Author(s)

O. L. V. Costa, A. C. Maiali & A. de C. Pinto

Abstract

In this paper we consider the mean-variance hedging problem of a continuous state space financial model with the rebalancing strategies for the hedging portfolio taken at discrete times. An expression is derived for the optimal self-financing mean-variance hedging strategy problem, considering any given payoff in an incomplete market environment. To some extent, the paper extends the work of ˇCerný [1] to the case in which prices may assume any value within a continuous state space, a situation that more closely reflects real market conditions. An expression for the \“fair hedging price” for a derivative with any given payoff is derived. Closed-form solutions for both the \“fair hedging price” and the optimal control for the case of a European call option are obtained. Numerical results indicate that the proposed method is consistently better than the Black and Scholes approach, often adopted by practitioners. Keywords: discrete-time mean-variance hedging, options pricing, optimal control. 1 Introduction The problem of hedging options has systematically been the focus of attention from both researchers and practitioners alike. The complex nature of most derivatives has led academics to often simplify the conditions under which trading occurs, proposing models which, albeit computational and mathematically treatable, do not capture all of the peculiarities of these instruments. When modelling the dynamics of an asset price, its derivatives and the corresponding hedging process, the choices of state space and time parameter are determined so as to simplify themodel’s complexity.However,with respect to hedging, the situation that more closely follows what is observed in real market conditions is the use

Keywords

discrete-time mean-variance hedging, options pricing, optimal control.