Asian and Australian options : A common perspective

We show that Australian options are equivalent to fixed or floating strike Asian options and consequently that by studying Asian options from the Australian perspective and vice versa, much can be gained. One specific application of this "Australian Approach" leads to a natural dimension reduction for the pricing PDE of Asian options, with or without stochastic volatility, featuring time independent coefficients. Another application lies in the improvement of Monte Carlo schemes, where the "Australian Approach" results in a path-independent method. We also show how the Milevsky and Posner (1998) result on the reciprocal $\Gamma$-approximation for Asian options can be quickly obtained by using the connection to Australian options. Further, we present an analytical (exact) pricing formula for Australian options and adapt a result of Carr, Ewald and Xiao (2008) to show that the price of an Australian option is increasing in the volatility and by doing this answering a standing question by Moreno and Navas (2008)