Laguerre Series for Asian and Other Options

Abstract

This paper has four goals: (a) relate ladder height distributions to option values; (b) show how Laguerre expansions may be used in the computation of densities, distribution functions, and option prices; (c) derive some new results on the integral of geometric Brownian motion over a finite interval; and (d) apply the preceding results to the determination of the distribution of the integral of geometric Brownian motion and the computation of Asian option values. The usual fixed-strike options on the average are treated, as well as options with payoffs expressed in terms of one over the average of the underlying security, which this author calls "reciprocal Asian options." In all cases the underlying asset is represented by geometric Brownian motion, the averages are performed continuously, and the options are of European type. Copyright Blackwell Publishers, Inc..

The square-root process and Asian options

“…Although the valuation of these options in the Black and Scholes framework has triggered the interest of financial mathematicians for over a decade in an intricate interplay between theoretical ( Yor [40], Geman and Yor [41], Schröder [36], Milevsky and Posner [28], [32], Dufresne [16], etc. ) and computational ( Caverhill and Clewlow [6], Fu et al [18], Rogers and Shi [34], Zvan [42], Vecer [38], etc. )…”

“…By solving a triangular set of partial differential equations, we prove that these moments take a simple explicit form, which can be practical. Those moments have indeed an important informational content, since they are proven to actually determine the distribution of these processes and are needed for moments-base expansions of the Laguerre type [16]. In the fourth section, we also derive the joint distribution of the process and its integral by analytic Laplace transform inversion, using a simplifying measure change relating the square-root process to a time-changed square Bessel process.…”

Australian Asian Options

“…The expressions for the mean and variance follow from straightforward application of (2)- (3) in Lemma 1 to S t as given by (12).…”

“…this is the case of Yor (1992Yor ( , 1993), Geman and Yor (1993), De Schepper et al (1994), Eydeland and Geman (1995), Fu et al (1999) and Shirakawa (1999), who use the theory of Bessel processes and the inversion of a Laplace transform. Alternatively, Ju (1997) employs A Fourier transform, while Dufresne (2000) suggests a Laguerre expansion.…”

Bessel Processes, the Integral of Geometric Brownian Motion, and Asian Options

“…A common objective in their valuation is to derive an explicit expression for a certain functional of A (ν) . The pursuit of this objective has evolved over the last fifteen years as an interplay between theoretical and computational perspectives, see [RS] or [Du00] for instance. Yor's work in [Y] clarified the structure of the Black-Scholes prices of Asian options by expressing them as certain triple integrals.…”