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MEAN VARIANCE HEDGING IN A GENERAL JUMP MARKET

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 Z...

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Bibliographic Details
Published in:International journal of theoretical and applied finance 2010-08, Vol.13 (5), p.789-820
Main Authors: XIONG, DEWEN, KOHLMANN, MICHAEL
Format: Article
Language:English
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Summary: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.
ISSN:0219-0249
1793-6322
DOI:10.1142/S0219024910006005