Free lunches and martingales in convex markets

Teemu Pennanen∗

April 11, 2007

Abstract

This paper presents a stochastic model for discrete-time trading in fi-nancial markets where trading costs are given by convex cost functions andportfolios are constrained by convex sets. The model does not assume theexistence of a cash account. In addition to classical frictionless marketsand markets with transaction costs or bid-ask spreads, our frameworkcovers markets with nonlinear illiquidity effects for large instantaneoustrades. In the presence of nonlinearities, the classical notion of free lunchturns out to have two equally meaningful generalizations, a marginal anda scalable one. Using techniques of convex analysis we give martingalecharacterizations of both.

Key words Illiquidity, Portfolio constraints, Free lunches, Martingales,Convex analysis

1 Introduction

When trading securities, marginal prices depend on the quantity traded.This is obvious already from the fact that different marginal prices areassociated with purchases and sales. Marginal prices depend not only onthe sign (buy/sell) but also on the size of the trade. When the trade affectsthe instantaneous marginal cost but not the costs of subsequent trades,the dependence acts like a nonlinear transaction cost. Such short-termprice impacts have been studied in several papers recently; see for exampleCetin, Jarrow, Protter [5], Rogers and Singh [33], Cetin and Rogers [6] andAstic and Touzi [1] and their references. Short-term effects are differentin nature from feedback effects where large trades have long-term priceimpacts that affect the marginal prices of transactions made at later times;see Kraus and Stoll [20] for comparison and empirical analysis of short-and long-term liquidity effects. Models for long-term price impacts havebeen developed e.g. in Platen and Schweizer [28], Bank and Baum [3].Krokhmal and Uryasev [21] and Kuhn [22] have proposed models thatencompass both short and long run liquidity effects.

∗Department of Business Technology, Helsinki School of Economics, PL 1210, 00101

Helsinki, Finland, pennanen@hse.fi

1

This paper presents a discrete time model for a general class of shortrun liquidity costs. We model the total costs of purchases (positive ornegative amounts) by random convex functions of the trade size. Con-vexity allows us to drop all assumptions about differentiability of the costfunction so that discontinuities in marginal costs can be modeled. Thisis essential e.g. in ordinary double auction markets, where marginal costsof market orders (instantaneous trades) are piecewise constant functions.Indeed, the marginal cost of a market order always has a jump of the sizeof the bid-ask spread at the origin and additional jumps at points whichcorrespond to limit orders of different prices. In addition to double auc-tion markets, our model covers traditional frictionless markets and manyother market models studied in the literature. Besides being natural fromthe modeling point of view, convexity has many important consequencesthat facilitate the analysis of a model.

Nonsmooth convex cost functions have been studied e.g. in Jouini andKallal [16] and in Kaval and Molchanov [19] but in these works, the costfunctions were sublinear, i.e. convex and positively homogeneous. In theterminology of microeconomic theory, positive homogeneity correspondsto markets with constant returns to scale, whereas convexity correspondsto diminishing returns to scale. In the case of diminishing returns toscale, the traditional notions of arbitrage and free lunch turn out havetwo natural generalizations. The first one is related to the possibility ofproducing something out of nothing and the second one to the possibilityof producing arbitrarily much out of nothing. Accordingly, we introducethe conditions of no marginal free lunch and no scalable free lunch. In thecase of constant returns to scale, as in [16, 19] and in traditional linearmodels (see Delbaen and Schachermayer [10] for a comprehensive treat-ment of the linear case), the two notions coincide. In general, however,a market model can allow for marginal free lunches without allowing forscalable ones. We extend the fundamental theorem of asset pricing togeneral convex market models by giving martingale characterizations ofboth notions of free lunch. In doing this, we do not assume the existenceof a cash account (a perfectly liquid asset that can be traded withoutrestrictions) and we allow for general convex portfolio constraints. Muchas in the classical (perfectly liquid and unconstrained) market models,the free lunch conditions turn out to be equivalent to the existence of“deflators” which turn certain marginal price process into “generalizedmartingales”. In the case of marginal free lunch, the marginal price pro-cess corresponds to mark-to-market prices, whereas in the case of scalablefree lunch, the marginal price is contained in the closure of the range ofall possible marginal prices.

Above, a generalized martingale refers to a stochastic process X =(Xt)

Tt=0 such that E[Xt+1 | Ft] − Xt ∈ Kt, for some adapted sequence

of closed convex cones Kt. When Kt = 0, this reduces to the usualnotion of a martingale. In the case of perfectly liquid markets with a cashaccount, no arbitrage and/or no free lunch conditions in the presenceof portfolio constraints have been characterized in terms of generalizedmartingales e.g. in Pham and Touzi [27], Napp [25], Evstigneev, Schurgerand Taksar [14] and Rokhlin [34, 36]. In [14, 34], the convex cones Kt werethe normal cones to non conical convex portfolio constraints at the origin.

2

In this paper, we obtain a similar characterization for the marginal no freelunch condition for general convex price processes and without assumingthe existence of a cash account. Moreover, we obtain a characterizationof the new notion of no scalable free lunch in which the normal cone isreplaced by the range of the normal cone mapping of the constraint set.

Many of the concepts and techniques used in this paper come fromconvex analysis, a field of mathematics that grew out of the study of op-timization and other extremum problems. Most of the techniques usedhere can be found e.g. in the monographs of Rockafellar [30, 31], Ekelandand Temam [13], Aubin [2] and Rockafellar and Wets [32]. Convexity wasused in a deterministic framework in the analysis of fixed income mar-kets in Dermody and Rockafellar [11, 12], where it was also observed thatprices behave in a nonlinear but convex way. Convex price processes (in aslightly less general form than here) in the presence of a cash account andwithout portfolio constraints were recently analyzed in an unpublishedmanuscript [26]. Convexity has played an essential role also in the cur-rency market models of Kabanov [18], Delbaen, Kabanov and Valkeila [9],Schachermayer [37] and in the recent model of Astic and Touzi [1]. Inthese models, feasible trades are described in terms of convex “solvencyregions” and arbitrage concepts are defined in terms of contingent claimswith physical delivery.

The rest of this paper is organized as follows. Starting with an ex-ample from double auction markets, Section 2 introduces general convexprice processes together with two sublinear price processes that describethe local and global behavior of a general process. Section 3 defines gen-eral convex portfolio constraints together with two conical portfolio con-straints that describe the local and global behavior of a general portfolioconstraint. Section 4 defines self-financing trading strategies and discusseslocal and global forms of arbitrage in terms of general contingent claimsin markets without a cash account. Section 5 defines the marginal andscalable free lunches and characterizes them in terms of deflators. Sec-tion 6 contains the main results which characterize free lunches in termsof martingales. Section 7 gives some applications of the martingale char-acterizations. Some useful results from convex analysis are collected inthe Appendix.

2 Convex price processes

Most modern stock exchanges are based on the so called double auctionmechanism to determine trades between market participants. In such anexchange, market participants submit offers to buy or sell shares withincertain limits on the unit price and quantity. The trading system main-tains a record, called the “limit order book”, of all the offers that have notbeen offset by other offers. At any given time, the lowest unit price overall selling offers in the limit order book (the “ask price”) is thus greaterthan the highest unit price over all buying offers (the “bid price”). Whenbuying in such a market, only a finite number of shares can be boughtat the ask price and when buying more, one gets the second lowest priceand so on. The marginal cost of buying is thus a positive, nondecreas-

3

ing, piecewise constant function of the number of shares bought. Whenselling shares, the situation is similar and the marginal revenue of sellingis a positive, nonincreasing, piecewise constant function of the number ofshares sold.

Interpreting negative purchases as sales, we can incorporate the in-stantaneous marginal cost and marginal revenue functions into a singlefunction x 7→ s(x) giving the marginal cost of buying a positive or a neg-ative number x of shares at a fixed point in time. Since the bid pricelimx 0 s(x) is lower than the ask price limx 0 s(x), s is a nonnegativenondecreasing function. If x is greater than the total number of sharesfor sale we set s(x) = +∞. The interpretation is that one cannot buymore than the total supply no matter how much one is willing to pay. Onthe other hand, if x is less than the negative of the total demand we sets(x) = 0 with the interpretation that one can not gain additional revenueby selling more than the total demand.

Given a marginal cost function s : R → [0, +∞] representing a limitorder book, we can define the associated total cost function

S(x) :=

Z x

0

s(w)dw,

which gives the total cost of buying x shares. By Theorem 24.2 of Rock-afellar [30], the total cost S : R

J → R ∪ +∞ associated with a non-decreasing marginal cost s : R 7→ [0, +∞] is an extended real-valued,lower semicontinuous convex function which vanishes at 0. If s happensto be finite everywhere, then by [30, Theorem 10.1], S is not only lowersemicontinuous but continuous.

