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Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application

Published online by Cambridge University Press:  06 April 2009

Louis O. Scott
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Abstract

In this paper, we examine the pricing of European call options on stocks that have variance rates that change randomly. We study continuous time diffusion processes for the stock return and the standard deviation parameter, and we find that one must use the stock and two options to form a riskless hedge. The riskless hedge does not lead to a unique option pricing function because the random standard deviation is not a traded security. One must appeal to an equilibrium asset pricing model to derive a unique option pricing function. In general, the option price depends on the risk premium associated with the random standard deviation. We find that the problem can be simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with the stock return. The resulting solution is an integral of the Black-Scholes formula and the distribution function for the variance of the stock price. We show that accurate option prices can be computed via Monte Carlo simulations and we apply the model to a set of actual prices.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 22 , Issue 4 , December 1987 , pp. 419 - 438
Copyright
Copyright © School of Business Administration, University of Washington 1987

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References

[1]Arnold, L.Stochastic Differential Equations: Theory and Applications. New York: John Wiley & Sons (1974).Google Scholar
[2]Black, F. “Studies of Stock Price Volatility Changes.” Proceedings of the Business and Economic Statistics Section, American Statistical Association (1976), 177–181.Google Scholar
[3]Black, F., and Scholes, M.. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy, 81 (05/06 1973), 637–659.CrossRefGoogle Scholar
[4]Blattberg, R. C., and Gonedes, N. J.. “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices.” Journal of Business, 47 (04 1974), 244–280.CrossRefGoogle Scholar
[5]Boyle, P. P. “Options: A Monte Carlo Approach.” Journal of Financial Economics, 4 (05 1977), 323–338.CrossRefGoogle Scholar
[6]Breeden, D. T. “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities.” Journal of Financial Economics, 7 (09 1979), 265–296.CrossRefGoogle Scholar
[7]Castanias, R. P. “Macroinformation and the Variability of Stock Market Prices.” Journal of Finance, 34 (05 1979), 439–450.CrossRefGoogle Scholar
[8]Christie, A. “The Stochastic Behavior of Common Stock Variances: Value, Leverage, and Interest Rate Effects.’ Journal of Financial Economics, 10 (12 1982), 407–432.CrossRefGoogle Scholar
[9]Clark, P. “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices.” Econometrica, 41 (01 1973), 135–155.CrossRefGoogle Scholar
[10]Cox, J. C.; Ingersoll, J. E.; and Ross, S. A.. “An Intertemporal General Equilibrium Model of Asset Prices.” Econometrica, 53 (03 1985), 363–384.CrossRefGoogle Scholar
[11]Cox, J. C., and Ross, S. A.. “The Valuation of Options for Alternative Stochastic Processes.” Journal of Financial Economics, 3 (03 1976), 145–166.CrossRefGoogle Scholar
[12]Dothan, M. U., and Reisman, H.. “Volatility Bias in Option Pricing.” Univ. of Minn, (revised 07 1985).Google Scholar
[13]Eisenberg, L. “Relative Pricing from No-Arbitrage Conditions: Random Variance Option Pricing.” Univ. of Ill. (11 1985).Google Scholar
[14]Epps, T. W., and Epps, M. L.. “The Stochastic Dependence of Security Price Changes and Transaction Volumes: Implications for the Mixture-of-Distribution Hypothesis.” Econometrica, 44 (03 1976), 305–321.CrossRefGoogle Scholar
