Derivative Pricing and Logarithmic Portfolio Optimization in Incomplete Markets

Derivative Pricing and Logarithmic Portfolio Optimization in Incomplete Markets Dissertation zur Erlangung des Doktorgrades der Mathematischen Fakultät der Albert-Ludwigs-Universität Freiburg i. Br. vorgelegt von Thomas Goll November 200

Dekan: Referenten: Prof. Dr. Ernst Kuwert Prof. Dr. Ludger Rüschendorf Prof. Dr. Dmitry Kramkov Datum der Promotion: 2. Februar 2002 Institut für Mathematische Stochastik Albert-Ludwigs-Universität Freiburg Eckerstraße D-7904 Freiburg i. Br.

Preface This thesis studies the problem of derivative pricing in incomplete markets and the problem of portfolio optimization for logarithmic utility. In a complete financial market any contingent claim is attainable and the unique price of a derivative compatible with the no arbitrage condition can be obtained by the unique equivalent martingale measure. In an incomplete market, when perfect hedging of contingent claims is not always possible, the no arbitrage criterion does not suffice to value contingent claims any more. Each equivalent martingale measure yields a possible price and it can happen that these prices fill the whole price interval of trivial bounds cf. Eberlein and Jacod 997. Therefore additional criteria have to be imposed for derivative pricing. One approach is to consider the martingale measure which minimizes a certain distance: like L 2 -distance e.g. Schweizer 996, Hellinger distance e.g. Keller 997, or entropy distance e.g. Frittelli 2000a. One aim of this thesis is to characterize such minimal distance martingale measures in general semimartingale models. We do not consider special distances but the whole class of f-divergence distances defined by strictly convex, differentiable functions, which includes all the distances above. Additionally, we allow to restrict the class of martingale measures by moment constraints. An application is derivative pricing when market prices of some derivatives are known and one is looking for a pricing measure consistent with observed market prices. For derivative pricing we propose the minimal distance martingale measure consistent with observed market prices. Some necessary and some sufficient conditions for these measures are derived. Moreover, we show that minimal distance martingale measures are equivalent to so-called minimax martingale measures, which are defined with respect to concave utility functions and were studied in various forms in the past e.g. Bellini and Frittelli 2000. Based on the characterization of minimal distance martingale measures we directly obtain a duality result for optimal portfolios, i.e. for portfolios maximizing the expected utility of terminal wealth with respect to utility functions of the kind u : R + R. This duality result does not completely characterize optimal portfolios, since minimal distance martingale measures need not exist for a complete duality result in the case without constraints see Kramkov and Schachermayer 999, but it allows us to show the relationship between derivative pricing by minimal distance martingale measures and utility-based derivative pricing cf. Davis 997. The idea of utility-based derivative pricing is that derivative prices should be such, that trading I

II the derivatives does not increase the maximal expected utility of an investor in comparison to just trading the underlyings. Therefore minimal distance martingale measures can be motivated economically. Moreover, our duality result enables us to determine minimal distance martingale measures explicitly in the case without constraints and exponential Lévy processes with respect to several classical distances. Another problem studied in this thesis is the determination of optimal portfolios for logarithmic utility in general semimartingale models. This is a classical problem in mathematical finance, which has been studied for several market models in the past e.g. Merton 97, Cvitanić and Karatzas 992. Recently Kramkov and Schachermayer 999 proved the existence of log-optimal portfolios for terminal wealth in a general semimartingale framework. This thesis provides a complete explicit characterization of log-optimal portfolios in terms of the semimartingale characteristics of the price process, containing earlier results as special cases. We use the so-called duality or martingale approach to obtain these results cf. Chapter 3 and the references therein. This approach represents the connecting link between the two parts of this thesis. What precisely are our results concerning log-optimal portfolios? We give a sufficient condition which turns out to be necessary as well. Moreover, the sufficient part is extended in two respects: Firstly, we allow for random convex constraints similar to Cvitanić and Karatzas 992. This includes, for instance, the case of short sale constraints. Secondly, the consumption clock may be stochastic as well. As an example we consider a life-insurance problem for which the time of consumption is bounded by the random lifetime. Furthermore, we discuss some special properties of log-optimal portfolios by extending earlier results to the general semimartingale framework. We also determine the minimal distance martingale measure with respect to the reverse relative entropy, which is the f-divergence corresponding to logarithmic utility, and discuss its special features. The main tools which are used in this thesis are the theory of f-divergences and the corresponding results on f-projections cf. Liese and Vajda 987 and especially semimartingale characteristics in the sense of Jacod and Shiryaev 987. Characteristics describe the local behaviour of a semimartingale and turn out to be an appropriate tool to characterize martingale measures and optimal portfolios. The thesis is structured as follows. In Chapter we introduce the concept of σ- localization, which is indispensable for the complete characterization of log-optimal portfolios. In particular, σ-martingales, which play an important role in incomplete financial markets, are considered and their differences to local martingales and martingales are pointed out. We show how martingale measures and the related classes of local martingale and σ-martingale measures can be characterized in terms of semimartingale characteristics. Chapter 2 contains an introduction to f-divergences. Some necessary and some sufficient conditions for minimal distance martingale measures consistent with observed market prices are derived. We introduce the notion of a minimax measure and show that minimal distance martingale measures are equivalent to minimax martingale measures. Chapter 3 shows how the results of Chapter 2

are related to the problem of portfolio optimization. The relationship between minimal distance martingale measures and utility-based derivative pricing is pointed out. Moreover, minimal distance martingale measures are determined explicitly for exponential Lévy processes with respect to several classical distances. In Chapter 4 we derive a complete explicit solution to the problem of portfolio optimization for logarithmic utility in terms of the semimartingale characteristics of the price process. We determine the martingale measure minimizing the reverse relative entropy and discuss its special features. The final chapter contains some interesting examples of log-optimal portfolios. As an example of a random consumption clock we consider a life-insurance problem. I thank my advisor Ludger Rüschendorf for encouraging me to start this project and for many valuable suggestions and comments. In particular, Chapters 2 and 3 are based on Goll and Rüschendorf 2000, 200. I am also very grateful to Jan Kallsen for many fruitful discussions. A result of them is our joint work in Chapters and 4, and Section 5.. In slightly different form Chapters and 4 have been submitted in Goll and Kallsen 200a. Section 5. is mostly based on Goll and Kallsen 2000, 200b. Special thanks go to Walter Schachermayer for his hospitality and fruitful discussions during my research stay at the Technische Universität Wien. Moreover, I thank Fehmi Özkan for his encouraging spirit and for fruitful discussions and Ulla Lang for correcting my English. I very much enjoyed the time at the Institut für Mathematische Stochastik and I thank all its members for contributing to this. III