In the above situation, the instantaneous marginal cost is nonnegative,or equivalently, the total cost is nondecreasing as a function of sharesbought. This corresponds to “free disposal” of the traded good, the stock.In the market model that we are about to present, the total cost is allowedto be a general lower semicontinuous convex function that vanishes at theorigin. In particular, it allows negative marginal prices in situations wherefree disposal is not a valid assumption. Moreover, instead of a single assetwe will allow for a finite set J of assets and the total cost will be a functionon the Euclidean space R

J of portfolios.Consider an intertemporal setting, where price functions are observed

at discrete points in time t = 0, . . . , T . Let (Ω,F , P ) be a probability spacewith a filtration (Ft)

Tt=0 of sub-σ-algebras of F giving the information

available to an investor at each time t = 0, . . . , T . For simplicity, we willassume that each Ft is P -complete. The Borel σ-algebra on R

J will bedenoted by B(RJ ).

Definition 1 A price process is a sequence S = (St)Tt=0 of extended real-

valued functions on RJ × Ω such that for t = 0, . . . , T ,

1. the function St(·, ω) is lower semicontinuous and vanishes at 0 forevery ω ∈ Ω,

2. St is B(RJ ) ⊗Ft-measurable.

A price process S is said to be convex, nondecreasing, nonlinear, polyhe-dral, sublinear, linear, . . . if the functions St(·, ω) have the correspondingproperty for every ω ∈ Ω.

4

The interpretation is that buying a portfolio xt ∈ RJ at time t and

state ω costs St(xt, ω) units of cash. The measurability property impliesthat if the portfolio xt is Ft-measurable then the cost ω 7→ St(xt(ω), ω)is also Ft-measurable (see e.g. [32, Proposition 14.28]). This just meansthat the cost is known at the time of purchase.

In Definition 1, we have used some terminology from convex analy-sis; see the appendix. The measurability property together with lowersemicontinuity in Definition 1 mean that St is an Ft-measurable normalintegrand in the sense of Rockafellar [29]; see also Rockafellar and Wets [32,Chapter 14]. This has many important implications which will be essen-tial in variational analysis of trading strategies. We emphasize that wepose no smoothness assumptions on the the functions St(·, ω). Convexityas well as measurability properties are well preserved under various kindsof operations, such as conjugation or subdifferentiation, which will playmajor roles in our analysis.

Given a convex price process S, we define

S′t(x,ω) = lim inf

x′→xinfα>0

St(αx′, ω)

α,

S∞t (x,ω) = sup

α>0

St(αx, ω)

α.

These are lower semicontinuous convex functions known as the subderiva-tive of St(·, ω) at the origin and the horizon function of St(·, ω), respec-tively; see [32] or the appendix. It follows from the convexity of St that

St(α1x, ω)

α1≤

St(α2x,ω)

α2∀x ∈ R

J

whenever α1 ≤ α2, so it is possible to define S′t and S∞

t also as certainlimits of the function x 7→ St(αx, ω)/α as α tends to zero and infinity,respectively; see [32]. In fact, S′

t(·, ω) is the lower semicontinuous hull ofthe directional derivative function of St(·, ω) at the origin; see [30, Theo-rem 23.1]. Whereas S′

t describes the local behavior of St near the origin,S∞

t describes the behavior of St infinitely far from 0. Under convexity,our definitions above coincide with the definitions of subderivative andhorizon functions in [32]; see Theorem 3.21 and Proposition 8.21 of [32].

If S is sublinear (i.e. convex and positively homogeneous), we simplyhave S′

t(x, ω) = S∞t (x,ω) = St(x,ω). In general, we have the following.

Proposition 2 Let S be a convex price process. The sequences S ′ =(S′

t)Tt=0 and S∞ = (S∞

t )Tt=0 define sublinear price processes in the sense

of Definition 1. Whereas S′ is the greatest sublinear price process lessthan S, S∞ is the least sublinear price process greater than S.

Proof. The properties in the first condition of Definition 1 are easy conse-quences of convexity; see Proposition 8.21 and Theorem 3.21 of [32]. Themeasurability properties follow from Theorem 14.56 and Exercise 14.54 of[32].

Given a convex price process S = (St)Tt=0 and an x ∈ R

J , the set ofsubgradients

∂St(x,ω) := v ∈ RJ |St(x

′, ω) ≥ St(x, ω) + v · (x′ − x) ∀x′ ∈ RJ

5

is a closed convex set which is Ft-measurable with respect to ω; see [32,Theorem 14.56]. The Ft-measurable set-valued mapping ∂St(x) : ω ⇒ R

J

given by ω 7→ ∂St(x, ω) gives the set of marginal prices at x. In particular,∂St(0) can be viewed as the set of mark-to-market prices which give themarginal prices associated with infinitesimal trades in a market describedby S. In the scalar case (when J is a singleton), ∂St(0, ω) is the closedinterval between the bid and ask prices, i.e. left and right directionalderivatives of St(·, ω) at the origin. If St(·, ω) happens to be differentiableat a point x, we have ∂St(x, ω) = ∇St(x, ω).

Whereas ∂St(0) : Ω ⇒ RJ gives the set of mark-to-market prices, the

mapping rge ∂St : Ω ⇒ RJ given by ω 7→ rge ∂St(·, ω) :=

S

x∈RJ ∂St(x,ω)gives the set of all possible marginal prices an investor may face whentrading at time t in a market described by S. These mappings are relatedto the sublinear price processes S′ and S∞ through the following.

Lemma 3 If S is a convex price process, then

∂St(0, ω) = v ∈ RJ |S′

t(x, ω) ≥ v · x ∀x ∈ RJ,

cl rge ∂St(·, ω) = v ∈ RJ |S∞

t (x,ω) ≥ v · x ∀x ∈ RJ.

In particular, if S is sublinear, we have ∂St(0, ω) = rge ∂St(·, ω).

Proof. The first expression follows directly from convexity (see [30, The-orem 23.2]). By Theorems 13.3 and 23.4 and Corollary 23.5.1 of [30],

S∞t (x, ω) = supx · v | v ∈ dom S∗

t (·, ω)

= supx · v | v ∈ dom ∂S∗t (·, ω)

= supx · v | v ∈ rge ∂St(·, ω),

so the second expression follows from [30, Theorem 13.1].

Besides double auction markets as described earlier, Definition 1 coversvarious more specific situations treated in the literature.

Example 4 If st is an RJ -valued Ft-measurable price vector for each

t = 0, . . . , T , then the functions

St(x,ω) = st(ω) · x

define a linear price process in the sense of Definition 1. This correspondsto a frictionless market (with possibly negative unit prices), where unlim-ited amounts of all assets can be bought or sold for prices st. In this case,S′ = S∞ = S and ∂St(0) = rge ∂St = st.

Proof. This is a special case of Example 5 below.

Example 5 If st and st are RJ -valued Ft-measurable price vectors with

st ≤ st, then the functions

St(x,ω) =X

j∈J

Sjt (x

j , ω),

where

Sjt (xj , ω) =

(

sjt (ω)xj if xj ≥ 0,

sjt (ω)xj if xj ≤ 0

6

define a sublinear price process in the sense of Definition 1. This corre-sponds to a market with transaction costs or bid-ask spreads, where un-limited amounts of all assets can be bought or sold for prices st and st,respectively. This situation was studied in Jouini and Kallal [16]. Wehave S′ = S∞ = S and ∂St(0) = rge ∂St = [st, st]. When s = s, onerecovers Example 4.

Proof. This is a special case of Example 6 below.

Example 6 If Zt is an Ft-measurable set-valued mapping from Ω to RJ ,

then the functionsSt(x, ω) = sup

s∈Zt(ω)

s · x

define a sublinear price process in the sense of Definition 1. This situationwas studied in Kaval and Molchanov [19] (in the case that the mappingsZt have convex compact values in the nonnegative orthant R

J+). We have

S′ = S∞ = S and ∂St(0) = rge ∂St = Zt. When Zt = [st, st] one recoversExample 5.

Proof. The functions St(·, ω) are clearly sublinear and vanish at 0. By[32, Example 14.51], St(x, ω) is also an Ft-measurable normal integrand.

In Examples 4, 5 and 6 the price process S is sublinear (actually linearin Example 4), so S′ = S∞ = S. The following example models illiquidityeffects encountered e.g. in stock exchanges.

