Exact Pricing of Asian Options: An Application of Spectral Theory
نویسنده
چکیده
Arithmetic Asian or average price (rate) options deliver payoffs based on the average underlying price over a pre-speciÞed time period. Asian options are an important family of derivative contracts with a wide variety of applications in currency, equity, interest rate, commodity, energy, and insurance markets. We derive two analytical formulae for the price of the arithmetic Asian option when the underlying asset price follows geometric Brownian motion. The mathematics of the Asian option turns out to be related to the Schrödinger equation with Morse (1929) potential. Our derivation relies on the spectral theory of singular Sturm-Liouville (Schrödinger) operators and associated eigenfunction expansions. The Þrst formula is an inÞnite series of terms involving Whittaker functions M and W . The second formula is a single real integral of an expression involving Whittaker function W plus (for some parameter values) a Þnite number of additional terms involving incomplete Gamma functions and Laguerre polynomials. The two formulae allow exact computation of Asian option prices. ∗Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, Phone: (847) 491 2084, Fax (847) 491 8005, E-mail: [email protected].
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