Passport options with stochastic volatility

A passport option is a call option on the profits of a trading account. In this article, the robustness of passport option pricing is investigated by incorporating stochastic volatility. The key feature of a passport option is the holders' optimal strategy. It is known that in the case of exponential Brownian motion the strategy is to be long if the trading account is below zero and short if the account is above zero. Here this result is extended to models with stochastic volatility where the volatility is defined via an autonomous SDE. It is shown that if the Brownian motions driving the underlying asset and the volatility are independent then the form of the optimal strategy remains unchanged. This means that the strategy is robust to misspecification of the underlying model. A second aim of this article is to investigate some of the biases which become apparent in a stochastic volatility regime. Using an analytic approximation, comparisons are obtained for passport option prices using the exponential Brownian motion model and some well-known stochastic volatility models. This is illustrated with numerical examples. One conclusion is that if volatility and price are uncorrelated, then prices are sometimes lower in a model with stochastic volatility than in a model with constant volatility.