Option Pricing with Random Volatilities in Complete Markets

Posted: 2 Jan 2007 Last revised: 27 Aug 2008

See all articles by Larry Eisenberg

Larry Eisenberg

New Jersey Institute of Technology

Robert A. Jarrow

Cornell University - Samuel Curtis Johnson Graduate School of Management

Abstract

This article presents the theory of option pricing with random volatilities in complete markets. As such, it makes two contributions. First, the newly developed martingale measure technique is used to synthesize results dating from Merton (1973) through Eisenberg, (1985, 1987). This synthesis illustrates how Merton's formula, the CEV formula, and the Black-Scholes formula are special cases of the random volatility model derived herein. The impossibility of obtaining a self-financing trading strategy to duplicate an option in incomplete markets is demonstrated. This omission is important because option pricing models are often used for risk management, which requires the construction of synthetic options. Second, we derive a new formula, which is easy to interpret and easy to program, for pricing options given a random volatility. This formula (for a European call option) is seen to be a weighted average of Black-Scholes values, and is consistent with recent empirical studies finding evidence of mean-reversion in volatilities.

Keywords: option pricing, synthetic options, martingale measure, European call option

JEL Classification: G13

Suggested Citation

Eisenberg, Laurence K. and Jarrow, Robert A., Option Pricing with Random Volatilities in Complete Markets. Review of Quantitative Finance and Accounting, Vol. 4, No. 1, pp. 5-17, 1994, Available at SSRN: https://ssrn.com/abstract=954171

Laurence K. Eisenberg (Contact Author)

New Jersey Institute of Technology ( email )

School of Management
Newark, NJ 07102-1982
United States
973 642-7263 (Phone)
973 596-3074 (Fax)

HOME PAGE: http://som.njit.edu/people/eisenberg.php

Robert A. Jarrow

Cornell University - Samuel Curtis Johnson Graduate School of Management ( email )

Department of Finance
Ithaca, NY 14853
United States
607-255-4729 (Phone)
607-254-4590 (Fax)

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