大阪大学　数理・データ科学教育研究センター
Center for Mathematical Modeling and Data Science,Osaka University

New results for the distribution of the average of the geometric Brownian motion

Jan Vecer (Charles University in Prague; Frankfurt School of Finance and Management)

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4. New results for the distribution of the average of the geometric Brownian motion

大阪大学 数理・データ科学セミナー 金融・保険セミナーシリーズ 第67回

New results for the distribution of the average of the geometric Brownian motion

Jan Vecer (Charles University in Prague; Frankfurt School of Finance and Management)

Finding the distribution of the arithmetic average of the stock price that follows a geometric Brownian motion is a widely studied problem in quantitative finance.
The average of the stock price appears in evaluations of the cash flows, the pricing of Asian options or in perpetuities, but there is no simple analytical formula that would give a density of the stock price average. In this talk, we extend the work of Marc Yor (1993) who found a representation of the density in terms of a Laplace transform and noticed that the corresponding pricing formula for the Asian option (option on the average of the price) resembles the Black-Scholes formula for the plain vanilla option. We are able to prove that the pricing formula indeed admits the Black-Scholes representation, but under a special and previously unknown martingale measure that is associated with the "average asset" taken as a numeraire.
The Black-Scholes representation of the pricing formula gives immediately the hedging portfolio as the probability that the contract is in the money under this new martingale measure. We give both the Laplace transform representation of the average price distribution under this new measure and the corresponding partial differential equation characterization. Both approaches give us the chance to obtain the distribution numerically. Interestingly, the distribution (under the all relevant martingale
measures) of the perpetual average admits a closed form solution (which is related to the inverse gamma distribution) and thus we get interesting new solutions of the Black-Scholes partial differential equations for the price of the average.