Bibliographical notes

In these notes, we do not make any attempt to give a systematic account of all the sources which have been relevant for the development of the field. We simply mention a number of references which had a direct influence on our decisions how to present the topics discussed in this book. More comprehensive lists of references can be found, e.g., Delbaen and Schachermayer [88], Jeanblanc, Yor and Chesney [164], Karatzas and Shreve [177], and McNeil, Frey, and Embrechts [208].

Chapter 1: The proof of Theorem 1.7 is based on Dalang, Morton, and Willinger [70]. Remark 1.18 and Example 1.19 are taken from Schachermayer [244]. The Taylor formulas in Example 1.25 and Example 1.3.3 go back at least to Lacroix [198]. Their financial interpretation was emphasized by Bick [28]. Exercise 1.5.3 was communicated to us by Martin Barbie. Section 1.6 is mainly based on [244], with the exception of Lemma 1.64, which is taken from Kabanov and Stricker [170]. Our proof of Lemma 1.68 combines ideas from[170] with the original argument in [244], as suggested to us by Irina Penner. For a historical overview of the development of arbitrage pricing and for an outlook to continuous-time developments, we refer to [88]. For some mathematical connections between superhedging of call options as discussed in Section 1.3 and bounds on stop-loss premiums in insurance see Chapter 5 of Goovaerts et al. [145].

Chapter 2: The results on the structure of preferences developed in this chapter are, to a large extent, standard topics in mathematical economics. We refer to textbooks on expected utility theory such as Fishburn [119], [120], Kreps [193], or Savage [243], and to the survey articles in [11], [16]. The ideas and results of Section 2.1 go back to classical references such as Debreu [80], Eilenberg [106], Milgram [211], and Rader [229]. The theory of affine numerical representations in Section 2.2 was initiated by von Neumann and Morgenstern [216] and further developed by Herstein and Milnor [155]. The characterization of certain equivalents with the translation in Proposition 2.46 goes back to de Finetti [118]; see also Kolmogorv [183] and Nagumo [215]. The drastic consequences of the assumption that a favorable bet is rejected at any level of wealth, as explained in Proposition 2.49, were stressed by Rabin [228]. The discussion of the partial orders ≽uni, ≽mon, and ≽bal in Sections 2.4 and 2.6 has a long history. Afirst version of Theorem 2.57 is already contained in Hardy, Littlewood, and Polya [148]. A complete treatment was given by Strassen [266]; this paper is also the source for Section 2.6. The economic interpretation of Theorem 2.57 was developed by Rothschild and Stiglitz [237], [238]. The analysis of robust preferences in Section 2.5 is mainly based on the ideas of Savage [243], Anscombe and Aumann [6], and Gilboa and Schmeidler [144]. See Gilboa [142] for an alternative axiomatic approach to the characterization in part (b) of Theorem 2.78, and Karni and Schmeidler [179] and Gilboa [143] for surveys of related developments. In the context of robust statistics, a special case of Proposition 2.84 appears in Huber [158]. The relaxed axiom of weak certainty independence and the corresponding extension of Theorem 2.87 is due to Maccheroni, Marinacci, and Rustichini [205]; see also [131].

