  We characterize the price of an Asian option, a financial contract, as a
fixed-point of a non-linear operator. In recent years, there has been interest
in incorporating changes of regime into the parameters describing the evolution
of the underlying asset price, namely the interest rate and the volatility, to
model sudden exogenous events in the economy. Asian options are particularly
interesting because the payoff depends on the integrated asset price. We study
the case of both floating- and fixed-strike Asian call options with arithmetic
averaging when the asset follows a regime-switching geometric Brownian motion
with coefficients that depend on a Markov chain. The typical approach to
finding the value of a financial option is to solve an associated system of
coupled partial differential equations. Alternatively, we propose an iterative
procedure that converges to the value of this contract with geometric rate
using a classical fixed-point theorem.
