Hedging options for a large investor and forward-backward SDE's

Abstract

In the classical continuous-time financial market model, stock prices have been understood as solutions to linear stochastic differential equations, and an important problem to solve is the problem of hedging options (functions of the stock price values at the expiration date). In this paper we consider the hedging problem not only with a price model that is nonlinear, but also with coefficients of the price equations that can depend on the portfolio strategy and the wealth process of the hedger. In mathematical terminology, the problem translates to solving a forward-backward stochastic differential equation with the forward diffusion part being degenerate. We show that, under reasonable conditions, the four step scheme of Ma, Protter and Yong for solving forward-backward SDE's still works in this case, and we extend the classical results of hedging contingent claims to this new model. Included in the examples is the case of the stock volatility increase caused by overpricing the option, as well as the case of different interest rates for borrowing and lending.