IV

Contents On σ-martingales and σ-martingale Measures. Introduction................................2 σ-localization and σ-martingales....................3 Equivalent σ-martingale Measures................... 8 2 Minimal Distance Martingale Measures 3 2. Introduction............................... 3 2.2 f-divergences and Minimal Distance Measures............ 4 2.3 Minimal Distance Martingale Measures................ 6 2.4 Minimal Distance and Minimax Measures............... 2 3 The Relationship to Portfolio Optimization 29 3. Introduction............................... 29 3.2 The Case Without Constraints..................... 30 3.3 Examples of Minimal Distance Martingale Measures......... 33 3.3. The Esscher Transform..................... 33 3.3.2 Distance Minimization for Power Functions.......... 35 3.3.3 The Minimal Entropy Martingale Measure.......... 37 3.4 The Case With Constraints....................... 39 4 Optimal Portfolios for Logarithmic Utility 43 4. Introduction............................... 43 4.2 Preliminaries.............................. 44 4.3 Explicit Solution in Terms of Characteristics.............. 5 4.4 Necessity and Existence in the Absence of Constraints........ 59 4.5 Some Properties and Special Cases................... 64 4.6 The Minimal Distance Martingale Measure.............. 67 5 Examples of Log-Optimal Portfolios 75 5. Deterministic Consumption Clock................... 75 5.2 Random Consumption Clock...................... 80 A Results from Stochastic Calculus 83 V

VI CONTENTS Bibliography 87

Chapter On σ-martingales and σ-martingale Measures. Introduction The aim of this chapter is twofold. On the one hand, it provides the concept of σ- localization, which is needed in the following chapters, especially in Chapter 4. On the other hand, we characterize equivalent martingale measures in terms of semimartingale characteristics. As a particular case of a σ-localized class of semimartingales we consider the class of σ-localized martingales, i.e. σ-martingales. The class of σ-martingales is a useful extension of the class of martingales. In general the stochastic integral with respect to a martingale is not a martingale again. If the stochastic integral is bounded from below, one knows that it is a local martingale cf. Ansel and Stricker 994, Corollary 3.5. This holds true if the integrator is a σ-martingale. Therefore, if one is interested in the gains processes of portfolio strategies, the requirement that the price process is a σ-martingale is as good as the requirement that the price process is a martingale cf. Delbaen and Schachermayer 998, Remark 2.4. Hence σ-martingales are a useful tool in mathematical finance. This becomes evident if one considers the general versions of the Fundamental Theorems of Asset Pricing in Delbaen and Schachermayer 998 and Cherny and Shiryaev 200..2 σ-localization and σ-martingales σ-martingales have been introduced by Chou 979 and were investigated further by Emery 980 under the name semimartingales de la classe Σ m. The term σ- martingale was introduced by Delbaen and Schachermayer 998, who used the concept of σ-martingales in mathematical finance for the first time. Note that some researchers prefer the older name martingale transforms, which was originally applied in discrete-time settings cf. Cherny and Shiryaev 200.

2 Chapter. On σ-martingales and σ-martingale Measures A σ-martingale can be interpreted quite naturally as a semimartingale with vanishing drift. Similar to local martingales, the set of σ-martingales can be obtained from the class of martingales by a localization procedure, but here localization has to be understood in a broader sense than usually cf. Jacod and Shiryaev 987, I.d; henceforth Jacod and Shiryaev 987 is abbreviated to JS. In the following we consider a filtered probability space Ω, F, F t t R+, P in the sense of JS, Definition I..2, i.e. the filtration is assumed to be right-continuous but is not necessarily complete. For any semimartingale X and any predictable set D Ω R +, we write X D := X 0 D 0 + D X, where D 0ω := D ω, 0 for ω Ω. In particular, we have X [[0,T ]] = X T for any stopping time T cf. JS, I.4.37. Definition. For any class C of semimartingales we define the σ-localized class C σ as follows: A process X belongs to C σ if and only if there exists an increasing sequence D n n N of predictable sets such that D n Ω R + up to an evanescent set and X Dn C for any n N. The σ-localization is most useful for classes of semimartingales, which are stable under stopping cf. JS, Definition I..34. Lemma.2 If C is stable under stopping, then C loc σ = C σ. Proof. We mimic the proof of a similar statement in JS, I..35. Let X C loc σ and D n n N a localizing sequence of predictable sets such that X Dn C loc for any n N. Since with C also C loc is stable under stopping cf. JS I..35a, we may assume that D n = D n [0, n]. For any n N there exists a localizing sequence of stopping times T n, p p N and p n N such that X Dn T n,p C for any p N and P T n, p n < n 2 n. We define a new sequence of predictable sets by D n := D n [0, T m, p m ]. m n Observe that D n n N is an increasing sequence. Let k N and ω, t Ω [0, k] with ω, t n N D n\ n N D n = lim sup n D n \ D n. Obviously, this holds up to evanescence also for ω, k instead of ω, t. Since P {ω Ω : ω, k D n \ D n } m n P T m, p m < n m n P T m, p m < m m n 2 m = 2 n, the Borel-Cantelli lemma yields P {ω Ω : ω, k lim supd n \ D n } = 0, n

