Time consistency of dynamic risk measures in markets with transaction costs

Abstract Set-valued dynamic risk measures are defined on with and with an image space in the power set of . Primal and dual representations of dynamic risk measures are deduced. Definitions of different time consistency properties in the set-valued framework are given. It is shown that the recursive form for multivariate risk measures as well as an additive property for the acceptance sets is equivalent to a stronger time consistency property called multi-portfolio time consistency.

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

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

[3]  Patrick Cheridito,et al.  Time-Inconsistency of VaR and Time-Consistent Alternatives , 2007 .

[4]  Raimund M. Kovacevic.  Conditional risk and acceptability mappings as Banach-lattice valued mappings , 2012 .

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

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

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

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

[9]  J. Bion-Nadal.  Time consistent dynamic risk processes , 2009 .

[10]  Zhao Zhang,et al.  Dynamic Coherent Acceptability Indices and Their Applications to Finance , 2010 .

[11]  Andreas Löhne,et al.  Vector Optimization with Infimum and Supremum , 2011, Vector Optimization.

[12]  Raimund M. Kovacevic.  Time consistency and information monotonicity of multiperiod acceptability functionals , 2009 .

[13]  Mher Safarian,et al.  Markets with Transaction Costs: Mathematical Theory , 2009 .

[14]  L. Pontrjagin,et al.  LINEAR DIFFERENTIAL GAMES OF PURSUIT , 1981 .

[15]  Ilya Molchanov,et al.  MULTIVARIATE RISK MEASURES: A CONSTRUCTIVE APPROACH BASED ON SELECTIONS , 2013, 1301.1496.

[16]  Andreas H. Hamel,et al.  Benson type algorithms for linear vector optimization and applications , 2013, J. Glob. Optim..

[17]  Emmanuel Lepinette,et al.  Vector-Valued Risk Measure Processes , 2013 .

[18]  Nizar Touzi,et al.  Vector-valued coherent risk measures , 2002, Finance Stochastics.

[19]  Beatrice Acciaio,et al.  Dynamic risk measures , 2011 .

[20]  Birgit Rudloff,et al.  An algorithm for calculating the set of superhedging portfolios in markets with transaction costs , 2011, 1107.5720.

[21]  Igor Cialenco,et al.  DYNAMIC CONIC FINANCE: PRICING AND HEDGING IN MARKET MODELS WITH TRANSACTION COSTS VIA DYNAMIC COHERENT ACCEPTABILITY INDICES , 2012, 1205.4790.

[22]  Damir Filipović,et al.  Separation and duality in locally L 0 -convex modules , 2009 .

[23]  A. Hamel,et al.  Risk minimization and set-valued average value at risk via linear vector optimization , 2012 .

[24]  Patrick Cheridito,et al.  COMPOSITION OF TIME-CONSISTENT DYNAMIC MONETARY RISK MEASURES IN DISCRETE TIME , 2011 .

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

[26]  Alet Roux.  The Fundamental Theorem of Asset Pricing Under Proportional Transaction Costs , 2007, 0710.2758.

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

[28]  Yuri Kabanov,et al.  Hedging and liquidation under transaction costs in currency markets , 1999, Finance Stochastics.

[29]  Andreas H. Hamel,et al.  Duality for Set-Valued Measures of Risk , 2010, SIAM J. Financial Math..

[30]  Saul Jacka,et al.  ON REPRESENTING CLAIMS FOR COHERENT RISK MEASURES , 2007 .

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

[32]  Y. Kabanov,et al.  Markets with transaction costs , 2010 .

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

[34]  Andreas H. Hamel,et al.  Set-valued risk measures for conical market models , 2010, 1011.5986.

[35]  Damir Filipović,et al.  Approaches to Conditional Risk , 2011, SIAM J. Financial Math..

[36]  H. Hadwiger,et al.  Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt , 1950 .

[37]  Paolo Guasoni,et al.  The fundamental theorem of asset pricing under transaction costs , 2011, Finance and Stochastics.

[38]  Andreas H. Hamel,et al.  Set-valued average value at risk and its computation , 2012 .

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

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

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

[42]  Andreas H. Hamel,et al.  A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory , 2009 .

[43]  Jocelyne Bion-Nadal,et al.  Dynamic risk measures: Time consistency and risk measures from BMO martingales , 2008, Finance Stochastics.