Optimal hedging strategies on asymmetric functions

Abstract.

We treat in this paper optimal hedging problems for contingent claims in an incomplete financial market, which problems are based on asymmetric functions. In summary, we consider the problem

$$\mathop{\rm min}\limits_{\vartheta\in\Theta} E[f(H - G_T(\vartheta))],$$

where H is a contingent claim, Θ, which is a suitable set of predictable processes, represents the collection of all admissible strategies, $G_T(\vartheta)$ is a portfolio value at the maturity T induced by an admissible strategy $\vartheta$ , and $f : \mathbf{R} \to \mathbf{R}_+$ is a differentiable strictly convex function with f(0) = 0. In particular, under the assumption that there exist two positive constants c ₀ and C ₁ such that, for any $x \in \mathbf{R}$ being far away from 0 sufficiently, $c_0|x|^p\leq f(x)$, and $|f^\prime(x)|\leq C_1|x|^{p-1}$, where 1 < p < ∞, we shall prove the unique existence of a solution and shall discuss its mathematical property.

The author would like to thank Jan Kallsen and Shigeo Kusuoka for their valuable comments and discussion, and is very grateful to an anonymous referee for helpful comments, which has greatly improved the paper. The financial support of the author has been partially granted by Grant-in-Aid for Young Scientists (B) No.16740062 and Scientific Research (C) No.19540144 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.