.2. σ-localization and σ-martingales 3 and hence Ω [0, k] n N D n up to an evanescent set. Since C is stable under stopping and X D n = X Dn T n,pn m n+ T m,pm, we deduce that X D n C for any n N. Hence X belongs to C σ. Of course, we have C C loc C σ for any class C of semimartingales. As a particular case, we obtain the set M σ of σ-martingales. Observe the special feature in our notion of a σ-martingale. To be precise, X M σ holds if and only if X X 0 is a semimartingale de la classe Σ m. Equivalent definitions of σ-martingales can be found in Emery 980, Proposition 2. We denote by D the set of semimartingales such that the stopped process X t is of class D for any t R + cf. JS I..46. Below we consider the corresponding localized classes D loc, D σ. The σ-martingale property can be easily read from the characteristics of a semimartingale cf. Kabanov 997, Lemma 3. In the following let X be a real-valued semimartingale with X 0 = 0. Let B, C, ν be a version of the characteristics of X relative to some truncation function h : R R which is of the form B = b A, C = c A, ν = A F,. where A A + loc and b are predictable processes, c is a predictable R +-valued process, and F is a transition kernel from Ω R +, P into R, B. Note that by JS, II.2.9, such a version of the characteristics of X always exists. We start with an auxiliary result. Lemma.3. X D σ if and only if x F dx < P A-almost { x >} everywhere on Ω R +. 2. X D loc if and only if x F dx LA if and only if X is a special { x >} semimartingale. Proof. 2. The second equivalence follows from JS, II.2.29 and II.2.3. Therefore it suffices to show that X D loc if and only if X is a special semimartingale. This equivalence is stated as an exercise in Jacod 979, Exercise 2., and is proved in the following. : Let X = M + A be a semimartingale decomposition of X, i.e. M is a local martingale and A belongs to V. Let S n be a sequence of stopping times such that X Sn D. Define the stopping time T n := inf{t > 0 : VarA t n} S n cf. JS, I..28b. Since VarA Tn n + A Tn n + A Tn + A Tn 2n + A Tn,

4 Chapter. On σ-martingales and σ-martingale Measures we see that A A loc. By JS, I.4.23, X is a special semimartingale. : By JS, I.4.23, any semimartingale decomposition X = M + A satisfies A A loc. Let S n be a localizing sequence of stopping times such that M Sn is a uniformly integrable martingale and let T n be a localizing sequence such that E Var A Tn <. If we set R n := S n T n, it follows that X Rn D and hence X D loc.. Obviously D is stable under stopping. Therefore, D σ = D loc σ holds by Lemma.2, and the result follows from Statement 2. Remark. The class D σ was introduced by Chou 979 under the name semimartingales de la classe Σ. It was investigated further by Emery 980. To be precise, X D σ holds if and only if X X 0 is a semimartingale de la classe Σ see Emery 980, Proposition 2. The following Lemma compares σ-martingales with local martingales and martingales. Lemma.4. X is a σ-martingale if and only if x F dx < and { x >} b + x hxf dx = 0.2 hold P A-almost everywhere on Ω R +. 2. X is a local martingale if and only if x F dx LA and Equation { x >}.2 holds P A-almost everywhere on Ω R +. 3. X is a martingale if and only if X D and Equation.2 holds P A-almost everywhere on Ω R +. Proof. 2. This follows immediately from JS, II.2.29, and Lemma.3.. This follows from Statement 2 because M σ = M loc σ cf. also Lemma A.5. 3. : Since martingales are local martingales which belong to D cf. JS, Proposition I..47 this is a consequence of Statement 2. : It follows from Lemma.3 that X is a special semimartingale. By JS, II.2.29, and Statement 2 we get that X is a local martingale. Due to JS, I..47, we are done. Consequently, martingales, local martingales, and σ-martingales can all be interpreted as processes with vanishing drift see Equation.2 that differ only in the extent of uniform integrability. If we drop Equation.2 and keep the integrability conditions in Lemma.4, then we end up with the sets D σ, D loc, and D. Since in discrete time one can choose A t := s t Ns, Lemma.4 shows that in this case the class of σ-martingales coincides with the class of local martingales see also Shiryaev 999, Ch. II, c. In continuous time this does not hold true. There are σ-martingales which are not local martingales. The following example was given by Emery 980.

.2. σ-localization and σ-martingales 5 Example.5 Let τ be an exponential distributed random variable with parameter λ =, i.e. P τ > t = e t. Let η be independent of τ with P η = = P η = =. 2 Define the process M by { 0 if t < τ, M t := η if t τ and set H t =. Obviously M is a martingale with respect to the filtration F t t t R+ generated by M. Moreover we have H L S M, i.e. the process H is M-integrable in the sense of Stieltjes. For X := H M we have { 0 if t < τ, X t := η τ if t τ. Sine we have E X T = for any F t -stopping time T with P T > 0 > 0, the semimartingale X fails to be a local martingale. But, due to Emery 980, Proposition 2, it is a σ-martingale. Let us consider the characteristics of the semimartingale X. Let B, C, ν be the characteristics of M relative to the truncation function hx := x { x }. It turns out that they are of the form B = b A, C = c A, ν = A F,.3 where A t := t, b = 0, c = 0, and F t G = [[0,τ]] 2 ε G + 2 ε G for any G B. Let B, C, ν be the characteristics of H M relative to h. By Lemma A.5, they are of the form.3 as well, but with b = hh t x H t hxf t dx, c = 0, F t G = G H t xf dx for any G B. Now it is easy to check that X fulfills the condition of Lemma.4, i.e. x F dx < and { x >} b + x hx F dx = 0 P A-a.e. on Ω R +. However, it turns out that { x >} and hence X is not a local martingale. x F dx = [[0,τ]] { t >} t / LA

