MEAN VARIANCE HEDGING IN A GENERAL JUMP MARKET

Dewen Xiong () and Michael Kohlmann ()
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Dewen Xiong: Department of Mathematics, Shanghai Jiaotong University, Shanghai (200240), P. R. China
Michael Kohlmann: Department of Mathematics and Statistics, University of Konstanz, D-78457, Konstanz, Germany

International Journal of Theoretical and Applied Finance (IJTAF), 2010, vol. 13, issue 05, 789-820

Abstract: We consider a financial market in which the discounted price process S is an ℝd-valued semimartingale with bounded jumps, and the variance-optimal martingale measure (VOMM) Qopt is only known to be a signed measure. We give a backward semimartingale equation (BSE) and show that the density process Zopt of Qopt with respect to P is a possibly non-positive stochastic exponential if and only if this BSE has a solution. For a general contingent claim H, we consider the following generalized version of the classical mean-variance hedging problem $$ \min_{\pi\in Adm} E\{(X^{w,\pi}_{\tilde\tau})^2 I_{\{{\tilde\tau}\leq T\}}+|H - X^{w,\pi}_T|^2 I_{\{{\tilde\tau} > T\}}\}, $$ where ${\tilde\tau} = \inf\{t > 0; Z^{\rm opt}_t=0\}$. We represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward martingale equation (BME) and an appropriate predictable process δ both with a straightforward intuitive interpretation.

Keywords: Optimality principle; signed VOMM; backward semimartingale equations; mean-variance hedging (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1142/S0219024910006005

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