Valuation of American maximum exchange rate quanto lookback options

In light of the globalization of finance, quanto options have become popular options to be traded in over-the-counter markets in many countries. The payoff of a quanto option depends on foreign currency, but the actual payment (actual payout) is determined by the domestic currency. Various types of quanto options exist, depending on the purpose. In this paper, we study valuation of American maximum exchange-rate quanto options, which apply the value of the maximum realized exchange rate until maturity. The American maximum exchange-rate quanto options can be represented as a two-dimensional inhomogeneous Black-Scholes partial differential equation(PDE) with a mixed boundary condition. We derive an analytic solution of this two-dimensional inhomogeneous Black-Scholes PDE using double Mellin transform techniques. Using our approach, we find an integral equation that is satisfied by American maximum exchange-rate quanto lookback options. Furthermore, we find that the integral equation is correct by solving it numerically, using a simple iterative method. We also compare our method's results to those obtained through simulation using the forward shooting grid method to verify its accuracy.