6 Chapter. On σ-martingales and σ-martingale Measures For a further illuminating example of a σ-martingale which is not a local martingale we refer to Delbaen and Schachermayer 998. In the following we summarize some relationships between σ-martingales, local martingales, and martingales. Corollary.6. X is a local martingale if and only if it is both a σ-martingale and a special semimartingale. 2. X is a martingale if and only if it is both a σ-martingale and in D. 3. X is a uniformly integrable martingale if and only if it is both a σ-martingale and of class D. Proof. This follows immediately from Lemma.4, Lemma.3, and JS, I..47. Corollary.7. Any locally bounded σ-martingale is a local martingale. 2. Any bounded σ-martingale is a uniformly integrable martingale. Proof.. By JS, I.4.24 and I.4.25, a locally bounded semimartingale is a special semimartingale. Hence the claim follows from Corollary.6. 2. This follows immediately from Statement and JS, I.47c. The following proposition is an extension of Jacod 979, Lemma 5.7, from local supermartingales to σ-supermartingales. We will need it for the proof of Theorem 4.6. Proposition.8 Let X be a non-negative σ-supermartingale with EX 0 <. Then X is a supermartingale. Proof. Let the characteristics of X be of the form.. Let D n n N be a sequence of predictable sets with D n Ω R + such that Dn X is a supermartingale for any n. Fix n N. By Lemma A.5, the characteristics of Dn X are of the form. as well, but with b := Dn b, c := Dn c, F := Dn F instead of b, c, F. Since Dn X is a supermartingale, it is a special semimartingale and its unique predictable part of bounded variation is given by cf. JS, II.2.29 b + x hxf dx A = b + x hxf dx Dn A. This process must be decreasing cf. Dellacherie and Meyer 982, Theorem VII.2, which implies that b + x hxf dx 0 P A-almost everywhere on D n and hence on Ω R + because n was arbitrarily chosen. The non-negativity of X implies that x 0F dx X F dx. { x >} { x >}

.2. σ-localization and σ-martingales 7 By JS, II.2.3, this implies { x >} x 0F dx LA. Since b + x hxf dx 0 and b and hxf dx belong to LA, it follows that x F dx LA. By Lemma.3 this means that X is a special { x >} semimartingale whose predictable part of bounded variation b + x hxf dx A is decreasing. Hence, X is in fact a nonnegative local supermartingale. Due to Jacod 979, 5.7, the claim follows. So far, we have concentrated on real-valued processes. As for local martingales, it makes sense to define R d -valued σ-martingales as semimartingales whose components are σ-martingales. Following Delbaen and Schachermayer 998 one may also call a R d -valued process X σ-martingale if there exists a R d -valued martingale M and a predictable, non-negative process ϕ such that ϕ LM i and X i = X0 i + ϕ M i for i =,..., d. By Cherny and Shiryaev 200, Theorem 5.6, and Emery 980, Proposition 2, the two definitions are consistent. The class of σ-martingales has a particular feature. In contrast to martingales or local martingales this class of semimartingales is stable relative to stochastic integration. We observe that Example.5 of Emery shows that unless the integrand is locally bounded the stochastic integral H M with respect to a local martingale M is in general not a local martingale again. Lemma.9 below shows that the σ-localized classes of local martingales and special semimartingales are stable relative to stochastic integration. Let us briefly touch the concept of stochastic integration with respect to a multidimensional semimartingale as general references we refer to Jacod 980 and Cherny and Shiryaev 200. For predictable processes ϕ = ϕ,..., ϕ d with ϕ i LS i the vector stochastic integral ϕ S is defined and coincides with the sum of the onedimensional stochastic integrals, i.e. ϕ S = d i= ϕi S i. However, the set of componentwise stochastic integrals is too small to have sufficient closedness properties see Cherny and Shiryaev 200 and the references therein. It turns out that a vector stochastic integral ϕ S can be defined for a broader class of predictable processes. For a R d -valued local martingale N and a R d -valued process A with components A i V Jacod 980 defined the classes of predictable processes L loc N and L S A. Now analogous to the one-dimensional case a predictable process ϕ belongs to the class LS if there exists a decomposition S = S 0 + N + A in a R d -valued local martingale N and a R d -valued process A with components A i V such that ϕ L loc N L SA, and the vector stochastic integral is defined as ϕ S = ϕ N+ϕ A. It is important to remark that if ϕ LS and S = S 0 + N + A is a decomposition

8 Chapter. On σ-martingales and σ-martingale Measures of the semimartingale S, one has in general not that ϕ L loc N L SA. Hence for a local martingale N there is the following inclusion L loc N LN. For example the integrand in Example.5 is in LM but not in t L loc M. Note that we have the following equivalence for a local martingale N and a process ϕ LN: ϕ N is a local martingale if and only if ϕ L loc N cf. Jacod 980, Proposition 2, Lemma.9, and Corollary.6. Lemma.9 Let X be a R d -valued semimartingale and H LX. Then the following statements hold:. If X i M σ for i =,..., d, then H X M σ. 2. If X i D σ for i =,..., d, then H X D σ. Proof.. The first statement follows immediately from Lemma.4 and Lemma A.5. 2. According Lemma.3 and Lemma A.5 it is sufficient to show that H x F dx <. { H x >} Define an increasing sequence of predictable sets by By Lemma.3 we have for any n Dn D n := {ω, t : H i tω n for i d}. { H x >} H x F dx n < d i= { x i > dn } x i F dx P A-almost everywhere on Ω R +, and the claim follows. Using Lemma.9 and Proposition.8 one can easily verify the following wellknown result cf. Ansel and Stricker 994, Corollary 3.5. Corollary.0 Let X be a R d -valued σ-martingale and H LX. If H X is bounded from below, then H X is a supermartingale..3 Equivalent σ-martingale Measures In the following we show how equivalent σ-martingale measures can be characterized in terms of semimartingale characteristics. To this end, let S = S,..., S d be a R d - valued semimartingale which stands for the discounted prices of securities,..., d.

