Numeraires and Martingale Measures in the Numeraires and Martingale Measures in the Black-Scholes Models

Shunlong Luo and Jiaan Yan

Institute of Applied Mathematics, Academia Sinica, Beijing, 100080, P. R. China (luosl@amath4.amt.ac.cn)

Qiang Zhang

Department of Economics and Finance, City University of Hong Kong, Hong Kong

Abstract

In the Black-Scholes models, complete or incomplete, for any positive self-financing wealth process, there is a simple and natural probability measure under which all security prices denominated by this self-financing wealth process are martingales. Thus, if we take this self-financing wealth process as the numeraire and the new probability as the equivalent martingale measure (risk-neutral measure), we obtain a simple arbitrage price system which has an intuitive probabilistic interpretation. Moreover, all this kind of arbitrage price systems with different numeraires are equivalent. In particular, if we choose the growth-optimal wealth as the numeraire (the so-called numeraire portfolio), then the objective probability itself becomes a martingale measure. This provides an intuitive, conceptually clear and an analytically tractable martingale method to price contingent claims when the market is incomplete. Applications to stochastic volatility models are developed.


File translated from TEX by TTH, version 1.94.
On 6 May 1999, 09:42.