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Mean-variance hedging for continuous processes: New proofs and examples. (English) Zbl 0894.90023

Summary: Let \(X\) be a special semimartingale of the form \(X= X_0+ M+\int d\langle M\rangle\widehat\lambda\) and denote by \(\widehat K= \int \widehat\lambda^{\text{tr}} d\langle M\rangle\widehat\lambda\) the mean-variance tradeoff process of \(X\). Let \(\Theta\) be the space of predictable processes \(\vartheta\) for which the stochastic integral \(G(\vartheta)= \int\vartheta dX\) is a square-integrable semimartingale. For a given constant \(c\in\mathbb{R}\) and a given square-integrable random variable \(H\), the mean-variance optimal hedging strategy \(\xi^{(c)}\) by definition minimizes the distance in \({\mathcal L}^2(P)\) between \(H- c\) and the space \(G_T(\Theta)\). In financial terms, \(\xi^{(c)}\) provides an approximation of the contingent claim \(H\) by means of a self-financing trading strategy with minimal global risk. Assuming that \(\widehat K\) is bounded and continuous, we first give a simple new proof of the closedness of \(G_T(\Theta)\) in \({\mathcal L}^2(P)\) and of the existence of the Föllmer-Schweizer decomposition. If moreover \(X\) is continuous and satisfies an additional condition, we can describe the mean-variance optimal strategy in feedback form, and we provide several examples where it can be computed explicitly. The additional condition states that the minimal and the variance-optimal martingale measures for \(X\) should coincide. We provide examples where this assumption is satisfied, but we also show that it will typically fail if \(\widehat K_T\) is not deterministic and includes exogenous randomness which is not induced by \(X\).

MSC:

91B28 Finance etc. (MSC2000)
60H05 Stochastic integrals
60G48 Generalizations of martingales
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