Chapter 3: Given a preference relation of von Neumann–Morgenstern type, the analysis of optimal portfolios in Section 3.1 or, more generally, of optimal asset profiles in Section 3.3 is a standard exercise, both in microeconomic theory and in convex optimization. Section 3.1 shows that the existence of a solution is equivalent to the absence to arbitrage; here we follow Rogers [235]. In the special case of exponential utility, the construction of the optimal portfolio is equivalent to the minimization of relative entropy as discussed in Section 3.2. This may be viewed as the financial interpretation of general results on entropy minimization in Csiszar [62], [63]. The methods for characterizing optimal asset profiles in Section 3.3 in terms of “first-order conditions” are well-known; see, for example, [107] or [177], where they are developed in greater generality. The optimization problem in Theorem 3.44, which is formulated in terms of the partial order ≽uni and involves the Hardy–Littlewood inequalities of Theorem A.28, is less standard. Our discussion is based on Dybvig [105], Jouini and Kallal [166] and on a proof of Dana and Meilijson, and we are obliged to Rose-Anne Dana for introducing us to this topic; see Dana [74, 72] for further developments. The characterization of a least-favorable measure in Proposition 3.53 is due to Huber and Strassen [159]. The application to robust portfolio choice is taken from Schied [250]. The concept of weak information in Example 3.49 was suggested by Baudoin [18]. As to the existence of Arrow–Debreu equilibria discussed in Section 3.6, we refer to the classical version in Debreu [79] and to the survey articles in [11]. In our financial context, equilibrium allocations do no longer involve commodity bundles in Euclidean space as in [79] but asset profiles described by random variables on a probability space. This formulation of the equilibrium problem goes back to Borch [33], where it was motivated by the problem of risk exchange in a reinsurance market. The systematic analysis of the equilibrium problem in an infinite-dimensional setting was developed by Bewley, Mas-Colell, and others; see [27] and, for example, [206], [207], [4]. In our introductory approach, the existence proof is reduced to an application of Brouwer’s fixed point theorem. Here we benefitted from discussions with Peter Bank; see also Dana [71] and Carlier and Dana [44]. Examples 3.61 and Example 3.62 are based on Bühlmann [38]; see also [39] and Deprez and Gerber [95]. As mentioned in Remark 3.68, the equilibrium discussion of interest rates requires an intertemporal setting; for a systematic discussion see, e.g., Duffie [101] and, in a different conceptual framework, Bank and Riedel [15].

Chapter 4: The axiomatic approach to coherent measures of risk and their acceptance sets was initiated by Artzner, Delbaen, Eber, and Heath [12], and most results of Section 4.1 are based on this seminal paper. The extension to convex risk measures was given independently by Heath [152], Heath and Ku [154], Föllmer and Schied [129], and Frittelli and Rosazza Gianin [138]. The robust representation theorems in Section 4.2 are taken from [130]; the discussion of convex risk measures on a space of continuous functions corrects an error in [130] and in the first edition of this book; see also Krätschmer [186] for a further analysis. Exercise 4.2.1 is taken from El Karoui and Ravanelli [109]. The representation theory on L∞ as presented in Section 4.3 was developed by Delbaen [81], [82]; for the connection to the general duality theory as explained in Remarks 4.18 and 4.44 see [81], [82], [138], [139]. Among the results on Value at Risk and its various modifications in Section 4.4, Proposition 4.47 and Theorem 4.67 are taken from [12] and [81]. Average Value at Risk is discussed, e.g., by Acerbi and Tasche [2], Delbaen [81], and Rockafellar and Uryasev [233]. Remark 4.49 was pointed out to us by Ruszczynski; see [219]. The notation V@R is taken from Pflug and Ruszczynski [223]. The representations of law-invariant risk measures given in Section 4.5 were first obtained in the coherent case by Kusuoka [197]; see also Kunze [195] and Frittelli and Rosazza Gianin [139, 140] for the extension to the general convex case. Theorem 4.70 in Section 4.6 was first proved in [197]. The representations of the core of a concave distortion in Theorem 4.79 and Corollary 4.80 are due to Carlier and Dana [45]. See also [46] for further applications. The alternative representation in Exercise 4.6.8 is due to Cherny [53]. The definition of the risk measures MINVAR, MAXVAR, MINMAXVAR, and MAXMINVAR and the representations given in Exercises 4.1.8, 4.6.1, 4.6.5, 4.6.6, and 4.6.7 are due to Cherny and Madan [54]. Exercises 4.1.7 and 4.6.4 are inspired by Example A.1 in Pflug and Kovacevic [222]. The study of Choquet integrals with respect to general set functions as used in Section 4.7 was started by Choquet [56]. The connections with coherent risk measureswere observed by Delbaen [81], [82]. The two implications in Theorem 4.88 are due to Dellacherie [89] and Schmeidler [254], respectively. The proof via Lemma 4.89, which we give here, is taken from Denneberg [94]. Theorem 4.93 is due to Kusuoka [197]. The equivalence between (b) and (d) in Theorem 4.94 was first proved in [56], item (c) was added in [254]. The proof given here is based on [94]. The first part of Section 4.8 is based on [129], the second on Carr, Geman, and Madan [47]. Exercise 4.8.3 is taken from Barrieu and El Karoui [17]. The representation of utility-based shortfall risk in Section 4.9 is taken from [129]. Theorem 4.115 is an extension of a classical result for Orlicz spaces; see Krasno-selskii and Rutickii [191]. Exercise 4.9.2 is taken from Bellini, Klar, and Müller [22], Exercise 4.9.1 from Frittelli and Scandolo [141]. The distorted Bayesian preferences in Exercise 4.9.3 were introduced in Klibanoff, Marinacci and Mukerji [181]. Divergence risk measures were considered by Ben-Tal and Teboulle [24, 25] under the name of optimized certainty equivalents. The proof of Theorem 4.122 is taken from Schied [252]. In the actuarial literature, risk measures ρ have appeared in the form of premium principles H = −ρ; see Deprez and Gerber [95] and, e.g., Denneberg [93] and Wang and Dhaene [273].

