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.
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