Asian Options on Harmonic Average

Abstract

The contracts written on the harmonic average of the underlying price are quite popular in the foreign exchange market. If X denotes the foreign currency and Y denotes the domestic currency, the payoff of the contract is a function of a price of an asset H which is defined asH(T) = \left[{\int_0^T [X_Y (t)]^{-1} \eta(t) dt} \right]^{-1} Y(T) = \left[ \frac{1}{\int_0^T Y_X(t) \eta(t) dt} \right] Y(T).The harmonic average resembles a quanto option: the price Y_X(t) is monitored with respect to the foreign currency X, but the payoff is settled in the domestic currency Y. Although the pricing problem appears to be rather complex, it can be ultimately simplified to a partial differential equation in one spatial variable after a numeraire change and using the time reversal argument.