An Asymptotic Analysis of an Optimal Hedging Model for Option Pricing with Transaction Costs

Davis, Panas, and Zariphopoulou (1993) and Hodges and Neuberger (1989) have presented a very appealing model for pricing European options in the presence of rehedging transaction costs. In their papers the 'maximization of utility' leads to a hedging strategy and an option value. The latter is different from the Black–Scholes fair value and is given by the solution of a three–dimensional free boundary problem. This problem is computationally very time–consuming. In this paper we analyze this problem in the realistic case of small transaction costs, applying simple ideas of asymptotic analysis. The problem is then reduced to an inhomogeneous diffusion equation in only two independent variables, the asset price and time. The advantages of this approach are to increase the speed at which the optimal hedging strategy is calculated and to add insight generally. Indeed, we find a very simple analytical expression for the hedging strategy involving the option's gamma.