Chapter 5: Martingales in Finance have a long history; see, e.g., Samuelson [241]. In the context of dynamic arbitrage theory, martingales and martingale measures are playing a central role, both in discrete and continuous time; for a historical overview we refer again to Delbaen and Schachermayer [88]. The first four sections of this chapter are based on Harrison and Kreps [149], Kreps [192], Harrison and Pliska [150], Dalang, Morton, and Willinger [70], Stricker [268], Schachermayer [244], Jacka [161], Rogers [235], Ansel and Stricker [10], and Kabanov and Kramkov [169]. The binomial model of Section 5.5 was introduced by Cox, Ross, and Rubinstein in [61]. Geometric Brownian motion, which appears in Section 5.7 as the diffusion limit of binomial models, was proposed since the late 1950s by Samuelson and others as a model for price fluctuations in continuous time, following the re-discovery of the linear Brownian motion model of Bachelier [14]; see Samuelson [242] and Cootner [60]. The corresponding dynamic theory of arbitrage pricing in continuous time goes back to Black and Scholes [31] and Merton [210]. A Black–Scholes type formula for option pricing appears in Sprenkle [265] in an ad hoc manner, without the arbitrage argument introduced by Black and Scholes. The approximation of Black–Scholes prices for various options by arbitrage-free prices in binomial models goes back to Cox, Ross, and Rubinstein [61]. A functional version of Theorem 5.54, based on Donsker’s invariance principle, can be found in [101].

Chapter 6: The dynamic arbitrage theory for American options begins with Bensoussan [23] and Karatzas [172]. A survey is given in Myeni [214]. The theory of optimal stopping problems as presented in Sections 6.1 and 6.2 was initiated by Snell [264]; see [217] for a systematic introduction. Stability under pasting as discussed in Section 6.4 has appeared under several names in various contexts; see Delbaen [83] for a number of references and for an extension to continuous time. Our discussion of upper and lower Snell envelopes in Section 6.5 uses ideas from Karatzas and Kou [175] and standard techniques from dynamic programming. The application to the time consistency of dynamic coherent risk measures in Remark 6.54 recovers results by Artzner, Delbaen, Eber, Heath, and Ku [13], Delbaen [83], and Riedel [231].

Chapter 7: Optional decompositions, or uniform Doob decompositions as we call them, and the resulting construction of superhedging strategies were first obtained by El Karoui and Quenez [108] in a jump-diffusion model. In a general semimar-tingale setting, the theory was developed by Kramkov [188], Föllmer and Kabanov [124], and Delbaen and Schachermayer [88]. From a mathematical point of view, the existence of martingale measures with marginals determined by given option prices in Theorem 7.25 is a corollary of Strassen [266]; continuous-time analoga were proved by Doob [98] and Kellerer [180]. For the economic interpretation, see, e.g., Breeden and Litzenberger [35]. The results on superhedging of exotic derivatives by means of plain vanilla options stated in Theorems 7.27, 7.31, 7.33, and Corollary 7.34 are due to Hobson [156] and Brown, Hobson, and Rogers [36]; they are related to martingale inequalities of Dubins and Gilat [99].

