Method, 83Least Squares Monte Carlo, 25Linear basis functions, 51LSM, 25

Markov property, 9

92

martingale measure, 10Matlab, 54mean squared error, 19mean-squared risk measure, 22Monte Carlo, 7, 25, 80multicollinearity, 48multivariate Ornstein-Uhlenbeck process, 63

nodes, 51

OLS, 46Optimal control, 7optimal hedge, 18optimal hedge ratio, 20Ordinary Least Squares, 46over-fitting, 46, 49

perfect hedge, 9portfolio process, 9profit and loss, 6put option, 78

Risk, 15risk, 15risk measure, 15risk-neutral measure, 10

Schwarz criterion, 49self-financing, 9self-financing in expectation, 25, 28SF, 9SFE, 25, 28short rate, 79simple contingent claim, 20standard errors, 48Stochastic Volatility Model, 13subhedge, 9superhedge, 9Swaption, 67

time-inconsistent, 32Training set, 46

Validation set, 46Value at Risk (VaR), 16vanilla, 78variance-optimal hedge, 18volatility smile, 14

wealth process, 9

93