Closed-form expansions of discretely monitored Asian options in diffusion models

In this paper we propose a closed-form asymptotic expansion approach to pricing discretely monitored Asian options in general one-dimensional diffusion models. Our expansion is a small-time expansion because the expansion parameter is selected to be the square root of the length of monitoring interval. This expansion method is distinguished from many other pricing-oriented expansion algorithms in the literature because of two appealing features. First, we illustrate that it is possible to explicitly calculate not only the first several expansion terms but also any general expansion term in a systematic way. Second, the convergence of the expansion is proved rigorously under some regularity conditions. Numerical experiments suggest that the closed-form expansion formula with only a few terms (e.g., four terms up to the third order) is accurate, fast, and easy to implement for a broad range of diffusion models, even including those violating the regularity conditions. Keywords: discretely monitored Asian options; the CEV model; the CIR process; the Black-Scholes model; the Brennan and Schwartz process; small-time expansion MSC2000 subject classification: Primary: 60H07, 91B24, 91B70; secondary: 60H30 OR/MS subject classification: Primary: asset pricing, diffusion, stochastic model applications; secondary: Markov processes
1. Introduction. Asian options, whose payoffs depend on the arithmetic average of the underlying asset prices over a prespecified time period, are among the most popular exotic options traded actively [...]