Pricing options on stocks denominated in different currencies: Theory and illustrations

Basket options have long been an important structured product. One can write a basket option on assets denominated in different currencies, but settle the option in one single currency at some fixed exchange rate. This special type of basket options can be found in many life insurance products that encompass an investment component. In order to value such options, we need to consider not only the joint dynamics of the returns on the underlying assets but also the quanto feature involved. In this paper, we use a regime-switching multivariate lognormal model for modeling returns on various assets and exchange rates. As the parameters of the model can change according to the state of a Markov chain, the model allows for stochastic volatility and correlations. We then demonstrate how domestic investors can choose a risk-neutral probability measure by the multivariate Esscher transform. This valuation methodology is illustrated with an hypothetical investment guarantee that is sold with a life insurance contract.