An ergodic BSDE approach to forward entropic risk measures: representation and large-maturity behavior

Using elements from the theory of ergodic backward stochastic differential equations (BSDEs), we study the behavior of forward entropic risk measures in stochastic factor models. We derive general representation results (via both BSDEs and convex duality) and examine their asymptotic behavior for risk positions of large maturities. We also compare them with their classical counterparts and provide a parity result.

[1]  M. Kobylanski.  Backward stochastic differential equations and partial differential equations with quadratic growth , 2000 .

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

[3]  M. Musiela,et al.  Portfolio choice under dynamic investment performance criteria , 2009 .

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

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

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

[7]  Marek Musiela.  Derivatives pricing , investment management and the term structure of exponential utilities : The case of binomial model , 2007 .

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

[9]  Huyen Pham,et al.  Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach , 2014, 1410.1125.

[10]  M. Musiela,et al.  Investment and Valuation Under Backward and Forward Dynamic Exponential Utilities in a Stochastic Factor Model , 2007 .

[11]  S. Nadtochiy,et al.  OPTIMAL INVESTMENT FOR ALL TIME HORIZONS AND MARTIN BOUNDARY OF SPACE‐TIME DIFFUSIONS , 2013, 1308.2254.

[12]  Thaleia Zariphopoulou,et al.  Representation of homothetic forward performance processes via ergodic and infinite horizon quadratic BSDE in stochastic factor models , 2015 .

[13]  D. Hobson,et al.  Utility indifference pricing : an overview , 2009 .

[14]  Marek Musiela,et al.  An example of indifference prices under exponential preferences , 2004, Finance Stochastics.

[15]  Nicole El Karoui,et al.  An Exact Connection between Two Solvable SDEs and a Nonlinear Utility Stochastic PDE , 2010, SIAM J. Financial Math..

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

[17]  Emanuela Rosazza Gianin,et al.  Risk measures via g-expectations , 2006 .

[18]  Ying Hu,et al.  Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces , 2007, SIAM J. Control. Optim..

[19]  W. Fleming,et al.  Risk-Sensitive Control on an Infinite Time Horizon , 1995 .

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

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

[22]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

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

[24]  Pierre-Yves Madec,et al.  A Probabilistic Approach to Large Time Behavior of Mild Solutions of HJB Equations in Infinite Dimension , 2015, SIAM J. Control. Optim..

[25]  Vicky Hendersony.  Valuation of Claims on Non-Traded assets using Utility Maximisation , 2001 .

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

[27]  Ying Hu,et al.  Exponential utility maximization and indifference valuation with unbounded payoffs , 2017 .

[28]  Vicky Henderson,et al.  Pseudo linear pricing rule for utility indifference valuation , 2014, Finance Stochastics.

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

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

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

[32]  Ronnie Sircar,et al.  Asymptotic Analysis of Forward Performance Processes in Incomplete Markets and Their Ill-Posed HJB Equations , 2016, SIAM J. Financial Math..

[33]  Gordan Zitkovic.  A dual characterization of self-generation and exponential forward performances , 2009 .

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

[35]  S. Peng,et al.  Filtration-consistent nonlinear expectations and related g-expectations , 2002 .

[36]  T. Zariphopoulou,et al.  A Class of Homothetic Forward Investment Performance Processes with Non-zero Volatility , 2014 .

[37]  Thaleia Zariphopoulou,et al.  Optimal asset allocation in a stochastic factor model-an overview and open problems , 2009 .

[38]  Thaleia Zariphopoulou,et al.  Representation of Homothetic Forward Performance Processes in Stochastic Factor Models Via Ergodic and Infinite Horizon BSDE , 2015, SIAM J. Financial Math..

[39]  M. Musiela,et al.  Stochastic Partial Differential Equations and Portfolio Choice , 2010 .

[40]  V. Henderson,et al.  VALUATION OF CLAIMS ON NONTRADED ASSETS USING UTILITY MAXIMIZATION , 2002 .

[41]  Gordan Zitkovic,et al.  Maturity-independent risk measures , 2007, 2008 47th IEEE Conference on Decision and Control.

[42]  Thaleia Zariphopoulou,et al.  An Approximation Scheme for Solution to the Optimal Investment Problem in Incomplete Markets , 2013, SIAM J. Financial Math..

[43]  M. Musiela,et al.  Optimal Asset Allocation under Forward Exponential Performance Criteria , 2008 .

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

[45]  Hanqing Jin,et al.  Time-Inconsistent Stochastic Linear-Quadratic Control , 2011, SIAM J. Control. Optim..

[46]  Marek Musiela,et al.  Portfolio Choice under Space-Time Monotone Performance Criteria , 2010, SIAM J. Financial Math..

[47]  Gianmario Tessitore,et al.  Ergodic BSDEs under weak dissipative assumptions , 2010, 1004.1571.

[48]  Gordan vZitkovi 'c.  A dual characterization of self-generation and exponential forward performances , 2009 .

[49]  Ronnie Sircar,et al.  Bounds and Asymptotic Approximations for Utility Prices when Volatility is Random , 2004, SIAM J. Control. Optim..

[50]  Ying Hu,et al.  Stability of BSDEs with Random Terminal Time and Homogenization of Semilinear Elliptic PDEs , 1998 .