Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-18T19:06:51.656Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic Volatility Option Pricing

Published online by Cambridge University Press:  06 April 2009

Clifford A. Ball  and
Antonio Roma
Clifford A. Ball
Affiliation:
Owen Graduate School of Management, Vanderbilt University, Nashville, TN 37203
Antonio Roma
Affiliation:
Università di Siena, Facoltà di Scienza Economiche e Bancarie, 53100 Siena, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper examines alternative methods for pricing options when the underlying security volatility is stochastic. We show that when there is no correlation between innovations in security price and volatility, the characteristic function of the average variance of the price process plays a pivotal role. It may be used to simplify Fourier option pricing techniques and to implement simple power series methods. We compare these methods for the alternative mean-reverting stochastic volatility models introduced by Stein and Stein (1991) and Heston (1993). We also examine the biases in the Black-Scholes model that are eliminated by allowing for stochastic volatility, and we correct some errors in the Stein and Stein (1991) analysis of this issue.

Type
Research Article
Information
Journal of Financial and Quantitative Analysis , Volume 29 , Issue 4 , December 1994 , pp. 589 - 607
Copyright
Copyright © School of Business Administration, University of Washington 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bhattacharya, R. N., and Waymire, E. C.. Stochastic Processes with Applications. New York, NY: John Wiley and Son (1990).Google Scholar
Brennan, M. J., and Schwartz, E. S.. “A Continuous Time Approach to the Pricing of Bonds.” Journal of Banking and Finance, 3 (07 1979), 133–155.CrossRefGoogle Scholar
Brennan, M. J., and Schwartz, E. S.. “An Equilibrium Model of Bond Pricing and a Test of Market Efficiency.” Journal of Financial and Quantitative Analysis, 17 (09 1982), 301–329.CrossRefGoogle Scholar
Canina, L., and Figlewski, S.. ‘The Information Content of Implied Volatility.” Review of Financial Studies, 6 (3, 1993), 659–681.CrossRefGoogle Scholar
Chesney, M., and Scott, L.. “Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model.” Journal of Financial and Quantitative Analysis, 24 (09 1989), 267–284.CrossRefGoogle Scholar
Constantinides, G. M. “A Theory of the Nominal Term Structure of Interest Rates.” Review of Financial Studies, 5 (4, 1992), 531–552.CrossRefGoogle Scholar
Cox, J. C.; Ingersoll, J. E.; and Ross, S. A.. “A Theory of the Term Structure of Interest Rates,” Econometrica, 53 (2, 1985), 385–407.CrossRefGoogle Scholar
Cox, D. R., and Miller, H. D.. “The Theory of Stochastic Processes.” Chapman and Hall Ltd. (1970).Google Scholar
Dothan, L. U. “On the Term Structure of Interest Rates.” Journal of Financial Economics, 6 (1978), 59–69.CrossRefGoogle Scholar
Garman, M. “A General Theory of Asset Valuation under Diffusion State Processes.” Working Paper, Univ. of California, Berkeley (1976).Google Scholar
Heston, S. L. “A Closed Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” Review of Financial Studies, 6 (2, 1993), 327–343.CrossRefGoogle Scholar
Hull, J., and White, A.. “The Pricing of Options on Assets with Stochastic Volatilities.” Journal of Finance, 42 (1987), 281–300.CrossRefGoogle Scholar
Hull, J., and White, A.. “An Analysis of the Bias in Option Pricing Caused by a Stochastic Volatility,” Advances in Futures and Options Research, 3 (1988), 29–61.Google Scholar
Ingersoll, J. E.Theory of Financial Decision Making. Savage, MD: Rowman and Littlefield (1987).Google Scholar
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
Karlin, S., and Taylor, H. M.. “A First Course in Stochastic Processes.” San Diego, CA: Academic Press (1975).Google Scholar
Karlin, S., and Taylor, H. M.. “A Second Course in Stochastic Processes.” San Diego, CA: Academic Press (1981).Google Scholar
Kendall, M., and Stuart, A. S. “The Advanced Theory of Statistics,” Volume 1. New York, NY: MacMillan Co., Inc. (1977).Google Scholar
Longstaff, F., and Schwartz, E.. “Interest-Rate Volatility and the Term Structure of Interest Rates: A Two-Factor General Equilibrium Model.” Journal of Finance, 47 (1992), 1259–1282.Google Scholar
Nelson, D. B., and Foster, D. P.. “Asymptotic Filtering Theory for Univariate ARCH Models.,” Econometrica, 62 (1, 1994), 1–43.CrossRefGoogle Scholar
Ricciardi, L. M., and Sacerdote, L.. “On the Probability Densities of an Ornstein Uhlenbeck Process with a Reflecting Boundary.” Journal of Applied Probability, 24 (1987), 355–369.CrossRefGoogle Scholar
Scott, L. O. ‘Option Pricing when the Variance Changes Randomly: Theory, Estimation and an Application.” Journal of Financial and Quantitative Analysis, 22 (12 1987), 419–438.CrossRefGoogle Scholar
Scott, L. O. “Stock Market Volatility and the Pricing of Index Options.” Working Paper, Univ. of Georgia (1992).Google Scholar
Stein, E. M., and Stein, C. J.. “Stock Price Distributions with Stochastic Volatility: An Analytic Approach.” Review of Financial Studies, 4 (4, 1991), 727–752.CrossRefGoogle Scholar
Watanabe, T. “A Nonlinear Filtering Technique for Estimating Stochastic Volatility Models with an Application to Daily Returns in the New York Stock Exchange.” Working Paper, Yale Univ. (1992).Google Scholar
Wiggins, J. B. “Option Values under Stochastic Volatility: Theory and Empirical Estimates.” Journal of Financial Economics, 19 (1987), 351–372.CrossRefGoogle Scholar