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

14 Pages Posted: 5 Nov 2010

See all articles by Hyeng Keun Koo

Hyeng Keun Koo

Ajou University

Ji Hee Yoon

University College London - Department of Economics

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

Koo, Hyeng Keun Keun 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 College London - Department of Economics ( email )

Drayton House, 30 Gordon Street
30 Gordon Street
London, WC1H 0AX
United Kingdom

Here is the Coronavirus
related research on SSRN

Paper statistics

Downloads
85
Abstract Views
755
rank
318,059
PlumX Metrics