Example 7 If st are RJ+-valued Ft-measurable vectors and ϕj are lower

semicontinuous convex functions on R with ϕj(0) = 0, then the functions

St(x,ω) =X

j∈J

sjt(ω)ϕj(x

j)

define a convex price process in the sense of Definition 1. The scalarcase (J is a singleton), with strictly positive s and strictly convex, strictlyincreasing and differentiable ϕj was studied in Cetin and Rogers [6].

If the functions ϕj are real-valued, we have

S′t(x,ω) =

X

j∈J

sjt(ω)ϕ′

j(xj), S∞

t (x,ω) =X

j∈J

sjt(ω)ϕ∞

j (xj),

and

∂St(0) =“

sjt(ω)∂ϕj(0)

”

j∈J, rge ∂St =

“

sjt(ω) rge ∂ϕj

”

j∈J.

Proof. This follows from [32, Corollary 14.46] and [32, Proposition 14.44(d)].

A fundamental similarity between our approach and that in Cetin andRogers [6] is that both are based on convexity and optimization argu-ments. An advantage of Definition 1 is that it allows for nonsmooth (andnonmonotone) cost functions. For example, the total cost function associ-ated with a double auction market is a polyhedral convex function which

7

is nondifferentiable e.g. at the origin (the bid price is always strictly lessthan the ask price).

Example 7 illustrates the rich “calculus” available for convex functions.We refer the reader to the classic monograph [30] for a thorough treatmentof the subject.

3 Convex portfolio constraints

In addition to nonlinearities in prices, one often encounters portfolio con-straints when trading in practice. As in Rokhlin [34], we will considergeneral convex portfolio constraints where at each t = 0, . . . , T the port-folio xt is restricted to lie in a convex set Dt which may depend on ω.

Definition 8 A portfolio constraint process is a sequence D = (Dt)Tt=0

of set-valued mappings from Ω to RJ such that for t = 0, . . . , T ,

1. Dt(ω) is closed and 0 ∈ Dt(ω) for every ω ∈ Ω,

2. the set-valued mapping ω 7→ Dt(ω) is Ft-measurable.

A portfolio constraint process is said to be convex, polyhedral, conical, . . . ifthe sets Dt(ω) have the corresponding property for every ω ∈ Ω.

Given a convex portfolio constraint process D = (Dt)Tt=0, we define

D′t(ω) = cl

[

α>0

αDt(ω),

D∞t (ω) =

\

α>0

αDt(ω).

These are closed convex cones known as the tangent (or contingent) coneof Dt(ω) at 0 and the horizon (or asymptotic) cone of Dt(ω), respectively;see [32]. Since α1Dt ⊂ α2Dt whenever α1 ≤ α2, it is possible to defineD′

t and D∞t also as limits of αDt as α > 0 tends to infinity and zero,

respectively. Whereas D′t describes the local behavior of Dt near the

origin, D∞t describes the behavior of Dt infinitely far from 0. Under

convexity, our definitions above coincide with the definitions of horizonand tangent cones in [32]; see Theorems 3.6 and 6.9 of [32].

If D is conical, we simply have D′t(ω) = D∞

t (ω) = Dt(ω). In general,we have the following.

Proposition 9 Let D be a convex portfolio constraint process. The se-quences D′ = (D′

t)Tt=0 and D∞ = (D∞

t )Tt=0 define conical convex portfolio

constraint processes in the sense of Definition 8. Whereas D′ is the small-est conical portfolio constraint process containing D, D∞ is the largestconical portfolio constraint process contained in D.

Proof. The properties in the first condition of Definition 8 are easy conse-quences of convexity; see Theorems 3.6 and 6.9 of [32]. The measurabilityproperties come from Exercise 14.21 and Theorem 14.26 of [32].

8

Given a convex portfolio constraint process D = (Dt)Tt=0 and an x ∈

RJ , the normal cone

NDt(ω)(x) :=

(

v ∈ RJ | v · (x′ − x) ≤ 0 ∀x′ ∈ Dt(ω) if x ∈ Dt(ω),

∅ otherwise

to Dt at x is a closed convex set which is Ft-measurable with respectto ω; see [32, Theorem 14.26]. The Ft-measurable set-valued mappingNDt

(x) : Ω ⇒ RJ defined by ω 7→ NDt(ω)(x) gives the set of price vectors

v ∈ RJ such that the portfolio x ∈ R

J maximizes the value v · x over allx ∈ Dt. In particular, NDt(ω)(0) gives the set of price vectors v ∈ R

J

such that the zero portfolio maximizes v · x over Dt(ω). Note that whenDt ≡ R

J (no portfolio constraints), we have NDt(x) = 0 for every

x ∈ RJ .

Whereas NDt(0) gives the set of normal vectors to Dt at the origin

the set-valued mapping rge NDt: Ω ⇒ R

J given by ω 7→ rge NDt(ω) :=S

x∈RJ NDt(ω)(x) gives the set of all possible normal vectors associatedwith a portfolio constraint Dt(ω). These mappings are related to theconical portfolio constraint processes D′ and D∞ through the following.

Lemma 10 If D is a convex portfolio constraint process, then

NDt(ω)(0) = v ∈ R |x · v ≤ 0 ∀x ∈ D′t,

cl rge NDt(ω) = v ∈ R |x · v ≤ 0 ∀x ∈ D∞t .

In particular, if D is conical, we have NDt(ω)(0) = rge NDt(ω).

Proof. The first expression follows directly from the definitions of D′ andNDt

. The second can be proved by writing NDt(ω) = ∂δDt(ω)(0) (see theappendix) and following the proof of the second expression of Lemma 3.

Example 11 The classical case without constraints corresponds to Dt(ω) =R

J for every ω ∈ Ω and t = 0, . . . , T . In this case, D′ = D∞ = RJ and

NDt(0) = rge NDt

= 0.

Example 12 Given a closed convex set K ⊂ RJ containing the origin,

the sets Dt(ω) = K define a (deterministic) convex portfolio constraintprocess in the sense of Definition 8. This case has been studied e.g. byCvitanic and Karatzas [7] and Pham and Touzi [27]. We have D′

t ≡ K′,D∞

t ≡ K∞, NDt(0) ≡ NK(0) and rge NDt

≡ rge NK .

In addition to obvious “short selling” constraints, the above model(even with conical K) can be used to model situations where one en-counters different interest rates for lending and borrowing cash. Indeed,this can be done by introducing two separate “cash accounts” whose unitprices appreciate according to the two interest rates and restricting the in-vestments in these assets to be nonnegative and nonpositive, respectively.

In the two examples above, the constraint process is deterministic. Inthe following example, a stochastic constraint process is constructed fromstochastic matrices.

9

Example 13 Given a closed convex cone K ⊂ RL and an (Ft)

Tt=0-adapted

sequence (Mt)Tt=0 of real L × J matrices, the sets

Dt(ω) = x ∈ RJ |Mt(ω)x ∈ K,

define a convex conical portfolio constraint process in the sense of Def-inition 8. This case (with polyhedral K) was studied by Napp [25] inconnection with linear price processes.

Proof. It is easily checked that the sets Dt(ω) are closed convex cones.That each Dt is Ft-measurable follows, by [32, Example 14.15], from Ft-measurability of Mt.

If Mt(ω) is the diagonal matrix with mark-to-market values of theassets J on the diagonal, the above example corresponds to the case wherethe mark-to-market values (instead of the number of shares held) of theinvestments are required to lie in the cone K.

In Example 13, the portfolio constraints are conical. A simple examplethat goes beyond the conical case or the deterministic case in Example 12is when there are stochastic bounds on the amounts invested (measuredeither in units of mark-to-market values). The special structure in thefollowing example simplifies the form of D′

t and NDt(0).

Example 14 Let H = (Kt)Tt=0 and M = (Mt)

Tt=0 be (Ft)

Tt=0-adapted

sequences of closed convex cones and closed convex sets, respectively, suchthat

Ht(ω) ⊂[

α>0

αMt(ω).

Then the sets Dt(ω) = Ht(ω)∩Mt(ω) define a convex portfolio constraintprocess in the sense of Definition 8. This was studied in Evstigneev,Schurger and Taksar [14] but instead of convexity of H and M they as-sumed linear price processes and the existence of a cash account. In theconvex case, D′

t = Ht, D∞t = Ht ∩ M∞

t , NDt(0) = H∗

t and cl rge NDt=

cl(H∗t + rge NMt

).

