Optimal hedging strategies on asymmetric functions

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.