.3. Equivalent σ-martingale Measures 9 Definition. Let Q loc P. We call Q an absolutely continuous martingale measure resp. absolutely continuous local martingale measure, absolutely continuous σ- martingale measure if S i is a martingale resp. local martingale, σ-martingale relative to Q for i =,..., d. We denote by M a M a loc, Ma σ the set of absolutely continuous martingale resp. local martingale, σ-martingale measures and by M e M e loc, Me σ the subclass of locally equivalent probability measures in M a resp. M a loc, Ma σ. Remark. In Delbaen and Schachermayer 994 the term equivalent local martingale measure is used in the above sense. Kramkov and Schachermayer 999 and Becherer 200, however, apply the same name to denote measures Q P such that + ϕ S is a Q-local martingale for any ϕ LS with + ϕ S 0. In order to avoid confusion, we prefer to call measures of such kind equivalent weak local martingale measures. One easily shows that any σ-martingale measure is a weak local martingale measure while the converse is not true. Nevertheless, the existence of an equivalent weak local martingale measure suffices to ensure the Condition NFLVR cf. Becherer 200, Proposition 2.3. Fix a probability measure Q loc P with density process Z. By N := L Z := Z Z we denote the stochastic logarithm of Z, i.e. the unique semimartingale N such that E N = Z cf. Kallsen and Shiryaev 200 for details on the stochastic logarithm; note that Z and Z are strictly positive by Q loc P and JS, Lemma III.3.6. Suppose that the characteristics B, C, ν of the R d+ -valued semimartingale S, N relative to h : R d+ R d+ are given in the form B = b A, C = c A, ν = A F,.4 where A A + loc is a predictable process, b is a predictable Rd+ -valued process, c is a predictable R d+ d+ -valued process whose values are non-negative, symmetric matrices, and F is a transition kernel from Ω R +, P into R d+, B d+. The Girsanov-Jacod-Mémin theorem as stated in JS, III.3.24, indicates how the characteristics change if P is replaced with Q. Here, we formulate this result in terms of the joint characteristics of S, N, which is convenient for applications. Lemma.2 The Q-characteristics of S, N are of the form.4, but with b = b + c,d+ + hxx d+ F dx, instead of b, c, F. c = c, d F x = + xd+ df

0 Chapter. On σ-martingales and σ-martingale Measures Proof. According to JS, III.3.3, there exists a predictable R d+ -valued process β such that β cβ LA and for i =,..., d. It follows that Secondly, we have c i,d+ A = S i,c, N c = c i β A Z c, S i,c = Z N c, S i,c = Z c i β A. Z t = Z t + N t = Z t + x d+ for µ S,N -almost all t, x R + R d+, which implies that EZU µ S,N = E + x d+ Z U µ S,N for any nonnegative P B d+ -measurable function U. Put differently, we have Y Z = M P µ S,N Z P in the sense of JS, III.3c for Y ω, t, x := + x d+. By JS, III.3.24, we are done. Remark. If Q loc P such that the density process is of the form Z = E N, Lemma.2 still holds true. In general the stochastic logarithm of the density process Z may not exist. But if one makes use of the stopping times R n := inf{t > 0 : Z t < } a n similar result as in Lemma.2 can be achieved on n [0, R n ] see JS, Ch. III., 5a. For the proof of Lemma.4 we need the following proposition, which extends JS, III.3.8, to the σ-martingale case. Proposition.3 A real-valued semimartingale X is a Q-martingale resp. Q-local martingale, Q-σ-martingale if and only if XZ is a P -martingale resp. P -local martingale, P -σ-martingale. Proof. By JS, III.3.8 it suffices to prove the assertion for σ-martingales. : Let D n n N be an σ-localizing sequence for the Q-σ-martingale X. Since Dn X is a Q-martingale this implies that Dn XZ is a P -martingale. By partial integration we have Dn XZ = Dn X Z + Dn Z X + Dn [X, Z] = Dn X Z + Dn XZ Dn X Z.

.3. Equivalent σ-martingale Measures Therefore Dn XZ is a P -local martingale. It follows from Lemma.2 that XZ is a P -σ-martingale. : This follows similarly. Just exchange the roles of P and Q and note that Z is the density process of P with respect to Q. We are now ready to characterize equivalent martingale measures and the related classes of equivalent local martingale measures and equivalent σ-martingale measures in terms of semimartingale characteristics: Lemma.4. S is a Q-σ-martingale if and only if { x i >} x i + x d+ F dx < for i =,..., d and b i + c i,d+ + x i + x d+ h i xf dx = 0 for i =,..., d.5 holds P A-almost everywhere on Ω R +. 2. S is a Q-local martingale if and only if { x i >} xi +x d+ F dx LA for i =,..., d and Condition.5 holds P A-almost everywhere on Ω R +. 3. S is a Q-martingale if and only if SZ D relative to P and Condition.5 hold P A-almost everywhere on Ω R +. Proof. Statements and 2 follow from Lemmas.2 and.4. 3. : Condition.5 follows from Statement 2. By Proposition.3, SZ is a P -martingale and hence in D cf. Lemma.4. : Fix i {,..., d}. Since S i Z and Z are special semimartingales, this is also true for Y := Z S i Z S i Z From Y = S i + N it follows that = Z Z S i + [S i, Z] = S i + [S i, N]. x i + x d+ { x i +x d+ >} ν = x { x >} ν Y A loc, where ν Y denotes the compensator of the measure of jumps of Y. By JS, II.2.3 we have that x i + x d+ { x i >} ν x i + x d+ { x i +x d+ >} ν + { x >} ν A loc. Statement 2 yields that S is a Q-local martingale, which in turn implies that SZ is a P -local martingale. By Corollary.6, we have that SZ is a P -martingale. Once more applying Proposition.3 yields the claim.

2 Chapter. On σ-martingales and σ-martingale Measures

Chapter 2 Minimal Distance Martingale Measures 2. Introduction Derivative pricing in incomplete markets basically means choosing an equivalent martingale measure. Hence the problem is to determine a criterion for selecting an appropriate martingale measure. One approach is to consider the martingale measure which minimizes a certain distance: like L 2 -distance Schweizer 996, Delbaen and Schachermayer 996, Hellinger distance Keller 997, Grandits 999, entropy distance Frittelli 2000a, Chan 999, Miyahara 999, Grandits and Rheinländer 999, Rheinländer 999 and others. In this chapter we consider the class of all f-divergence distances defined by strictly convex, differentiable functions f, which includes all the distances above see e.g. Liese and Vajda 987. We obtain some necessary and some sufficient conditions for projections of the underlying measure on the set of martingale measures in general semimartingale market models. Moreover, we consider the case that there is some additional information on the prices of some derivatives. We allow to restrict the class of martingale measures to those which yield derivative prices which are consistent with observed market prices. For derivative pricing we propose the minimal distance martingale measure consistent with observed market prices. This notion is equivalent to the least favourable consistent pricing measure which is studied for a discrete setting in Kallsen 200. Related ideas of consistent derivative pricing are studied by Kallsen cf. Kallsen 998a,b and Avellaneda 998. In Kallsen 998a,b derivative prices are considered which are derived from utility maximization and which are consistent with given demand vectors in some derivatives of the market. In Avellaneda 998 probability measures are characterized which minimize the relative entropy distance between a prior measure and the class of all probability measures consistent with the observed market prices. However, the calibrated measure obtained in this way is not necessarily a martingale measure cf. Samperi 2000 in 3