Proof. Convexity of D is obvious whereas its measurability follows from[32, Proposition 14.11]. The expressions for D′ and NDt

follow fromthe assumption that H is contained in the cone generated by K and theexpression for D∞ follows from [32, Proposition 3.9] and the fact that His conical. Finally, the expression for cl rge NDt

follows from the bipolartheorem (see the appendix) and the fact that, by Lemma 10, it is thepolar of D∞.

4 Trading strategies and arbitrage

A trading strategy, or a portfolio process, is an RJ -valued stochastic pro-

cess x = (xt)Tt=0 that is adapted to the filtration (Ft)

Tt=0, i.e. the portfolio

xt at time t, is Ft-measurable. We will not distinguish between strategiesthat differ only on a set of measure zero. More specifically, we will restricttrading strategies to the set of equivalence classes of essentially bounded(Ft)

Tt=0-adapted functions

N∞ = (xt)Tt=0 |xt ∈ L∞(Ω,Ft, P ; RJ).

10

A trading strategy x ∈ N∞ is said to be self-financing if for each t =0, . . . , T , the cost of updating the portfolio xt−1 to xt is nonpositive, i.e.if

St(xt − xt−1) ≤ 0 P -a.s. t = 0, . . . , T.

Here and in what follows, given an Ft-measurable function ut : Ω → RJ ,

St(ut) denotes the extended real-valued random variable ω 7→ St(ut(ω), ω).By [32, Proposition 14.28], St(ut) is Ft-measurable for any Ft-measurableut.

Although we measure prices in cash, we do not assume that thereexists a “cash account”, an asset which can be traded without frictionsand whose price is always strictly positive (see, however, the discussionafter Example 12). Defining self-financing trading strategies in terms ofstage-wise cash-flows is similar to Buhlmann, Delbaen, Embrechts andShiryaev [4] and Napp [25] where linear price processes were studied. Itis important to distinguish between cash flows at different dates if one isnot allowed to transfer wealth freely in time through a cash account.

An arbitrage opportunity is self-financing trading strategy that satisfiesportfolio constraints, starts and ends in zero portfolios and allows, withpositive probability, one to withdraw money from the portfolio at somepoint in time. In other words, an arbitrage opportunity is a trading strat-egy x ∈ N∞

0 that satisfies P -almost surely for t = 0, . . . , T both xt ∈ Dt

andSt(xt − xt−1) ≤ 0 t = 0, . . . , T,

where the inequality is strict with positive probability for some t. Hereand in what follows, we set x−1 = 0 and

N∞0 := x ∈ N∞ |xT = 0.

A price process S and a constraint process D are said to satisfy the noarbitrage condition if they do not allow for arbitrage opportunities.

As in frictionless markets (see [10]). the no arbitrage condition canbe written in terms of contingent claims. Here, however, a contingentclaim is a security that gives its owner a random cash-flow, not only atthe terminal date T , but possibly at all dates t = 0, . . . , T . We denote

M∞ = (Ft)Tt=0 |Ft ∈ L∞(Ω,F , P ; R),

the set of essentially bounded contingent claims. The no arbitrage condi-tion can be written as

C ∩M∞+ = 0,

where M∞+ ⊂ M∞ is the set of nonnegative bounded contingent claims

and

C = F ∈ M∞ | ∃x ∈ N∞0 : xt ∈ Dt, St(xt−xt−1)+Ft ≤ 0, t = 0, . . . , T

is the set of bounded contingent claims that can be hedged (super-replicated)with zero initial investment in a market described by the price process Sunder the portfolio constraint process D.

Having inequalities in the definition of C instead of equalities corre-sponds to free disposal of cash and it will be essential to our convexity

11

based approach. Free disposal of cash is a rather harmless assumptionsince it just corresponds to the fact that one can throw away cash at anypoint in time.

The following simple result will be crucial in all that follows.

Lemma 15 We always have M∞− ⊂ C. If S and D are convex, then C

is convex. If S is sublinear and D is conical, then C is a cone.

Proof. We have L∞− ⊂ C since St(0) = 0. The rest will follow from the

fact that C is the image of the set

E = (x, F ) ∈ N∞0 ×M∞ |xt ∈ Dt, St(xt−xt−1)+Ft ≤ 0, t = 0, . . . , T,

under the projection (x,F ) 7→ F . The set E is convex (cone) wheneverS is a convex (sublinear) price process and D is convex (cone). Indeed, ifS is sublinear (convex), D is a convex cone (a convex set), (xi, F i) ∈ Eand αi > 0 (such that α1 + α2 = 1), we have α1x1

t + α2x2t ∈ Dt P -almost

surely for t = 0 . . . , T and

St[(α1x1

t + α2x2t ) − (α1x1

t−1 + α2x2t−1)] = St[α

1(x1t − x1

t−1) + α2(x2t − x2

t−1)]

≤ α1St(x1t − x1

t−1) + α2St(x2t − x2

t−1)

≤ α1F 1t + α2F 2

t ,

or in other words, (α1x1+α2x2, α1F 1+α2F 2) ∈ E, so E is a cone (a convexset). The claimed properties now follow from the fact that projections ofconvex sets (convex cones) are convex sets (convex cones).

When C is a cone, the set C∩M∞+ is also a cone, which means that all

arbitrage opportunities (if any) can be scaled by arbitrary positive num-bers to yield arbitrarily “large” arbitrage opportunities. In general thisis not true and we can distinguish between two kinds of arbitrage oppor-tunities: the original ones defined as above and those that can be scaledby arbitrary positive numbers without leaving the set C. Accordingly,we will say that a price process S and a constraint process D satisfy thecondition of no scalable arbitrage if

\

α>0

αC

!

∩M∞+ = 0.

Obviously, the no arbitrage condition implies the no scalable arbitragecondition and when C is a cone, the two conditions coincide. In gen-eral, however, a market model may allow for arbitrage but still be free ofscalable arbitrage.

Note that the no arbitrage condition C ∩ M∞+ = 0 can be written

as

[

α>0

αC

!

∩M∞+ = 0.

Since C is a convex set containing the origin, we have α1C ⊂ α2C when-ever α1 ≤ α2. The no arbitrage condition is thus purely local in nature: itdepends only on the form of C near the origin. Similarly, the no scalable

12

arbitrage condition is of global character: it is independent of the form ofC on bounded subsets of M∞.

A simple condition guaranteeing the no scalable arbitrage condition is

infx∈RJ

St(x) > −∞ P -a.s., t = 0, . . . , T.

Indeed, the elements of C are uniformly bounded from above by the func-tion ω 7→ − infx St(x,ω) so if this is finite,

T

α>0 αC is contained in M−.The condition inf St(x) > −∞ means that the revenue one can generateby an instantaneous transaction at given time and given state is boundedfrom above. In the case of a double auction markets, it simply correspondsto the fact that the “bid-side” of the limit order book has finite depth (sell-ing more than the total demand at any given time instant generates noadditional revenue).

5 Free lunches and deflators

The above notions of arbitrage are of purely algebraic character. However,already in the linear unconstrained case (see [10]), topological propertiesof the set C become crucial when looking for characterizations of theno arbitrage condition in terms of continuous linear functionals on M∞

(e.g. martingale measures absolutely continuous wrt P ). Accordingly, weequip M∞ with the weak topology σ(M∞,M1) induced by the linearfunctionals

F 7→ E(F · y),

wherey ∈ M1 := (yt)

Tt=0 | yt ∈ L1(Ω,Ft, P ; R)

and

F · y =TX

t=0

Ftyt.

We will say that a price process S and a constraint process D satisfythe condition of no marginal free lunch if

C′ ∩M∞+ = 0, (NMFL)

and no scalable free lunch if

C∞ ∩M∞+ = 0, (NSFL)

respectively. Here C′ is the tangent cone to C at 0 and C∞ is the horizoncone of C, i.e.

C′ = cl[

α>0

αC and C∞ =\

α>0

α cl C.

In general, C∞ ⊂ C ⊂ C′ so (NMFL) implies (NSFL). When C is a cone,we have C′ = C∞ = cl C and (NMFL) and (NSFL) are equivalent.

When the price process is linear, there are no portfolio constraints andthere exists a cash account, the no arbitrage condition implies the condi-tion of no free lunch; see e.g. [8] or [10]. This was extended in [27] to the

13

case of conical and deterministic portfolio constraints under some nonde-generacy condition on the linear price process. A further generalizationwas obtained in [25], where portfolio constraints were allowed to be ran-dom. It would be interesting to see whether the no arbitrage condition/noscalable arbitrage condition is equivalent to (NMFL)/(NSFL) in the caseof general convex price processes and convex portfolio constraints. Weleave this for future investigations and concentrate here instead on theimplications of the no free lunch conditions.

