Passport Options with Stochastic Volatility

A passport option is a call option on the profits of a trading account. In this article we investigate the robustness of passport option pricing 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 we extend this result to models with stochastic volatility where the volatility is defined via an autonomous SDE. It is shown that for certain models of this type, the form of the optimal strategy remains unchanged. This means that pricing 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 we are able to obtain comparisons for passport option prices using the exponential Brownian motion model and some well known stochastic volatility models. This is illustrated by a number of numerical examples. One conclusion is that fair prices are generally lower in a model with stochastic volatility than in a model with constant volatility.