Risk-minimizing hedging of contingent claims in incomplete models of financial markets

Abstract

The paper is devoted to a specific optimization problem associated with the hedging of contingent claims in continuous-time incomplete models of financial markets. Generally speaking, we place ourselves within the standard framework of the theory of continuous trading, as exposed in Harrison and Pliska [13]. Our aim is twofold. Firstly, we present a relatively concise exposition of the risk-minimizing methodology (due essentially to Follmer and Sondermann [12], Follmer and Schweizer [11] and Schweizer [33]) in a multi-dimensional continuous-time framework. Let us mention here that this approach is based on the specific kind of minimization of the additional cost associated with a hedging strategy at all times before the terminal date T. Secondly, we provide some new results which formalize some concepts introduced in Hofman et a/.[l5], in particular, the general results of the first, part are specialized to the case of multi-dimensional Ito processes. Finally, in Section 6 the general theory is illustrated by means of an example dealing with the risk-minimizing hedging of a stock index option in an incomplete framework. This example is motivated bv the work of Lamberton and Lapeyre [22] who have! solved a related, but simpler, problem of a risk-minimizing hedging under the martingale measure.