Optimal Hedging in Discrete and Continuous Time

42 Pages Posted: 14 Dec 2009 Last revised: 25 Jun 2010

See all articles by Bruno Remillard

Bruno Remillard

Department of Decision Sciences, HEC Montreal

Sylvain Rubenthaler

Université de Nice Sophia Antipolis

Date Written: December 11, 2009

Abstract

In this article we find the optimal solution of the hedging problem in discrete time by minimizing the mean square hedging error, when the underlying assets are multidimensional, extending the results of Schweizer (1995). We also find explicit expressions for the optimal hedging problem in continuous time when the underlying assets are modeled by a regime-switching geometric L'evy process. It is also shown that the continuous time solution can be approximated by discrete time Hidden Markov models processes. In addition, in the case of the regime-switching geometric Brownian motion, the optimal prices are the same as the prices under an equivalent martingale measure, making that measure a natural choice. However, the optimal hedging strategy is not the usual delta hedging but it can be easily computed by Monte Carlo methods. These results presented here are different from those of Follmer & Schweizer (1991) on hedging and Prigent (2003) on weak convergence, who both considered the problem of local quadratic risk minimizing strategies instead of the global quadratic risk minimizing strategies. Even in the discrete case, these two settings were shown to lead, in general, to different solutions by Schweizer (1995).

Keywords: hedging, option pricing, regime-switching, Levy processes

JEL Classification: G12, C51, C61, C22

Suggested Citation

Remillard, Bruno and Rubenthaler, Sylvain, Optimal Hedging in Discrete and Continuous Time (December 11, 2009). Available at SSRN: https://ssrn.com/abstract=1522090 or http://dx.doi.org/10.2139/ssrn.1522090

Bruno Remillard (Contact Author)

Department of Decision Sciences, HEC Montreal ( email )

3000 Côte-Sainte-Catherine Road
Montreal, QC H2S1L4
Canada
514-340-6794 (Phone)

Sylvain Rubenthaler

Université de Nice Sophia Antipolis ( email )

Nice
France

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