Optimal arbitrage under model uncertainty.

In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is the value function of an associated stochastic control problem, namely, the maximal probability with which an auxiliary multidimensional diffusion process, controlled in a manner which affects both its drift and covariance structures, stays in the interior of the positive orthant through the end of the time-horizon. This value function is also characterized in terms of a stochastic game, and can be used to generate an investment rule that realizes such best possible outperformance of the market.

[1]  Joerg Vorbrink,et al.  Financial markets with volatility uncertainty , 2010, 1012.1535.

[2]  I. Karatzas,et al.  On optimal arbitrage , 2010, 1010.4987.

[3]  Jakša Cvitanić,et al.  Super-replication in stochastic volatility models under portfolio constraints , 1999, Journal of Applied Probability.

[4]  P. Meyer,et al.  La mesure de H. Föllmer en théorie des surmartingales , 1972 .

[5]  Johan Tysk,et al.  Bubbles, convexity and the Black–Scholes equation , 2009, 0908.4468.

[6]  KarouiNicole El,et al.  Compactification methods in the control of degenerate diffusions: existence of an optimal control , 1987 .

[7]  Denis Talay,et al.  Worst case model risk management , 2002, Finance Stochastics.

[8]  Lihe Wang.  On the regularity theory of fully nonlinear parabolic equations , 1990 .

[9]  E. Bayraktar,et al.  Optimal Stopping for Non-linear Expectations , 2009, 0905.3601.

[10]  Nicolai V. Krylov,et al.  Nonlinear Elliptic and Parabolic Equations of the Second Order Equations , 1987 .

[11]  Svante Janson,et al.  Feynman–kac Formulas for Black–Scholes–Type Operators , 2006 .

[12]  P. Dorato,et al.  Numerical solution of an optimal control problem with a probability criterion , 1972 .

[13]  Walter Schachermayer,et al.  The Existence of Absolutely Continuous Local Martingale Measures (1995) , 1995 .

[14]  Rüdiger Frey,et al.  Superreplication in stochastic volatility models and optimal stopping , 2000, Finance Stochastics.

[15]  H. Föllmer,et al.  Robust Preferences and Robust Portfolio Choice , 2009 .

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

[17]  Game approach to the optimal stopping problem† , 2005 .

[18]  M. Yor,et al.  Equivalent and absolutely continuous measure changes for jump-diffusion processes , 2005, math/0508450.

[19]  G. Meyer.  THE BLACK SCHOLES BARENBLATT EQUATION FOR OPTIONS WITH UNCERTAIN VOLATILITY AND ITS APPLICATION TO STATIC HEDGING , 2006 .

[20]  N. Krylov.  ON THE SELECTION OF A MARKOV PROCESS FROM A SYSTEM OF PROCESSES AND THE CONSTRUCTION OF QUASI-DIFFUSION PROCESSES , 1973 .

[21]  Constantinos Kardaras,et al.  The numéraire portfolio in semimartingale financial models , 2007, Finance Stochastics.

[22]  I. Karatzas,et al.  Optimal Stopping for Dynamic Convex Risk Measures , 2009, 0909.4948.

[23]  Steven Orey,et al.  Reaching zero rapidly , 1987 .

[24]  P. Lions.  Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part I , 1983 .

[25]  P. Protter,et al.  Analysis of continuous strict local martingales via h-transforms , 2007, 0711.1136.

[26]  D. Duffy.  Second‐Order Parabolic Differential Equations , 2013 .

[27]  Shlomo Levental,et al.  A Necessary and Sufficient Condition for Absence of Arbitrage with Tame Portfolios , 1995 .

[28]  S. Shreve,et al.  Robustness of the Black and Scholes Formula , 1998 .

[29]  Terry Lyons,et al.  Uncertain volatility and the risk-free synthesis of derivatives , 1995 .

[30]  H. Föllmer,et al.  Robust projections in the class of martingale measures , 2006 .

[31]  Tiziano Vargiolu,et al.  Superreplication of European multiasset derivatives with bounded stochastic volatility , 2002, Math. Methods Oper. Res..

[32]  J. Ruf,et al.  HEDGING UNDER ARBITRAGE , 2010, 1003.4797.

[33]  K. Parthasarathy,et al.  Probability measures on metric spaces , 1967 .

