A Non-Standard Construction of Multi-Dimensional Brownian Motions and Option Pricing
Hyeng Keun Koo
Ji Hee Yoon
University of Wisconsin - Madison
November 3, 2010
Anderson (1976) was the first to give a non-standard construction of a Brownian motion. His approach was to use the binomial model in a discrete time with infinitesimal time steps. Pricing an option in a model similar to the Black-Scholes model with the nonstandard Brownian motion can be done by using a binomial tree technique by Cox, Ross, and Rubinstein (1979). Furthermore, the standard part of the price is equal to the Black-Scholes price (Cutland, Kopp, and Willinger 1991). However, an important obstacle arises when his approach is applied to a multi-dimensional option pricing, namely, the financial market is not complete when there are n assets driven by n independent Brownian motions. In this paper we provide a new approach which resolves this problem. The financial market in the non-standard world becomes complete with n assets driven by an n-dimensional non-standard Brownian motion. We apply the construction to pricing and hedging of an exchange option.
Number of Pages in PDF File: 14
Keywords: nonstandard analysis, multi-dimensional Brownian motion, option pricing, exchange option
JEL Classification: C63, G1
Date posted: November 5, 2010
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