Chapter 8: The analysis of quantile hedging was triggered by a talk of D. Heath in March 1995 at the Isaac Newton Institute on the results in Kulldorf [194], where an optimization problem for Brownian motion with drift is reduced to the Neyman–Pearson lemma. Section 8.1 is based on Föllmer and Leukert [126]; see also Karatzas [173], Cvitanic and Spivak [69], Cvitanic and Karatzas [68], and Browne [37]. The results in Section 8.2 on minimizing the shortfall risk are taken from Föllmer and Leukert [127]; see also Leukert [203], Cvitanic and Karatzas [67], Cvitanic [64], and Pham [224]. Theorem 8.26 is due to Sekine [259]. The proof given here is taken from Schied [251] and Theorem 8.27 from [249].

Chapter 9: In continuous-time models, dynamic arbitrage pricing with portfolio constraints was considered by Cvitanic and Karatzas [65], [66]. Proposition 9.6 and Exercise 9.1.1 are based on Jacod and Shiryaev [162]. In a discrete-time model with convex constraints, absence of arbitrage was characterized by Carrassus, Pham, and Touzi [41]. In a general semimartingale setting, Föllmer and Kramkov [125] proved a uniform Doob decomposition and superhedging duality theorems for a predictably convex set of admissible trading strategies and for American contingent claims; see also Karatzas and Kou [175].

Chapter 10: The idea of quadratic risk minimization for hedging strategies goes back to Föllmer and Sondermann [134], where the optimality criterion was formulated with respect to a martingale measure. Extensions to the general case and the construction of minimal martingale measures were developed by Föllmer and Schweizer [133] and Schweizer; see, e.g., [255], [256]. Our exposition also uses arguments from Föllmer and Schweizer [132], Schäl [248], and Li and Xia [204]. Variance-optimal hedging was introduced by Duffie and Richardson [103] and further developed by Schweizer and others; the discrete-time theory as presented in Section 10.3 is based on Schweizer [257]. Melnikov and Nechaev [209] give an explicit formula for a variance-optimal strategy without condition (10.24); in fact, they show that their formula always defines a variance-optimal strategy if one does not insist on the square-integrability of the gains process at intermediate times. For a survey on related results, we refer to [258].

Chapter 11: Representations of conditional risk measures as in Section 11.1 were discussed in Artzner, Delbaen, Eber, Heath, and Ku [13], Bion-Nadal [29, 30], Burgert [40], Cheridito, Delbaen, and Kupper [48, 49, 50], Delbaen [83], Detlefsen and Scandolo [96], Föllmer and Penner [128], Penner [220], and Riedel [231]. Exercise 11.1.2 is taken from [50]. The characterization of time consistency in Section 11.2 is due to [13] and [83] in the coherent case and to [50], [29], [30], [220], and [128] in the general convex case; for a detailed survey we refer to Acciao and Penner [1].

Appendix. Our exposition of convex sets and functions in Section A.1 is mainly based on Rockafellar [232]. Additional background on absolutely continuous measures as considered in Section A.2 can be found in the books [20, 21] by Bauer. In analysis, quantile functions as discussed in Section A.3 are studied under the notion of an increasing rearrangement of a measurable function. We refer to Hardy, Littlewood, and Pólya [148] and to Chong and Rice [55] for further results and an historical overview. For Lemma A.26 see Pflug [221], Rockafellar and Uryasev [233], and [2]. Exercise A.3.2 is due to Rüschendorf [239]; see also Ferguson [115]. The statement of Lemma A.32 is due to Ryff [240]. The results in Section A.4 go back to Neyman and Pearson [218]. Comprehensive treatments of the material from Sections A.7 and A.6 can be found in Bourbaki [34] and Dunford and Schwartz [104].