BFS 2002

Contributed Talk

The Laplace transform approach to valuing exotic options: the case of the Asian option

Michael Schroeder, Peter Carr

Asian options are widely used financial derivatives which provide non-linear payoffs on the arithmetic average of an underlying asset. Their valuation has intrigued finance theorists for over a decade by now, even in the Black-Scholes context. Pursuing the Laplace transform approach proposed by Yor and Geman in 1993, this talk discusses our derivation of series and asymptotic expansions for computing benchmark prices for Asian options. And our point of view is that these formulas are consequences of the manifold interrelations this valuation problem has with other central parts of mathematics.
The punchline of the talk so is as follows. Since the Yor-Geman Laplace transforms are not those of the Asian option prices and, moreover, have been derived only under the restriction that the riskneutral drift is not less than half the variance, we re-address the notion of Laplace transforms of Asian option prices in a first part of the talk. While we will not discuss how in joint work with Peter Carr we have been able to lift the second of these restrictions, we will discuss how ideas of Peter Carr's have salvaged the relevance of the Yor-Geman results for valuing Asian options.
In a second part of the talk we will sketch the ideas, based on methods from complex analysis and special functions, of our analytic inversion of the Laplace transform, and compare the resulting integral for the Asian option price with Yor's 1992 triple integral for it.
In a third part, we will discuss series and asymptotic expansions for computing our integral. And we will indicate some of the background ideas from the analysis on the Poincare upper half plane and modular forms for this.