[15]Feller, W.An Introduction to Probability Theory and Its Applications II, Second Ed.New York: John Wiley & Sons (1971).Google Scholar
[16]Geske, R., and Roll, R.. “On Valuing American Call Options with the Black-Scholes European Formula.” Journal of Finance, 39 (06 1984), 443–456.Google Scholar
[17]Geske, R., and Shastri, K.. “Valuation by Approximation: A Comparison of Alternative Option Valuation Techniques.” Journal of Financial and Quantitative Analysis, 20 (03 1985), 45–72.CrossRefGoogle Scholar
[18]Gradshteyn, I. S., and Ryzhik, I. M.. Table of Integrals, Series, and Products Corrected and Enlarged Edition. New York: Academic Press (1980).Google Scholar
[19]Hansen, L. P. “Large Sample Properties of Generalized Method of Moments Estimators.” Econometrica, 50 (07 1982), 1029–1054.CrossRefGoogle Scholar
[20]Harris, L. “Transaction Data Tests of the Mixture of Distributions Hypothesis.” Journal of Financial and Quantitative Analysis, 22 (06 1987), 127–141.CrossRefGoogle Scholar
[21]Hull, J., and White, A.. “The Pricing of Options on Assets with Stochastic Volatilities.” Journal of Finance, 42 (06 1987), 281–230.CrossRefGoogle Scholar
[22]Johnson, H., and Shanno, D.. “Option Pricing when the Variance is Changing.” Journal of Financial and Quantitative Analysis, 22 (06 1987), 143–151.CrossRefGoogle Scholar
[23]Jones, E. P. “Option Arbitrage and Strategy with Large Price Changes.” Journal of Financial Economics, 13 (03 1984), 91–114.CrossRefGoogle Scholar
[24]Karlin, S., and Taylor, H. M.. A Second Course in Stochastic Processes. New York: Academic Press (1981).Google Scholar
[25]Kon, S. J. “Models of Stock Returns—A Comparison.” Journal of Finance, 39 (03 1984), 147–166.Google Scholar
[26]Levy, H., and Markowitz, H. M.. “Approximating Expected Utility by a Function of Mean and Variance.” American Economic Review, 69 (06 1979), 308–317.Google Scholar
[27]MacBeth, J., and Merville, L.. “An Empirical Examination of the Black-Scholes Call Option Pricing Model.” Journal of Finance, 34 (12 1979), 1173–1186.Google Scholar
[28]Manaster, S., and Rendleman, R. J.. “Option Prices as Predictors of Equilibrium Stock Prices.” Journal of Finance, 37 (09 1982), 1043–1058.CrossRefGoogle Scholar
[29]Merton, R. C. “The Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science, 4 (Spring 1973), 141–183.CrossRefGoogle Scholar
[30]Merville, L. J., and Pieptea, D. R.. “On the Stochastic Nature of the Stock Price Variance Rate and Strike Price Bias in Option Pricing.” The Univ. of Tex. at Dallas (11 1985).Google Scholar
[31]Roll, R., and Ross, S. A.. “An Empirical Investigation of the Arbitrage Pricing Theory.” Journal of Finance, 35 (12 1980), 1073–1103.CrossRefGoogle Scholar
[32]Ross, S. A. “The Arbitrage Theory of Capital Asset Pricing.” Journal of Economic Theory, 13 (12 1976), 341–360.CrossRefGoogle Scholar
[33]Ross, S. A. “Return, Risk, and Arbitrage.” In Risk and Return in Finance, Vol. I, Friend, I. and Bicksler, J. L., eds. Cambridge, MA: Ballinger (1977).Google Scholar
[34]Rubinstein, M. “Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes.” Journal of Finance, 40 (06 1985), 455–480.CrossRefGoogle Scholar
[35]Smith, C. W. “Option Pricing: A Review.” Journal of Financial Economics, 3 (01–03 1976), 3–51.CrossRefGoogle Scholar
[36]Tauchen, G. E., and Pitts, M.. “The Price Variability-Volume Relationship on Speculative Markets.” Econometrica, 51 (03 1983), 485–505.CrossRefGoogle Scholar
[37]Whaley, R. “Valuation of American Call Options on Dividend-Paying Stocks: Empirical Tests.” Journal of Financial Economics, 10 (03 1982), 29–58.CrossRefGoogle Scholar
[38]Wiggins, J. B. “Stochastic Volatility Option Valuation: Theory and Empirical Estimates.” Cornell Univ. (1986).Google Scholar