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Asymptotics for the discrete-time average of the geometric Brownian motion and Asian options

Part of: Limit theorems Mathematical finance

Published online by Cambridge University Press:  26 June 2017

Dan Pirjol  and
Lingjiong Zhu
Dan Pirjol*
Affiliation:
J. P. Morgan
Lingjiong Zhu*
Affiliation:
Florida State University
*
* Postal address: J. P. Morgan, New York, NY 10172, USA. Email address: dpirjol@gmail.com
** Postal address: Department of Mathematics, Florida State University, 1017 Academic Way, Tallahassee, FL 32306, USA. Email address: zhu@math.fsu.edu
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Abstract

The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black–Scholes model, with both fixed and floating strike.

Keywords

Asian optioncentral limit theoremBerry–Esseen boundlarge deviations

MSC classification

Primary: 91G20: Derivative securities
Secondary: 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems) 60F05: Central limit and other weak theorems 60F10: Large deviations

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 446 - 480
Copyright
Copyright © Applied Probability Trust 2017 

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