An Analysis of Asian options

Abstract

The objective of this paper is to provide an analytic theory for pricing of Asian options of European type. We present a partial differential equation describing the fair price process of an Asian option. This appears as

$$ \left( {\partial _t - A - x \cdot \nabla _y } \right)u = 0 $$

and the associated payoff function as the end value. Here the operator A is the d-dimensional Black-Scholes operator, and B = x·∇ _y represents the path dependence in terms of the price averaging in Asian options. The main result will be to prove, that a solution of this partial differential equation exists, is unique, and depends continuously on the data in appropriate function spaces, i.e., that the problem is well posed. On our way we are going to employ semigroup methods, in particular the Lumer-Phillips theorem.