A direct application of the Kreps-Yan theorem yields the followingversion of the “fundamental theorem of asset pricing”.

Theorem 16 A convex S and a convex D satisfy

1. (NMFL) iff there is a strictly positive y ∈ M1 which is nonpositiveon C′.

2. (NSFL) iff there is a strictly positive y ∈ M1 which is nonpositiveon C∞.

Proof. In the context of 〈M∞,M1〉, the Kreps-Yan theorem says that,if K is a closed convex cone with M∞

− ⊂ K and K ∩ M∞+ = 0, then

there is a strictly positive y ∈ M1 such that E(F · y) ≤ for every F ∈ K.It thus suffices to take K = C ′ and K = C∞.

The proof of the Kreps-Yan theorem combines a separation argu-ment with an “exhaustion” argument based on the Lindelof property ofthe underlying space. It is well documented in Delbaen and Schacher-mayer [10], Follmer and Schied [15], Jouini, Napp and Schachermayer [17]and Rokhlin [35].

The processes y ∈ M1 in Theorem 16 may be interpreted as defla-tors (stochastic discount factors, pricing kernels, price systems, . . . ) thatassign strictly positive values E(F · y) to every F ∈ M∞

+ \ 0 but non-positive values to all claims in C ′ or C∞. In linear markets with a cashaccount, they give rise to probability measures which are equivalent tothe original measure P and under which discounted marginal prices aremartingales; see Example 24 in Section 7.

6 The profit function and martingales

We can interpret the deflators given in Theorem 16 more directly in termsof the price process S and the constraint process D by analyzing the profitfunction π : M1 → R ∪ +∞ defined as the support function of C,

π(y) = supF∈C

E(F · y).

The term “profit function” is suggested by microeconomic theory; see e.g.Aubin [2] or Mas-Collel, Whinston and Green [23]. Being a pointwisesupremum of a collection of continuous linear functionals, π is a lowersemicontinuous positively homogeneous convex function on M1. Further-more, since M∞

− ⊂ C, we have that π is nonnegative and

dom π := y ∈ M1 |π(y) < +∞ ⊂ M1+.

Theorem 16 can be written in terms of the profit function as follows.

14

Theorem 17 A convex S and a convex D satisfy

1. (NMFL) iff there is a strictly positive y with π(y) ≤ 0.

2. (NSFL) iff there is a strictly positive y ∈ cl dom π.

Proof. By Theorem 16, it suffices to show that the polar cones of C ′ andC∞ can be expressed as

(C′)∗ = y ∈ M1 |π(y) ≤ 0 and (C∞)∗ = cl dom π,

respectively. The first one follows from linearity and continuity of x 7→〈x, v〉:

v ∈ V | σC(v) ≤ 0 = v ∈ V | 〈x, v〉 ≤ 0 ∀v ∈ C

= v ∈ V | 〈x, v〉 ≤ 0 ∀x ∈[

α>0

(αC)

= v ∈ V | 〈x, v〉 ≤ 0 ∀x ∈ C ′.

The second follows from the fact that f∞ = σdom f∗ for any closed con-vex function f ; see the appendix. Indeed, applying this to the indicatorfunction δC , gives δC∞ = σdom π = δ(dom π)∗ (because dom π is a cone,by sublinearity of π), so the second expression follows from the bipolartheorem; see the appendix.

The profit function can be written in terms of integral functionalsassociated with S and D. First, we define the indicator function δDt(ω) :R

J → R ∪ +∞ of Dt(ω) by

δDt(ω)(x) =

(

0 if x ∈ Dt(ω),

+∞ otherwise.

Given a y ∈ M1+, the functions

(x,ω) 7→ yt(ω)St(x,ω) and (x, ω) 7→ δDt(ω)(x)

are then convex normal integrands on RJ × Ω; see the appendix and

[32, Example 14.32 and Corollary 14.46]. It follows that the integralfunctionals

xt 7→ E(ytSt)(xt) and xt 7→ EδDt(xt),

are extended real-valued convex functions on L∞(Ω,Ft, P ; RJ); see theappendix. Here (ytSt)(xt) and δDt

(xt) denote the measurable functions

ω 7→ yt(ω)St(xt(ω), ω) and ω 7→ δDt(ω)(xt(ω)),

respectively, and the expectations are defined to equal +∞ if both thepositive and negative parts of the integrands integrate to +∞. Note that

EδDt(xt) =

(

0 if xt(ω) ∈ Dt(ω) P -almost surely,

+∞ otherwise.

We will say that a price process S = (St)Tt=0 is bounded if there is an

M ∈ R such thatsup|x|≤ρ

St(x) ≤ M P -a.s.

for every t = 0, . . . , T . Here and in what follows, | · | denotes the Euclideannorm on R

J .

15

Lemma 18 Assume that S and D are convex and S is bounded. Thenfor y ≥ 0,

π(y) = − infx∈N∞

0

(

TX

t=0

E(ytSt)(xt − xt−1) +

T−1X

t=0

EδDt(xt)

)

and π(y) = +∞ otherwise.

Proof. Since M∞− ⊂ C, we have π(y) = +∞ unless y ≥ 0. For y ≥ 0,

the definition of C gives

π(y) = supx∈N∞

0, F∈M∞

TX

t=0

E(Ftyt) |xt ∈ Dt, St(xt − xt−1) + Ft ≤ 0

= supx∈N∞

0

TX

t=0

supFt∈L∞

t

E(Ftyt) |xt ∈ Dt, Ft ≤ −St(xt − xt−1)

= supx∈N∞

0

−TX

t=0

E(ytSt)(xt − xt−1) |xt ∈ Dt

= supx∈N∞

0

(

−TX

t=0

E(ytSt)(xt − xt−1) −T−1X

t=0

EδDt(xt)

)

,

where the third equality follows from the boundedness of S.

We will next derive a dual expression for π. To this end, we pair N∞

withN 1 = (vt)

Tt=0 | vt ∈ L1(Ω,Ft, P ; RJ)

through the bilinear form (x, v) 7→ E(x · v). The dual expression willinvolve the integral functionals

vt 7→ E(ytSt)∗(vt) and vt 7→ EσDt

(vt)

associated with the normal integrands

(ytSt)∗(v, ω) := sup

x∈RJ

x · v − yt(ω)St(x, ω),

and

σDt(ω)(v) := supx∈RJ

x · v |x ∈ Dt(ω).

That the above expression do define normal integrands follows from [32,Theorem 14.50].

Lemma 19 Assume that S and D are convex and S is bounded. Thenfor y ≥ 0,

π(y) = minv∈N1

(

TX

t=0

E(ytSt)∗(vt) +

T−1X

t=0

EσDt(E[vt+1|Ft] − yt)

)

,

and π(y) = +∞ otherwise.

16

Proof. Again, π(y) = +∞ unless y ≥ 0, so let y ≥ 0. Defining, A :N∞

0 → N∞, k : N∞ → R and h : N∞0 → R by

Ax = (x0, x1 − x0, . . . ,−xT−1),

k(x) = E

TX

t=0

ytSt(xt),

h(x) = ET−1X

t=0

δDt(xt),

we can writeπ(y) = − inf

x∈N∞

0

h(x) + k(Ax),

which fits the format of the Fenchel-Rockafellar duality theorem (see theappendix). By [31, Theorem 22], the boundedness condition implies thatk is continuous with respect to the Mackey topology τ (N∞,N 1) and then,by the Fenchel-Rockafellar duality theorem,

π(y) = minv∈N1

k∗(v) + h∗(−A∗v), (1)

where A∗ : N 1 → N 10 is the adjoint of A and k∗ : N 1 → R and h∗ : N 1

0 →R are the conjugates of k and h, respectively. It is not hard to check that

A∗v = (v0 − E[v1|F0], . . . , vT−1 − E[vT |FT−1]) .

Writingk(x) = k0(x0) + . . . + kT (xT ),

where kt : L∞(Ω,Ft, P ; RJ ) → R is given by

kt(xt) = E[yt(ω)St(xt(ω), ω)],

we getk∗(v) = k∗

0(v0) + . . . + k∗T (vT ),

where, by [31, Theorem 21], k∗t (vt) = E(ytSt)

∗(vt(ω), ω). By [31, Theo-rem 21] again,

h∗(v) = E

T−1X

t=0

σDt(vt).

Plugging the above expressions for A∗, k∗ and h∗ in (1) gives desiredexpression.

