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.
Similar content being viewed by others
References
Cox, J. and S. Ross, “The Valuation of Options for Alternative Stochastic Processes.”Journal of Financial Economics 3, 145–166, (1976).
Eisenberg, L., “Random Variance Option Pricing and Spread Valuation,” Working Paper, University of Illinois, (1985).
Eisenberg, L.,Random Variance Option Pricing, Ph.D. thesis, University of Pennsylvania, 1987.
Harrison, J.M. and S. Pliska, “Martingales and Stochastic Integrals in the Theory of Continuous Trading.”Stochastic Processes and Their Applications 11, 215–260, (1981).
Heath, D., R. Jarrow, and A. Morton, “Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation.”Econometrica, 60 (1), 77–105, (1992).
Hull, J. and A. White, “The Pricing of Options on Assets with Stochastic Volatilities.”Journal of Finance 42, 271–301, (1987).
Jarrow, R. and D. Madan, “A Characterization of Complete Security Markets on a Brownian Filtration.”Mathematical Finance 1(3), 31–44, (1991).
Jarrow, R. and A. Rudd,Option Pricing, Homewood, Illinois: Richard Irwin, Inc., 1983.
Jarrow, R. and J. Wiggins, “Option Pricing and Implicit Volatilities: A Review and a New Perspective.”Journal of Economic Surveys 3, 59–81, (1989).
Johnson, H. and D. Shanno, “Option Pricing When the Variance is Changing.”Journal of Financial and Quantitative Analysis 22, 143–153, (1987).
Jones, E.P., “Option Arbitrage and Strategy with Large Price Changes.”Journal of Financial Economics 13, 91–113, (1984).
Karatzas, I. and S. Shreve,Brownian Motion and Stochastic Calculus, New York: Springer-Verlag, 1988.
Madan, D., F. Milne, and H. Shefrin, “The Multinomial Option Pricing Model and Its Brownian and Poisson Limits.”The Review of Financial Studies 2(2), 251–266, (1989).
Merton, R., “Theory of Rational Option Pricing.”Bell Journal of Economics and Management Science 4, 141–183, (1973).
Merville, L. and D. Pieptea, “Stock-Price Volatility, Mean-Reverting Diffusion, and Noise.”Journal of Financial Economics 24, 193–214, (1989).
Scott, Louis, “Option Pricing When Variance Changes Randomly: Theory, Estimation, and An Application.”Journal of Financial and Quantitative Analysis 22, 419–438, (1987).
Stein, E. and J. Stein, “Stock Price Distributions with Stochastic Volatility: An Analytic Approach.”The Review of Financial Studies 4(4), 727–752, (1991).
Wiggins, J., “Option Values Under Stochastic Volatility: Theory and Empirical Estimates.”Journal of Financial Economics 19, 351–372 (1987).
Author information
Authors and Affiliations
Additional information
Helpful comments from an anonymous referee are greatly appreciated.
Rights and permissions
About this article
Cite this article
Eisenberg, L., Jarrow, R. Option pricing with random volatilities in complete markets. Rev Quant Finan Acc 4, 5–17 (1994). https://doi.org/10.1007/BF01082661
Issue Date:
DOI: https://doi.org/10.1007/BF01082661