4 Chapter 2. Minimal Distance Martingale Measures this context. The chapter is organized as follows. In Section 2.2 we recall the definition of f- divergences and some relevant results about f-projections. Section 2.3 shows how these results can be applied to minimal distance martingale measures. Some necessary and some sufficient conditions for minimal distance martingale measures are derived. Section 2.4 considers minimax measures with respect to concave utility functions and convex sets of probability measures. We show that minimax measures are equivalent to minimal distance measures with respect to f-divergences induced by the convex conjugate of the utility function. 2.2 f-divergences and Minimal Distance Measures In the following we define f-divergences and recall some relevant results about f- projections. For general reference we refer to Liese and Vajda 987 or Vajda 989. Let Ω, F, P be a probability space. Definition 2. Let Q P and let f : 0, R be a convex function. Then the f-divergence between Q and P is defined as { f dq fq P :=, if the integral exists, else, where f0 = lim x 0 fx. Examples of f-divergence distances are the Kullback-Leibler or entropy distance for fx = x logx, the total variation distance for fx = x, the Hellinger distance for fx = x, the reverse relative entropy for fx = logx, and many others. In the following we assume that f is a continuous, strictly convex and differentiable function. By f 0 we denote the right-hand derivative at 0. Let K be a convex set of probability measures on Ω, F dominated by P. A measure Q K is called an f-projection of P on K if Remarks. fq P = inf fq P =: fk P. Q K fx. The f-divergence f P takes on values in the interval [f, f0 + lim ]. x x It holds fq P = f if and only if Q = P. See Liese and Vajda 987, Theorem.2. 2. If fk P <, then there exists at most one f-projection of P on K. See Liese and Vajda 987, Proposition 8.2.

2.2. f-divergences and Minimal Distance Measures 5 fx 3. If K is closed in the variational distance topology and lim =, then there x x exists an f-projection of P on K. See Liese and Vajda 987, Proposition 8.5. Let F be a convex cone of real valued random variables on Ω, F, i.e., { k i= } α i f i : α i 0, f i F = F, and define the moment family determined by inequality constraints with respect to F as K F := {Q P : F L Q and E Q f 0 for all f F }. 2. The first part of the following Theorem was given in Rüschendorf 984, Theorem 5. The second and the third part were given in Rüschendorf 987, Theorem 2: Theorem 2.2 Let Q P satisfy fq P <. i Assume that Q K. Then Q is the f-projection of P on K if and only if dq 0 f dq dq < 2.2 for all Q K with fq P <. ii Assume that Q K F and c := E Q f dq is finite. If Q is the f-projection on K F, then dq f c F, the L Ω, F, Q -closure of F. iii Assume that Q K F and c := E Q f dq is finite. If f dq c F, then Q is the f-projection on K F. Remarks.. In Theorem 5 in Rüschendorf 984 the assumption f dq L Q was also stated for part i but was not used for the proof of this part. 2. For every Q K with fq P < the value f dq dq dq coincides with the directional derivative of the function f P see Vajda 989, Lemma 9.3.i. Hence the condition in Theorem 2.2 i can be understood as a condition on the directional derivative in Q.

6 Chapter 2. Minimal Distance Martingale Measures 3. Theorem 2.2 ii is a generalization of a theorem of Csiszár 975 on the entropy distance. This result was applied in recent papers on mathematical finance for the characterization of minimal entropy martingale measures by Frittelli 2000a, Grandits and Rheinländer 999, and Rheinländer 999. An important question is when the f-projection of P on K is equivalent to P. Frittelli 2000a derived a sufficient condition in the case of the relative entropy, which corresponds to fx = x logx. He showed that in this case the f-projection of P on K is necessarily equivalent to P if there is a measure Q K with Q P and fq P <. Based on Theorem 2.2 i this result can be extended to general f- projections. Corollary 2.3 Let f 0 =. Assume the existence of a measure Q K such that Q P and fq P <. If Q is the f-projection of P, then Q P. Proof. Suppose Q is not equivalent to P, i.e. P dq = 0 > 0. Because Q P this implies Q dq = 0 > 0. Since f 0 = this leads to a contradiction to the necessary condition on an f-projection of Theorem 2.2 i. 2.3 Minimal Distance Martingale Measures In this section we apply Theorem 2.2 to characterize f-projections on the set of martingale measures which may fulfill some additional constraints. Our mathematical framework is as follows. Ω, F, F t t R+, P is a filtered probability space in the sense of JS, Definition I..2, i.e. the filtration is assumed to be right-continuous but is not necessarily complete. We consider a finite time horizon T and so we assume that F T = F. Let S be a R d -valued semimartingale with deterministic S 0. Let R := {R,..., R n } be a finite, possibly empty set of F T -measurable random variables and r {0,..., n}. We define the set of absolutely continuous martingale measures under constraints as M a := {Q M a : R L Q, E Q R i 0 for i r and E Q R i = 0 for r + i n}. The class M a stands for the set of martingale measures consistent with some information on the prices of the derivatives R,..., R n. Notice that price information of the form E Q R i [q i, p i ] can also be described by inequality constraints as in the definition of M a. The sets M a loc, M a σ, M e, M e loc M e σ are defined analogously to M a. In the following we characterize the f-projection of P on M a. As we will see, the crucial property is a representation of the Radon-Nikodym-density of the following type: dq f n = c + ϕ S T + µ i R i 2.3 i=