Our first main result is the following pointwise characterization of(NMFL).

Theorem 20 If S and D are convex and S is bounded, they satisfy(NMFL) iff there exist a strictly positive y ∈ M1 and an (Ft)

Tt=0-adapted

RJ -valued process s = (st)

Tt=0 such that

st ∈ ∂St(0) and E[yt+1st+1 | Ft] − ytst ∈ NDt(0)

P -almost surely for t = 0, . . . , T .

17

Proof. Combining Theorem 17 and Lemma 19, we see that S and Dsatisfy (NMFL) iff there is a strictly positive y ∈ M1 and a v ∈ N 1 suchthat

TX

t=0

E(ytSt)∗(vt) +

T−1X

t=0

EσDt(E[vt+1|Ft] − vt) ≤ 0.

On the other hand, since St(0, ω) = 0 and 0 ∈ Dt(ω) for every ω ∈ Ω, wehave (ytSt)

∗(v, ω) ≥ 0 and σDt(ω) (v) ≥ 0 for every v ∈ RJ and ω ∈ Ω.

(NMFL) is thus satisfied iff there is a strictly positive y ∈ M1 and av ∈ N 1 such that P -almost surely

vt(ω) ∈ argmin(ytSt)∗(·, ω) and E[vt+1|Ft](ω)−vt(ω) ∈ argmin σDt(ω).

Defining st = vt/yt and using the identities

argmin(ytSt)∗(·, ω) = yt(ω)∂St(0, ω),

andargmin σDt(ω) = ∂δDt(ω)(0) = NDt(ω)(0)

(see appendix) we obtain the condition in the statement.

When Dt(ω) ≡ RJ (no portfolio constraints), we have NDt(ω) = 0

and the condition in Theorem 20 says that there exists a sequence s =(st)

Tt=0 of mark-to-market prices and a deflator y = (yt)

Tt=0 such the de-

flated mark-to-market price process ys = (ytst)Tt=0 is a martingale. If

there exists a cash account, one can use the deflator y to define an equiva-lent measure Q under which discounted mark-to-market prices are a mar-tingale; see Example 24. When Dt ≡ R

J+, we have NDt

(0) ≡ RJ− and the

second inclusion means that ys is a super-martingale. The general normalcone condition in Theorem 20 is essentially the same as the one obtainedin Rokhlin [34] in the case of linear price process with a cash account.

When the price process S happens to be smooth, Theorem 20 takesthe following simpler form.

Corollary 21 If S and D are convex and S is bounded and smooth, theysatisfy (NMFL) iff there exists a strictly positive y ∈ M1 such that

yt∇St(0) − E[yt+1∇St+1(0) | Ft] ∈ NDt(0)

P -almost surely for t = 0, . . . , T .

The condition in Corollary 21 resembles the martingale condition inTheorem 3.2 of Cetin, Jarrow and Protter[5] which says that (in a marketwith a cash account and with one risky asset without portfolio constraints)the value of the supply curve at the origin is a martingale under a measureequivalent to P . It should be noted however, that the supply curve of [5]is not the same as the marginal price ∇St(·, ω) (or ∂St(·, ω), when St isnonsmooth) considered here. Indeed, the supply curve of [5] gives the “thestock price, per share, at time t ∈ [0, T ] that the trader pays/receives foran order of size x ∈ R”; see [5, Section 2.1]. In our notation, the supplycurve of [5] thus corresponds to the function x 7→ St(x, ω)/x which agreeswith ∇St(x, ω) in the limit x → 0 (if St(x, ω) is smooth at the origin) butis different in general.

18

In order to give a pointwise characterization of (NSFL) analogous toTheorem 22, we will need some additional regularity conditions on S andD. We will say that a price process S = (St)

Tt=0 is Lipschitz continuous if

there is an L ∈ R such that

|St(x) − St(z)| ≤ L|x − z| P -a.s.

for every x, y ∈ RJ and t = 0, . . . , T . Since St(0) = 0, Lipschitz continuous

price processes are automatically bounded. For sublinear price processesthe converse is true. In general, a price process S is Lipschitz continuousiff the price process S∞ is bounded; see [30, Theorem 10.5].

Our second main result is the following pointwise characterization of(NSFL).

Theorem 22 If S and D are convex and S is Lipschitz continuous, anecessary condition for (NSFL) is that there exist a strictly positive y ∈M1 and an (Ft)

Tt=0-adapted R

J -valued process s = (st)Tt=0 such that

st ∈ cl rge ∂St and E[yt+1st+1 | Ft] − ytst ∈ cl rge NDt

P -almost surely for t = 0, . . . , T . This is also sufficient if for some a ∈ R

St ≥ S∞t − a and Dt ⊂ D∞

t + aB,

P -almost surely for t = 0, . . . , T .

Proof. By Theorem 17, S and D satisfy (NSFL) iff there is a strictlypositive y ∈ cl domπ. It suffices to show that, if S is Lipschitz continuous,then

cl domπ ⊂ y ∈ M1+ | ∃v ∈ N 1 : vt ∈ yt cl rge ∂St, E[vt+1|Ft]−vt ∈ cl rge NDt

with an equality under the additional conditions on S and D. Indeed, theprocess st = vt/yt will then have the desired properties.

By Lemma 19, dom π = y ∈ M1 | ∃v ∈ N 1 : (y, v) ∈ dom θ, where

θ(y, v) =

TX

t=0

E(ytSt)∗(vt) +

T−1X

t=0

EσDt(E[vt+1|Ft] − vt) + δN1

+(y).

The integrands in the first sum can be written as

(ytSt)∗(v, ω) =

8

>

<

>

:

yt(ω)S∗t (v/yt(ω), ω) if yt(ω) > 0,

0 if yt(ω) = 0 and v = 0,

+∞ otherwise,

whereS∗

t (v, ω) = supx∈RJ

x · v − St(x, ω).

By [30, Corollary 13.3.3], the Lipschitz continuity of S means that thedomains domS∗

t (·, ω) are uniformly bounded. Defining

D = (y, v) ∈ M1+ ×N 1 | vt ∈ yt domS∗

t , E[vt+1|Ft] − vt ∈ dom σDt,

19

we thus have dom θ ⊂ D and dom π ⊂ y ∈ M1 | ∃v ∈ N 1 : (y, v) ∈ D.Under the additional conditions on S and D, dom θ = D = clD. Indeed,the conditions imply that

S∗t (v) ≤ sup

x∈RJ

x · v − S∞t (x) + a = δcl dom S∗

t(v) + a,

σDt(v) ≤ sup

x∈RJ

x · v |x ∈ D∞t + aB

= supx∈RJ

x · v |x ∈ D∞t + sup

x∈RJ

x · v | x ∈ aB

= δcl dom σDt(v) + a|v|

P -almost surely. Here we have used the fact that, for a closed convexfunction f , f∞ and δcl dom f are conjugates to each other; see the appendix.The second inequality, also uses the fact that δ∞

C = δC∞ when C ⊂ RJ is

closed and convex. If (y, v) ∈ D then vt = 0 on Ω0 = ω ∈ Ω | y(ω) = 0(since dom S∗

t is bounded) and

θ(y, v) =

TX

t=0

Z

Ω\Ω0

yt(ω)S∗t (vt(ω)/yt(ω), ω)dP +

T−1X

t=0

EσDt(E[vt+1|Ft] − vt)

≤TX

t=0

Z

Ω\Ω0

yt(ω)adP +

T−1X

t=0

Ea|E[vt+1|Ft] − vt|

≤ a‖y‖L1 + 2a‖v‖L1 ,

so (v, y) ∈ dom θ. As to the closedness of D, let (vν)∞ν=1 be a sequencein dom S∗

t (·, ω) converging to a point v ∈ RJ . The above bound on S∗

t

together the lower semicontinuity of S∗t (·, ω) give

S∗t (v, ω) ≤ lim inf S∗

t (vν , ω) ≤ a,

so dom S∗t (·, ω) is closed. Similarly, dom σDt(ω) is closed for every ω ∈ Ω

and then D is closed. Indeed, defining

D1 = (y, v) ∈ M1+ ×N 1 | vt ∈ yt dom S∗

t ,

D2 = (y, v) ∈ M1+ ×N 1

0 | − vt ∈ domσDt,

A∗v = (v0 − E[v1|F0], . . . , vT−1 − E[vT |FT−1]) ,

we can write D = D1 ∩ (A∗)−1D2, where D1 and D2 are closed by Theo-rem 29 (see the discussion following the theorem).

