Robustness of the Black-Scholes approach in the case of options on several assets

Abstract. In this paper we analyse a stochastic volatility model that is an extension of the traditional Black-Scholes one. We price European options on several assets by using a superstrategy approach. We characterize the Markov superstrategies, and show that they are linked to a nonlinear PDE, called the Black-Scholes-Barenblatt (BSB) equation. This equation is the Hamilton-Jacobi-Bellman equation of an optimal control problem, which has a nice financial interpretation. Then we analyse the optimization problem included in the BSB equation and give some sufficient conditions for reduction of the BSB equation to a linear Black-Scholes equation. Some examples are given.

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

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

[3]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[4]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[5]  Jean-Charles Rochet,et al.  Changes of numéraire, changes of probability measure and option pricing , 1995, Journal of Applied Probability.

[6]  William Margrabe.  The Value of an Option to Exchange One Asset for Another , 1978 .

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

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

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

[10]  H. Johnson.  Options on the Maximum or the Minimum of Several Assets , 1987, Journal of Financial and Quantitative Analysis.

[11]  N. Krylov.  Controlled Diffusion Processes , 1980 .

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

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

[14]  René M. Stulz,et al.  Options on the minimum or the maximum of two risky assets : Analysis and applications , 1982 .

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

[16]  N. S. Rasmussen.  Hedging with a Misspecified Model , 2001 .