2.3. Minimal Distance Martingale Measures 7 with i c = E Q f dq < ii µ,..., µ n R such that µ i 0 and µ i E Q R i = 0 for i r iii ϕ predictable and S-integrable such that ϕ S is a martingale under Q. It turns out that this representation is a necessary and under slight additional assumptions also a sufficient condition for the f-projection of P on M a. If we consider in the following a measure Q M a σ and the corresponding class of integrands with respect to Q we write LS, Q respectively L loc S, Q. Notice that if Q P one has LS = LS, P LS, Q and the stochastic integral with respect to P is a version of the stochastic integral with respect to Q see Cherny and Shiryaev 200, Theorem 4.4. The following theorem shows that the representation 2.3 is a necessary condition for the f-projection of P on M a. Theorem 2.4 Let Q M a satisfy fq P <. If Q is the f-projection of P on M a and c := E Q f dq is finite, then dq f = c + ϕ S T + n µ i R i Q -a.s. with µ,..., µ n R such that µ i 0 and µ i E Q R i = 0 for i r, and with some process ϕ L loc S, Q such that ϕ S is a martingale under Q. Proof. First we introduce a set G of random variables which determines M a as a moment family. We define the set G as G := {ϕ S T : ϕ i = Y i ]si,t i ], s i < t i, Y i bounded F si -measurable}. Let G be the convex cone generated by the set i= G {R i : i n} { R i : r + i n}. Then we have the following characterization of M a : M a = {Q P : G L Q and E Q g 0 for all g G}. The necessary condition in Theorem 2.2 ii yields: f dq c G, where G denotes the L Ω, F, Q -closure of G. By Yor 978, Corollary 2.5.2 for a multidimensional version see Delbaen and Schachermayer 999, Theorem.6, the L Q - closure of the vector space generated by G is contained in {ϕ S T : ϕ L locs, Q such that ϕ S is a Q -martingale}.

8 Chapter 2. Minimal Distance Martingale Measures According to Jacod 979, Proposition., this result is valid without the assumption of a complete filtration. Extending Proposition I.3.3 in Schaefer 97 from vector spaces to the class of closed convex cones one gets dq f = c + ϕ S T + n µ i R i Q -a.s., i= where µ i 0 for i r. By the definition of c this implies that µ i E Q H i = 0 for i r. The following theorem is a variant of Theorem 2.4. It shows that the necessary condition in Theorem 2.4 is also valid for the set M a loc under the additional assumption that S is locally bounded. Note that in this case the set M a loc coincides with the set M a σ cf. Corollary.7. Theorem 2.5 Let S be locally bounded. Let Q M a loc satisfy fq P <. If Q is the f-projection of P on M a loc and c := E Q f dq is finite, then dq f = c + ϕ S T + n µ i R i Q -a.s. 2.4 with µ,..., µ n R such that µ i 0 and µ i E Q R i = 0 for i r, and with some process ϕ L loc S, Q such that ϕ S is a martingale relative to Q. Proof. Let G loc be the convex cone generated by {ϕ S T : ϕ i = Y i ]si,t i ] [[0, T i ]], s i < t i, Y i bounded F si -measurable, T i γ i } i= {R i : i n} { R i : r + i n}, where γ i := { T i stopping time : S i T i is bounded}. Then the convex cone G loc determines M a loc as a moment family M a loc = {Q P : G loc L Q and E Q g 0 g G loc }. The representation 2.4 is then obtained as in the proof of Theorem 2.4. Remark. In the unconstrained case, i.e. R =, the results of Theorem 2.4 and Theorem 2.5 are explicitly given in Goll and Rüschendorf 200. Closely related results can be found in Kramkov and Schachermayer 999, Schachermayer 999, and Grandits and Rheinländer 999. From Theorem 2.2 i we obtain the following sufficient condition for f-projections of P on M a σ.

2.3. Minimal Distance Martingale Measures 9 Theorem 2.6 Let Q M a σ with fq P < such that for some process ϕ LS and µ,..., µ n R the following conditions hold: i ii dq f = c + ϕ S T + n µ i R i P -a.s., i= ϕ S is bounded from below P -a.s., iii E Q ϕ S T = 0, iv µ i 0, µ i E Q R i = 0 for i r. Then Q is the f-projection of P on M a σ. Proof. It follows from Corollary.0 and condition ii that ϕ S is a supermartingale relative to any Q M a σ. Therefore, we have dq E Q f = c + E Q ϕ S T + c = E Q f dq Now the result follows from Theorem 2.2 i.. n µ i E Q R i The following proposition shows that one can transform the minimization problem inf fq P with respect to M a into a minimization problem with respect to M a Q M a including some penalty terms for violating the constraints. The coefficients γ i in the penalty terms can be interpreted as Lagrange multipliers. Proposition 2.7 Let S and R,..., R n be bounded and inf Ef dq <. Assume the existence of a measure Q 0 M a such that E Q0 R i > 0 for i r Q M a and fq 0 P <. Furthermore assume that there exists a neighbourhood V of 0,..., 0 R n r such that for all v V there exists an element Q M a with fq P < and E Q R r+,..., E Q R n = v. Then Q M a is the f-projection of P on M a, i.e., i= fq P = inf fq P, Q M a if and only if there are γ,..., γ n R such that fq P + n i= and γ i 0, γ i E Q R i = 0 for i r. { γ i E Q R i = inf fq P + Q M a n } γ i E Q R i i=

20 Chapter 2. Minimal Distance Martingale Measures Proof. Step : Since S is bounded, it follows that the set M a of absolutely continuous martingale measures is closed in variation. Since R,..., R n are bounded this is also true for M a := {Q M a : E Q R i = 0 for r + i n}. Since M a, M a are convex, they are also closed in σl, L if one identifies Q M a with its Radon-Nikodym density with respect to P see, for example, Schaefer 97, Proposition IV.3.. We define a function B : M a R r by BQ = E Q R,..., E Q R r. Obviously, the component mappings of B are convex and continuous with respect to σl, L. Since the weak convergence of densities is equivalent with the convergence of the corresponding measures in the weak topology of setwise convergence, we get from Liese and Vajda 987, Theorem.47, that the functional f P is lower semicontinuous. The optimization problem is the following: This problem can be written in the form inf fq P. Q M a inf fq P, Q M a BQ 0 where BQ 0 is understood componentwise. The assumption on the existence of an inner point Q 0 in M a allows to apply a Lagrange multiplier theorem see Ekeland and Temam 976, Theorem III.5., which results in the following equivalence: Q M a is the f-projection of P on M a if and only if there are γ i 0 such that fq P + r i= { γ i E Q R i = inf fq P + Q M a r γ i E Q R i }, 2.5 and γ i E Q R i = 0 for i r. Step 2: Next we follow a similar line of argument to handle the equality constraints on the right-hand side of 2.5. Since the component mappings of B are continuous and linear, the mapping J : M a R, defined by i= JQ := fq P + r γ i E Q R i, i= is lower semicontinuous and convex. We define a function B : M a R n r by B Q := E Q R r+,..., E Q R n.

