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.

Keywords: stochastic volatility, superreplication, stochastic optimal control, Hamilton-Jacobi-Bellman

JEL Classification: G13