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Published May 1996 | Published
Journal Article Open

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.

Additional Information

© 1996 Institute of Mathematical Statistics. Received December 1994; revised January 1996. Research supported in part by NSF Grant DMS-95-03582 and Army Research Office Grant DAAH 04-95-1-0528. Research supported in part by NSF Grant DMS-93-01516. The authors would like to thank Professors Jiongmin Yong of Fudan University, China, and Martin Schweizer of Technische Universität Berlin, Germany, for helpful discussions. In particular, Martin Schweizer and the two anonymous referees pointed out that our original, longer proof of Lemma 2.3 is not needed. Thanks are also due one of the referees for many useful suggestions, in particular, for simplifying some of the proofs by using results from [8].

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August 20, 2023
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