Discrete-time optimal hedging for multi-asset path-dependent European contingent claims

In this paper, we consider the problem of discrete-time optimal hedging for a European contingent claim (ECC) written on multiple assets where the underlying assets are assumed to follow a vector Ito differential equation. Specifically, since the underlying asset is assumed to be a continuous-time process all discrete-time hedging strategies are non-replicable and lead to hedging errors. First, we present a framework for finding hedging strategies that minimize the variance of hedging errors due to discrete-time hedging. The general framework is valid for all ECCs whose underlying assets are martingales and the minimum variance hedging strategies are in terms of conditional covariance matrices. Next, we specialize the conditional covariance matrix formulas to the case of geometric Brownian motion. These results extend the existing formula for single asset European call and put options to simple and path-dependent ECCs written on multiple assets.