2.4. Minimal Distance and Minimax Measures 2 The optimization problem can be written as inf JQ Q M a inf JQ. Q M a B Q=0 The perturbation function Φ : M a R n r R is chosen as { JQ, if Q M ΦQ, v := a and B Q = v, otherwise. For v R n r we define hv := inf ΦQ, v. By assumption, h0 is finite. Q Ma We observe that M a is closed, J is lower semicontinuous and convex, and B is linear and continuous. Therefore, the function h is convex see Ekeland and Temam 976, Lemma III.5.2 and Lemma III.2.. By Rockafellar 970, Theorem 23.4, h is subdifferentiable in 0. Hence, due to Ekeland and Temam 976, Proposition III.3.2, we obtain the following equivalence: Q M a solves inf JQ if and only if there are γ r+,..., γ n R such that Q M a fq P + n { γ i E Q R i = inf fq P + Q M a i= n γ i E Q R i }. Remark. Following the line of argument of the proof of Theorem 7. in Goll and Rüschendorf 200 one verifies that under the additional assumption f dq L Q the coefficients γ,..., γ n in Proposition 2.7 correspond to µ,..., µ n, where µ i are the coefficients of the characterization of Q in Theorem 2.4. 2.4 Minimal Distance and Minimax Measures Again Ω, F, P is a probability space, and by K we denote a convex set of probability measures on Ω, F dominated by P. K may be thought of as a subclass of M a σ. A utility function u: R R { } is assumed to be strictly increasing, strictly concave, continuously differentiable in domu := {x R ux > } and to satisfy u = lim u x = 0, x 2.6 u x = lim u x = x x 2.7 i=

22 Chapter 2. Minimal Distance Martingale Measures for x := inf{x R ux > }. This implies either domu = x, or domu = [ x,. By I we denote the inverse of the derivative of u. Assumption 2.6 implies that I0 =. The convex conjugate function u : R + R of u is defined by u y := sup{ux xy} x R and satisfies u y = uiy yiy. In the following we introduce minimax measures. In the case where K is the set of equivalent martingale measures they were first introduced in a stronger form in He and Pearson 99a. Recently they were studied in another modified form by Bellini and Frittelli 2000, Frittelli 2000b, and in a finite market setting by Kallsen 998a. The minimax martingale measure has an economic interpretation. It produces prices which are least favourable for an investor with a given utility profile, i.e. the maximal expected utility with respect to prices based on a martingale measure is minimal. For a brief discussion of the economical significance of the minimax martingale measure see He and Pearson 99b, who studied so-called minimax local martingale measures. To be precise, they did not consider a set of measures for their minimax problem, and hence their framework is slight different from ours. For Q K and x > x define U Q x := sup{euy : Y L Q, E Q Y x, EuY < }. 2.8 The following lemma gives a well-known representation of U Q x cf. Karatzas and Shreve 998. Lemma 2.8 Let Q K such that E Q Iλ dq < for all λ > 0. Then { i U Q x = inf } E u λ dq λ>0 + λx. ii There is a unique solution for λ in the equation E Q Iλ dq = x, denoted by λ Q x 0,, and U Q x = E[uIλ Q x dq ]. Proof. We set Z := dq. Let Y L Q with E Q Y x and EuY <. Then we have for λ > 0: EuY EuY + λx E Q Y Eu λz + λx = EuIλZ + λx E Q IλZ. The inequalities hold as equalities if and only if Y is given as Iλ Q xz. Since we have E Q IλZ < for all λ > 0, one can conclude that E Q IλZ is a continuous, monotonically decreasing function of λ with values in x,. This guarantees the

2.4. Minimal Distance and Minimax Measures 23 existence of λ Q x. Finally one has to check that E[uIλ Q xz] <. From the inequality one gets that The inequality ux xy uiy yiy E[uIλZ λziλz] <. [uiλz] [uiλz λziλz] + [λziλz] implies that the condition E[uIλ Q xz] < is fulfilled. Remarks.. The random variable Iλ Q x dq can be interpreted as optimal contingent claim which is financeable under the pricing measure Q. 2. Notice that if for Q K there exists λ > 0 with Eu λ dq <, then U Qx < for all x > x. Moreover, if for Q K with U Q x < the assumption of Lemma 2.8 is fulfilled, then Eu λ Q x dq <. 3. For log x, xp, p e x the corresponding convex conjugate functions are log x, p x p p p, x + x log x. Hence for ux = e x the u -divergence distance is the relative entropy, for ux = log x the reverse relative entropy, and for ux = x the Hellinger distance. Definition 2.9 A measure Q = Q x K is called minimax measure for x and K if it minimizes Q U Q x over all Q K, i.e., U Q x = Ux := inf Q K U Qx. Remark. In general the minimax measure Q will depend on x. Fortunately for the standard utility functions like ux = xp p, \ {0}, ux = log x and p ux = e px p 0, the minimax measure is independent of x. Our weak notion of minimax measures allows to formulate a complete equivalence to minimal distance measures with respect to related f-divergence distances. This is the reason why we did not use the stronger forms of this notion in He and Pearson 99a or Bellini and Frittelli 2000. Later on we will see that under weak conditions the weak notion of a minimax measure coincides with the stronger notion in He and Pearson 99a and also with that of Bellini and Frittelli 2000.