Dynamic utility indierence valuation via convex risk measures

The (subjective) indierence value of a payo in an incomplete finan- cial market is that monetary amount which leaves an agent indierent between buying or not buying the payo when she always optimally ex- ploits her trading opportunities. We study these values over time when they are defined with respect to a dynamic monetary concave utility functional, i.e., minus a dynamic convex risk measure. For that pur- pose, we prove some new results about families of conditional convex risk measures. We study the convolution of abstract conditional con- vex risk measures and show that it preserves the dynamic property of time-consistency. Moreover, we prove that the dynamic risk measure (or utility functional) associated to superhedging in a market with trading constraints is time-consistent. By combining these results, we deduce that the corresponding indierence valuation functional is again time- consistent. As an auxiliary tool, we establish a variant of the represen- tation theorem for conditional convex risk measures which is in terms of equivalent probability measures. Since backward stochastic dierential equations (BSDEs) induce time-consistent DMCUFs, we also show how the valuation approach works in a BSDE setting.

[1]  B. Roorda,et al.  COHERENT ACCEPTABILITY MEASURES IN MULTIPERIOD MODELS , 2005 .

[2]  N. Karoui,et al.  Optimal derivatives design under dynamic risk measures , 2004 .

[3]  Peter Grandits,et al.  On the minimal entropy martingale measure , 2002 .

[4]  Olivier Ledoit,et al.  Gain, Loss, and Asset Pricing , 2000, Journal of Political Economy.

[5]  Freddy Delbaen,et al.  REPRESENTING MARTINGALE MEASURES WHEN ASSET PRICES ARE CONTINUOUS AND BOUNDED , 1992 .

[6]  Pauline Barrieu,et al.  Inf-convolution of risk measures and optimal risk transfer , 2005, Finance Stochastics.

[7]  F. Delbaen,et al.  Coherent and convex risk measures for bounded cadlag processes , 2003 .

[8]  Steven E. Shreve,et al.  Satisfying convex risk limits by trading , 2005, Finance Stochastics.

[9]  Hélyette Geman,et al.  Pricing and hedging in incomplete markets , 2001 .

[10]  Giacomo Scandolo,et al.  Conditional and dynamic convex risk measures , 2005, Finance Stochastics.

[11]  M. Owen,et al.  Utility based optimal hedging in incomplete markets , 2002 .

[12]  S. Peng.  Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob–Meyers type , 1999 .

[13]  M. Frittelli,et al.  Putting order in risk measures , 2002 .

[14]  David Heath,et al.  Coherent multiperiod risk adjusted values and Bellman’s principle , 2007, Ann. Oper. Res..

[15]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[16]  J. Neveu,et al.  Discrete Parameter Martingales , 1975 .

[17]  Martin Schneider,et al.  Recursive multiple-priors , 2003, J. Econ. Theory.

[18]  P. Imkeller,et al.  Utility maximization in incomplete markets , 2005, math/0508448.

[19]  Nicole El Karoui,et al.  Pricing Via Utility Maximization and Entropy , 2000 .

[20]  Frank Riedel,et al.  Dynamic Coherent Risk Measures , 2003 .

[21]  F. Delbaen,et al.  Coherent and convex monetary risk measures for bounded càdlàg processes , 2004 .

[22]  Mingxin Xu,et al.  Risk measure pricing and hedging in incomplete markets , 2006 .

[23]  F. Delbaen.  Coherent Risk Measures on General Probability Spaces , 2002 .

[24]  J. Cochrane,et al.  Beyond Arbitrage: 'Good Deal' Asset Price Bounds in Incomplete Markets , 1996 .

[25]  Aleš Černý,et al.  The Theory of Good-Deal Pricing in Financial Markets , 1998 .

[26]  Hedging bounded claims with bounded outcomes , 2006 .

[27]  F. Delbaen,et al.  Exponential Hedging and Entropic Penalties , 2002 .

[28]  Alexander Schied,et al.  Convex measures of risk and trading constraints , 2002, Finance Stochastics.

[29]  Christophe Stricker,et al.  Couverture des actifs contingents et prix maximum , 1994 .

[30]  Vyacheslav V. Kalashnikov,et al.  Complementarity, Equilibrium, Efficiency and Economics , 2002 .

[31]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[32]  Uwe Küchler,et al.  Coherent risk measures and good-deal bounds , 2001, Finance Stochastics.

[33]  H. Föllmer,et al.  Optional decompositions under constraints , 1997 .

[34]  Marco Frittelli,et al.  Introduction to a theory of value coherent with the no-arbitrage principle , 2000, Finance Stochastics.

[35]  S. Peng,et al.  Adapted solution of a backward stochastic differential equation , 1990 .

[36]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[37]  Georg Ch. Pflug,et al.  A risk measure for income processes , 2004 .

[38]  Patrick Cheridito,et al.  Coherent and convex monetary risk measures for unbounded càdlàg processes , 2004, Finance Stochastics.

[39]  Aleš Černý.  Generalized Sharpe Ratios and Asset Pricing in Incomplete Markets , 2000 .

[41]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[42]  J. Mémin,et al.  Espaces de semi martingales et changement de probabilité , 1980 .

[43]  Jeremy Staum,et al.  Fundamental Theorems of Asset Pricing for Good Deal Bounds , 2004 .

[44]  S. Weber.  Distribution-Invariant Dynamic Risk Measures , 2003 .

[45]  Walter Schachermayer,et al.  A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time , 1992 .

[46]  F. Delbaen.  The Structure of m–Stable Sets and in Particular of the Set of Risk Neutral Measures , 2006 .

[47]  Ying Hu,et al.  A Converse Comparison Theorem for BSDEs and Related Properties of g-Expectation , 2000 .

[48]  S. Peng.  Nonlinear Expectations, Nonlinear Evaluations and Risk Measures , 2004 .

[49]  Dirk Becherer,et al.  Rational hedging and valuation of integrated risks under constant absolute risk aversion , 2003 .

[50]  Michael Kohlmann,et al.  Optimal Superhedging under Nonconvex Constraints { A BSDE Approach , 2004 .

[51]  Hui Wang,et al.  Utility maximization in incomplete markets with random endowment , 2001, Finance Stochastics.

[52]  Michael Mania,et al.  Dynamic exponential utility indifference valuation , 2005 .

[53]  T. Koopmans.  Stationary Ordinal Utility and Impatience , 1960 .