Now, if y ∈ cl dom π, there is a sequence (yν , vν)∞ν=1 ⊂ D such thatyν → y. Since the Lipschitz continuity of S implies that dom S∗

t (·, ω)are uniformly bounded, the whole sequence (yν , vν)∞ν=1 clusters in theσ(M1×N 1,M∞×N∞)-topology at some (y, v). Indeed, the boundednessimplies that there is an M ∈ R such that |vν | ≤ M |yν | P -almost surely,which in turn implies that the sequence (vν)∞ν=1 is uniformly integrableand hence σ(N 1,N∞)-precompact by the Dunford-Pettis theorem. SinceD is convex, its strong closure clD is also σ-closed (see the appendix)so (y, v) ∈ clD. Since the strong L1-convergence implies P -almost sureconvergence along a subsequence, we thus get

clD ⊂ (y, v) ∈ M1+×N 1 | vt ∈ yt cl dom S∗

t , E[vt+1|Ft]−vt ∈ cl dom σDt,

20

so the first claim follows from the fact that cl dom f∗ = cl rge ∂f for anylower semicontinuous convex function f ; see the appendix.

The second claim follows from the fact that cl dom π ⊃ y ∈ M1 | ∃v ∈N 1 : (y, v) ∈ cl dom θ (simply by continuity of the projection (y, v) 7→ y)where, as verified above, dom θ = D = clD under the additional condi-tions on S and D.

If Dt(ω) ≡ RJ (no portfolio constraints), we have rge NDt(ω) = 0 and

the condition in Theorem 22 says that there exists a deflator y = (yt)Tt=0

and a price process s = (st)Tt=0 belonging P -almost surely to the closure

of the set rge ∂St of all possible marginal prices such that the deflatedprice process ys = (ytst)

Tt=0 is a martingale. When S is polyhedral, as in

double auction markets, the set rge ∂St itself is closed by Theorem 23.10and Corollary 23.5.1 of [30]. When S is sublinear and D is conical, we have(see Sections 2 and 3) S = S′ = S∞, D = D′ = D∞, rge ∂St = ∂St(0) andrge NDt

= NDt(0), so that Theorem 22 coincides with Theorem 20. This

is simply a reflection of the fact that, in that case, (NMFL) and (NSFL)are the same.

The following describes a situation where the conditions for (NSFL)become particularly simple.

Corollary 23 Let S be linear with a bounded marginal price process sand assume that Dt = Kt + Bt for Ft-measurable closed convex conesKt and Ft-measurable uniformly bounded closed convex sets Bt. Then(NSFL) holds iff there exists a strictly positive y ∈ M1 such that

E[yt+1st+1 | Ft] − ytst ∈ K∗t

P -almost surely for t = 0, . . . , T .

Proof. By Corollary 9.1.2 and Theorem 8.4 of [30], D∞t = Kt, so by

Lemma 10, rge NDt= K∗

t . We also have, ∂St(x, ω) = st(ω) for everyx ∈ R

J , so the claim follows from Theorem 22

The above situation corresponds to the market models studied in [14,34] where (NMFL) was characterized. There, price processes were linearbut constraints were allowed to be nonconical.

7 Examples

Theorems 20 and 22 can be applied to various combinations of S and D. Inthe frictionless case, it reduces to the following classical characterization;see Delbaen and Schachermayer [10] for a comprehensive treatment ofarbitrage in frictionless markets.

Example 24 Let S and D be as in Examples 4 and 11, that is, St(x,ω) =st(ω) · x and Dt = R

J . If s is bounded, then S and D satisfy (NMFL) iffthey satisfy (NSFL) iff there exists a strictly positive y ∈ M1 such thatys = (ytst)

Tt=0 is a P -martingale.

If one of the assets, say s0, is P -almost surely strictly positive, thenthe discounted price process s = s/s0 is a martingale under the probabilitymeasure Q ∼ P with density dQ

dP= yT s0

T /E(yT s0t ).

21

Proof. The equivalence of (NMFL) and (NSFL) follows from Lemma 15.According to Examples 4 and 11, ∂St(0) = st and NDt

(0) = 0, so themartingale characterization follows from Theorem 20 (or Theorem 22). Ifs0 is strictly positive, we can define qt = yts

0t and s = s/s0 and write

the martingale property as qtst = E[qt+1st+1|Ft]. Since s0 ≡ 1 we get, inparticular, that qt = E[qT |Ft] and it follows that the measure Q definedby dQ

dP= qT /EqT is a martingale measure for the discounted price process

s; see e.g. [15, Proposition A.12].

Theorems 20 and 22 can be applied also to more involved models,some of which seem new in the literature. The following combines sub-linear price processes of Kaval and Molchanov [19] with conical portfolioconstraints.

Example 25 Let S be as in Example 6, that is,

St(x, ω) = sups∈Zt(ω)

s · x,

and let D be conical and assume that Z = (Zt)Tt=0 is uniformly bounded.

Then S and D satisfy (NMFL) iff they satisfy (NSFL) iff there exist astrictly positive y ∈ M1 and an (Ft)

Tt=0-adapted R

J -valued process s =(yt)

Tt=0 such that

st ∈ Zt and E[yt+1st+1 | Ft] − ytst ∈ D∗t

P -almost surely for t = 0, . . . , T .

Proof. We have

St(x, ω) = sups∈Zt(ω)

s · x ≤ sups∈Zt(ω)

|s||x|,

so the boundedness of Z implies that S is bounded and we may applyTheorem 20 (or 22). It suffices to note that, according to Example 6,∂St(0) = Zt and NDt

(0) = D∗t .

Example 25 can be specialized in an obvious way to cases where S isgiven by bid-ask processes as in Example 5.

In Examples 24 and 25, S is sublinear and D is conical, so (NMFL)and (NSFL) are equivalent. In the nonlinear situation of Example 7, thisis no longer the case. In the following example, we assume for simplicitythat there are no portfolio constraints.

Example 26 Let D ≡ RJ and let S be as in Example 7, that is

St(x, ω) =X

j∈J

sjt(ω)ϕj(x

j),

and assume that s = (st)Tt=0 is bounded and ϕj are real-valued.

1. A necessary and sufficient condition for (NMFL) is that there is astrictly positive y ∈ M1 and (Ft)

Tt=0-adapted real-valued processes

gj = (gjt )

Tt=0 for j ∈ J such that

gjt ∈ ∂ϕj(0) P -a.s. t = 0, . . . , T,

22

and ytsjtg

jt are martingales. In particular, if the functions ϕj are

smooth at the origin, then (NMFL) is equivalent to the existence ofa strictly positive y ∈ M1 such that ys is a martingale.

2. If the intervals rge ∂ϕj are bounded, then a necessary condition for(NSFL) is that there exist a strictly positive y ∈ M1 and (Ft)

Tt=0-

adapted real-valued processes gj = (gjt )

Tt=0 for j ∈ J such that

gjt ∈ cl rge ∂ϕj(0) P -a.s. t = 0, . . . , T,

and ytsjtg

jt are martingales. This is also sufficient if there exists an

a ∈ R such thatϕj ≥ ϕ∞

j − a.

Proof. If s is bounded and the functions ϕj are real-valued, then S isbounded, so the first claim follows from Theorem 20 and the expressionsin Example 7.

By [30, Theorem 24.7], the boundedness of rge ϕj implies that thefunctions ϕj are Lipschitz continuous which, in turn, implies the Lipschitzcontinuity of S. By Example 7,

S∞(x) =X

j∈J

sjϕ∞j (xj),

so the condition ϕj ≥ ϕj − a implies (multiply by sj and sum over j ∈ J)

S(x) ≥ S∞(x) − a‖s‖L∞ .

The second claim claims now follows from Theorem 22 and the expressionsin Example 7.

8 Conclusions

We have characterized the (NMFL) and (NSFL) conditions in terms of L1-deflators in the case where contingent claims are defined as L∞-functions.With small changes, one could reverse this to allow for L1-contingentclaims to obtain L∞-deflators. Instead of defining free lunches in termsof contingent claims with cash-delivery, one could define free lunches interms of contingent claims with physical delivery much as in the cur-rency market models in Kabanov [18], Delbaen, Kabanov and Valkeila [9],Schachermayer [37] or in the nonconical model of Astic and Touzi [1]. Thiswould, in fact, simplify certain parts of the theory but then the conditionof free-disposal of cash would have to be substituted for something lessnatural. Another interesting topic for future research would be long-termprice impacts. A natural way to model that would be to make the costfunctions St dependent on the past trades in addition to the current ones.At the moment, however, it is less clear how such a dependence shouldact. As long as the dependencies are convex, much of the analysis pre-sented here would seem to carry over. Modern variational analysis and thetheory of normal integrals (see [32]) would also allow for generalizationsto nonconvex situations.

