Adv. Math. Econ. 11, 1-10 (2008)

Optimal hedging strategies on asymmetric functions

Takuji Arai

Department of Economics, Keio University, 2-15-45 Mita, Minato-ku, Tokyo, 108-8345, Japan

Received: June 26, 2007 Revised: November 7, 2007

JEL classification: G10

Mathematical Subject Classification (2000): 91B28, 52A41, 60H05

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

min_{ϑ∈Θ} E[f(H − G_{T} (ϑ))],

where H is a contingent claim, Θ, which is a suitable set of predictable processes, represents the collection of all admissible strategies, G_{T} (ϑ) is a portfolio value at the maturity T induced by an admissible strategy ϑ, and f : R → 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 ∈ R being far away from 0 sufficiently, c_{0}|x|^{p} ≤ f(x), and |f′(x)| ≤ C_{1}|x|^{p−1}, where 1 < p < ∞, we shall prove the unique existence of a solution and shall discuss its mathematical property.

Key words: Mathematical finance, incomplete market, convex function, semimartingale, stochastic integral