COMPOSITION OF TIME-CONSISTENT DYNAMIC MONETARY RISK MEASURES IN DISCRETE TIME

In discrete time, every time-consistent dynamic monetary risk measure can be written as a composition of one-step risk measures. We exploit this structure to give new dual representation results for time-consistent convex monetary risk measures in terms of one-step penalty functions. We first study risk measures for random variables modelling financial positions at a fixed future time. Then we consider the more general case of risk measures that depend on stochastic processes describing the evolution of financial positions or cumulated cash flows. In both cases the new representations allow for a simple composition of one-step risk measures in the dual. We discuss several explicit examples and provide connections to the recently introduced class of dynamic variational preferences.

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

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

[3]  Evan L. Porteus,et al.  Temporal Resolution of Uncertainty and Dynamic Choice Theory , 1978 .

[4]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework , 1989 .

[5]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[6]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

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

[8]  F. Delbaen.  Coherent risk measures , 2000 .

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

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

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

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

[13]  Tan Wang,et al.  Conditional preferences and updating , 2003, J. Econ. Theory.

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

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

[16]  Marco Frittelli,et al.  Dynamic convex risk measures , 2004 .

[17]  G. P. Szegö,et al.  Risk measures for the 21st century , 2004 .

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

[19]  Terry J. Lyons,et al.  Stochastic finance. an introduction in discrete time , 2004 .

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

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

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

[23]  F. Delbaen,et al.  Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes , 2004, math/0410453.

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

[25]  M. Frittelli,et al.  RISK MEASURES AND CAPITAL REQUIREMENTS FOR PROCESSES , 2006 .

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

[27]  Alexander Shapiro,et al.  Conditional Risk Mappings , 2005, Math. Oper. Res..

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

[29]  H. Föllmer,et al.  Convex risk measures and the dynamics of their penalty functions , 2006 .

[30]  Massimo Marinacci,et al.  Dynamic variational preferences , 2006, J. Econ. Theory.

[31]  S. Weber,et al.  DISTRIBUTION‐INVARIANT RISK MEASURES, INFORMATION, AND DYNAMIC CONSISTENCY , 2006 .

[32]  Susanne Klöppel,et al.  DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES , 2007 .

[33]  Leonard Rogers,et al.  VALUATIONS AND DYNAMIC CONVEX RISK MEASURES , 2007, 0709.0232.

[34]  N. El Karoui,et al.  Pricing, Hedging and Optimally Designing Derivatives via Minimization of Risk Measures , 2007, 0708.0948.

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

[36]  Freddy Delbaen,et al.  Representation of the penalty term of dynamic concave utilities , 2008, Finance Stochastics.

[37]  Hans Föllmer,et al.  Risk assessment for uncertain cash flows: model ambiguity, discounting ambiguity, and the role of bubbles , 2010, Finance Stochastics.