Partially exact and bounded approximations for arithmetic Asian options

Roger Lord

Abstract

ABSTRACT

This paper considers the pricing of European Asian options in the Black– Scholes framework. All approaches we consider are readily extendable to the case of an Asian basket option. We consider three methods for evaluating the price of an Asian option, and contribute to all three. First, we show the link between the approaches of Rogers and Shi (1995), Andreasen (1998), Hoogland and Neumann (2000a,b) and Ve¡ce¡r (2001). For the latter formulation we propose two reductions, which increase the numerical stability and reduce the calculation time. Second, we show how a closed-form expression can be derived for Curran’s and Rogers and Shi’s lower bound for the general case of multiple underlyings. Third, we considerably sharpen Thompson’s (1999a,b) upper bound such that it is tighter than all known upper bounds. Finally, we consider analytical approximations and combine the traditional moment matching approximations with Curran’s conditioning approach. The resulting class of partially exact and bounded approximations can be proven to lie between a sharp lower and upper bound. Using numerical examples we demonstrate that they outperform all current state-of-the-art bounds and approximations.