A Non-Standard Construction of Multi-Dimensional Brownian Motions and Option Pricing

14 Pages Posted: 5 Nov 2010  

Hyeng Keun Koo

Ajou University

Ji Hee Yoon

University of Wisconsin - Madison

Date Written: November 3, 2010

Abstract

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.

Keywords: nonstandard analysis, multi-dimensional Brownian motion, option pricing, exchange option

JEL Classification: C63, G1

Suggested Citation

Keun Koo, Hyeng and Yoon, Ji Hee, A Non-Standard Construction of Multi-Dimensional Brownian Motions and Option Pricing (November 3, 2010). Available at SSRN: https://ssrn.com/abstract=1703323 or http://dx.doi.org/10.2139/ssrn.1703323

Hyeng Keun Keun Koo

Ajou University ( email )

206 Worldcup-ro
Yeongtong-gu
Suwon, 443-749
Korea, Republic of (South Korea)
82-31-219-2706 (Phone)
82-31-219-1616 (Fax)

Ji Hee Yoon (Contact Author)

University of Wisconsin - Madison ( email )

716 Langdon Street
Madison, WI 53706-1481
United States

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