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Hedging Options for a Large Investor and Forward-Backward SDE's

Cvitanić, Jakša and Ma, Jin (1996) Hedging Options for a Large Investor and Forward-Backward SDE's. Annals of Applied Probability, 6 (2). pp. 370-398. ISSN 1050-5164. http://resolver.caltech.edu/CaltechAUTHORS:20120209-085609546

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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.


Item Type:Article
Related URLs:
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http://www.jstor.org/stable/2245176PublisherUNSPECIFIED
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].
Funders:
Funding AgencyGrant Number
NSFDMS-95-03582
Army Research Office (ARO)DAAH-04-95-1-0528
NSFDMS-93-01516
Subject Keywords:Forward-backward stochastic differential equations, contingent claims, hedging strategy, large investor.
Classification Code:AMS 1991 subject classifications: Primary 90A09, 60H30; secondary 90A12, 93A20.
Record Number:CaltechAUTHORS:20120209-085609546
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120209-085609546
Official Citation:Full Access Hedging Options for a Large Investor and Forward-Backward SDE's(pp. 370-398) Jakša Cvitanić, Jin Ma Stable URL: http://www.jstor.org/stable/2245176
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29219
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:09 Feb 2012 17:50
Last Modified:26 Dec 2012 14:49

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