23

A Some convex analysis

We collect here some useful results from convex analysis, most of whichcan be found in [30, 31, 32]. As in [31], we consider two X and V locallyconvex topological vector spaces in separating duality, with a duality pair-ing denoted by (x, v) → 〈x, v〉; see Schaefer and Wolff [38] for a thoroughtreatment of dual pairs of vector spaces.

Much of convex analysis revolves around the interplay between setsand functions, where an extended real-valued function f : X → R :=R ∪ +∞,−∞ can be characterized by its epigraph

epi f := (x, α) | f(x) ≤ α ⊂ X × R

and a set C ⊂ X can be characterized by its indicator function

δC(x) =

(

0 if x ∈ C,

+∞ otherwise.

A function f is convex/lower semicontinuous/sublinear/polyhedral if epi fis a convex/closed/conical/polyhedral subset of X × R. Conversely, aset C is convex/closed/polyhedral iff δC is a convex/lower semicontinu-ous/polyhedral function. If f : X → R is convex/sublinear/polyhedralthen the domain

dom f := x ∈ X | f(x) < ∞

of f is convex/conical/polyhedral.

A.1 Duality

The conjugate f∗ : V → R of an f : X → R is defined

f∗(v) = supx∈X

〈x, v〉 − f(x).

Being a pointwise supremum of a collection of σ(V, X)-continuous affinefunctions, f∗ is σ(V,X)-lower semicontinuous. The conjugate g∗ : X → R

of a function g : V → R is defined similarly and it is σ(X, V )-lowersemicontinuous. The conjugate

σC(v) := supx∈X

〈x, v〉 − δC(x) ≡ supx∈C

〈x, v〉

of the indicator function δC of a set C ⊂ X is known as the supportfunction of C. If C is a cone, i.e.

x ∈ C =⇒ αx ∈ C ∀α ≥ 0,

then σC = δC∗ , where C∗ := v ∈ V | 〈x, v〉 ≤ 0 ∀x ∈ C is the polar coneof C.

The convex/lower semicontinuous hull of a function f is denoted byco f/lsc f . The closure cl f of f is defined

(cl f)(x) =

(

(lsc f)(x) if (lsc f)(x′) > −∞ for every x′ ∈ X,

−∞ otherwise.

It is easily checked that co δC = δco C and cl δC = δcl C where co C andcl C are the convex hull and the closure of C.

24

Theorem 27 For any f : X → R, f∗∗ = cl co f .

Proof. See [31, Theorem 5].

The above theorem implies in particular that a lower semicontinuousconvex function on X which does not attain the value −∞ is automaticallyσ(X, V )-lower semicontinuous. Applying Theorem 27 to the indicatorfunction of a cone C ⊂ X, gives the bipolar theorem, C∗∗ = cl co C. Thefollowing theorem is a special case of the general duality framework foroptimization problems that was built around the notion of conjugation in[31].

Theorem 28 (Fenchel-Rockafellar duality) Let U and Y be an otherpair of topological vector spaces in separating duality and let A : X → Ube continuous and linear. If f : X → R and k : U → R are convex, then

infx∈X

h(x) + k(Ax) ≥ supy∈Y

−k∗(y) − h∗(−A∗y).

If there is a point x ∈ dom h such that k is continuous at Ax, then equalityholds and the sup is attained.

Proof. This is [31, Example 11’] up to some sign changes.

A.2 Local and global characterizations

Given a convex set C and a point x ∈ C, the tangent cone to C at x isthe closed convex cone

TC(x) = cl[

α>0

α(C − x).

The horizon cone of C is the closed convex cone

C∞ =\

α>0

α cl C.

Given a convex function f : X → R and a point x ∈ dom f , the subderiva-tive of f at x is the sublinear function df(x) whose epigraph is the tangentcone to epi f at the point (x, f(x)). The horizon function of f is the sub-linear function f∞ whose epigraph is the horizon cone of epi f . Here wehave followed the terminology and notation of [32]. In the terminology of[30], C∞ is the “recession cone” of the set cl C and f∞ is the recessionfunction of lsc f ; see [32, Theorem 3.6]. By [32, Proposition 8.21], df(x)is the lower semicontinuous hull of the directional derivative function off at x.

If f is closed and convex then, by [30, Theorem 13.3] (the proof ofwhich works also in the present infinite-dimensional setting),

f∞ = σdom f∗ .

By [31, Theorem 11],df(x) = σ∂f(x),

where∂f(x) = v | f(x′) ≥ f(x) + 〈v, x′ − x〉 ∀x′ ∈ X

25

is the set of subgradients f at x. When f is closed, [31, Corollary 12A]gives the inversion formula ∂f∗ = (∂f)−1, where ∂f : X ⇒ V denotes theset-valued mapping x 7→ ∂f(x) (and ∂f∗ is defined similarly).

The set of subgradients is defined so that the generalized Fermat’s rule

argmin f = x | 0 ∈ ∂f(x)

always holds. For closed convex functions, we can use the above inver-sion formula to rewrite this as argmin f = ∂f∗(0), or dually, ∂f(0) =argmin f∗.

The normal cone of a convex set C ⊂ X at a point x is the closedconvex cone given by

NC(x) := ∂δC(x) =

(

v ∈ V | 〈v, x′ − x〉 ≤ 0 ∀x′ ∈ C if x ∈ C,

∅ otherwise.

If C is a closed convex set containing the origin, then by the inversionformula,

NC(0) = ∂δC(0) = ∂(σC)∗(0) = argmin σC .

Note that if C is a cone then NC(0) = C∗, the polar cone of C.If X is finite dimensional, then by [30, Theorem 23.4], ∂f(x) is nonempty

at every point x in the relative interior of dom f , and thus, cl dom f =cl dom∂f , where

dom ∂f := x ∈ X | ∂f(x) 6= ∅.

Dually, cl dom f∗ = cl dom ∂f∗ where, by the inversion formula,

dom ∂f∗ = rge ∂f :=[

x∈X

∂f(x).

A.3 Measurability

A set-valued mapping F : Ω ⇒ RJ is said to be Ft-measurable if F−1(O) :=

ω ∈ Ω |F (ω) ∩ O 6= ∅ ∈ Ft whenever O ⊂ RJ is open. An extended

real-valued function f : RJ×Ω → R is said to be an Ft-measurable normal

integrand if the set-valued mapping ω 7→ epi f(·, ω) is closed-valued andFt-measurable. By [32, Corollary 14.28], this implies that f : R

J ×Ω → R

is B(RJ ) ⊗Ft-measurable, where B(RJ ) denotes the Borel σ-field on RJ .

If (Ω,Ft) is complete wrt some measure, the converse holds. If f is anFt-measurable normal integrand, then the function ω 7→ f(x(ω), ω) is Ft-measurable for every Ft-measurable x. Then, in particular, the integralfunctional

x 7→ Ef(x(ω), ω)

is well-defined extended real-valued function on Lp(Ω,Ft, P ; RJ ) for anyp ≥ 0. We use the convention that the functional equals +∞ if both thepositive and negative parts integrate to +∞. This integral functional isconvex if f(·, ω) is convex for every ω ∈ Ω. If f is a normal integrand,then the conjugate integrand

f∗(v, ω) := supx∈RJ

x · v − f(x, ω)

26

is a convex normal integrand; see [29]. Rockafellar [29] gave the followingimportant conjugation formula for integral functionals on Lp(Ω,Ft, P ; RJ )for p ∈ [1, +∞].

Theorem 29 (Rockafellar) Let f be a convex normal integrand andassume that the integral functionals

x 7→ Ef(x(ω), ω) and v 7→ Ef∗(v(ω), ω)

are finite somewhere in Lp and Lq, respectively. Then these integral func-tionals are conjugate to each other with respect to the pairing

〈x, v〉 7→ E[x(ω) · v(ω)],

and hence, they are lower semicontinuous with respect to the weak topolo-gies σ(Lp, Lq) and σ(Lq, Lp), respectively.

Applying the above result to the indicator function of a measurableclosed convex set ω 7→ D(ω), we get that the set

D = x ∈ Lp |x(ω) ∈ D(ω) P -a.s.

is σ(Lp, Lq)-closed in Lp. We refer the reader to Rockafellar [29], Rockafel-lar and Wets [32, Chapter 14] and to Molchanov [24] for a comprehensivetreatment of measurable set-valued mappings and normal integrands.

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