Option Pricing and Hedging in the Presence of Cross-Hedge Risk

This paper addresses the question of option pricing and hedging when the underlying asset is not available for dynamic trading, and some other asset is used as a substitute. We first provide an overview of the various hedging methodologies that can be used in this incomplete market setting, distinguishing between self-financing and non-self-financing strategies. Focussing on a local risk-minimization criterion, we present an analytical expression for the optimal hedging strategy and the corresponding option price. We also provide a quantitative measure of the residual risk over the life of the option. We find that the use of the optimal strategy induces a much smaller replication error compared to the replication error induced by a naive Black-Scholes strategy, especially for low levels of the correlation between the underlying asset and the substitute. In the absence of transaction costs, we also find that cross hedge risk is more substantial than the risk induced by discrete trading for reasonable parameter values. While this result implies that trading in the substitute can only be rationalized for exceedingly high correlations, the presence of (higher levels of) transaction costs is likely, however, to make trading in the actual underlying asset a prohibitively costly alternative.
Keywords: Option pricing; Cross-hedge risk; Incomplete markets.
JEL codes: G12; G13