[34]  Ananda Weerasinghe,et al.  Controlling a Process to a Goal in Finite Time , 1989, Math. Oper. Res..

[35]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[36]  David Heath,et al.  Minimizing or maximizing the expected time to reach zero , 1987 .

[37]  Tiziano Vargiolu,et al.  Robustness of the Black-Scholes approach in the case of options on several assets , 2000, Finance Stochastics.

[38]  Dimitri P. Bertsekas,et al.  Stochastic optimal control : the discrete time case , 2007 .

[39]  Erhan Bayraktar,et al.  Optimal Stopping for Non-Linear Expectations - Part I , 2009 .

[40]  D. W. Stroock,et al.  Multidimensional Diffusion Processes , 1979 .

[41]  W. Fleming,et al.  Deterministic and Stochastic Optimal Control , 1975 .

[42]  P. Lions.  Optimal control of diffusion processes and hamilton–jacobi–bellman equations part 2 : viscosity solutions and uniqueness , 1983 .

[43]  I. Karatzas,et al.  Probabilistic Aspects of Arbitrage , 2010 .

[44]  C. Kardaras,et al.  Robust maximization of asymptotic growth , 2010, 1005.3454.

[45]  W. Fleming,et al.  Convex duality approach to the optimal control of diffusions , 1989 .

[46]  I. Karatzas,et al.  Relative arbitrage in volatility-stabilized markets , 2005 .

[47]  Soumik Pal.  Analysis of market weights under volatility-stabilized market models , 2009, 0904.0656.

[48]  William D. Sudderth,et al.  Continuous-Time Red and Black: How to Control a Diffusion to a Goal , 1985, Math. Oper. Res..

[49]  Marcel Nutz,et al.  Random G-expectations. , 2010, 1009.2168.

[50]  F. Riedel.  Optimal Stopping With Multiple Priors , 2009 .

[51]  Hans Föllmer,et al.  ON THE REPRESENTATION OF SEMIMARTINGALES , 1973 .

[52]  H. Soner,et al.  Quasi-sure Stochastic Analysis through Aggregation , 2010, 1003.4431.

[53]  P. Lions.  Some Recent Results in the Optimal Control of Diffusion Processes , 1984 .

[54]  Existence , uniqueness and smoothness for the Black-Scholes-Barenblatt equation , 2001 .

[55]  S. Peng.  Nonlinear Expectations and Stochastic Calculus under Uncertainty , 2010, Probability Theory and Stochastic Modelling.

[56]  M. Avellaneda,et al.  Pricing and hedging derivative securities in markets with uncertain volatilities , 1995 .

[57]  U. Haussmann,et al.  On the existence of optimal controls , 1990 .

[58]  ching-tang wu,et al.  Duality theory for optimal investments under model uncertainty , 2005 .

[59]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[60]  J. Urbas,et al.  NONLINEAR ELLIPTIC AND PARABOLIC EQUATIONS OF THE SECOND ORDER , 1989 .

[61]  E. Fernholz.  Stochastic Portfolio Theory , 2002 .

[62]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[63]  N. Krylov.  A supermartingale characterization of a set of stochastic integrals , 1989 .

[64]  Alexander Schied,et al.  Optimal investments for risk- and ambiguity-averse preferences: a duality approach , 2006, Finance Stochastics.

[65]  Walter Schachermayer,et al.  The no-arbitrage property under a change of numéraire , 1995 .

[66]  国田 寛.  Stochastic flows and stochastic differential equations , 1990 .

[67]  Hans Föllmer,et al.  The exit measure of a supermartingale , 1972 .

[68]  L. Denis,et al.  A THEORETICAL FRAMEWORK FOR THE PRICING OF CONTINGENT CLAIMS IN THE PRESENCE OF MODEL UNCERTAINTY , 2006, math/0607111.

[69]  H. Soner,et al.  Dual Formulation of Second Order Target Problems , 2010, 1003.6050.

[70]  N. Krylov.  A supermartingale characterization of sets of stochastic integrals and applications , 2002 .

[71]  N. Krylov.  Smoothness of the value function for a controlled diffusion process in a domain , 1990 .

[72]  P. Protter.  Stochastic integration and differential equations , 1990 .

[73]  Lihe Wang.  On the regularity theory of fully nonlinear parabolic equations: II , 1992 .