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The Valuation of Volatility Options

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  • Jérôme B. Detemple

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  • Carlton Osakwe
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    Abstract

    This paper examines the valuation of European- and American-style volatility options based on a general equilibrium stochastic volatility framework. Properties of the optimal exercise region and of the option price are provided when volatility follows a general diffusion process. Explicit valuation formulas are derived in four particular cases. Emphasis is placed on the MRLP (mean-reverting in the log) volatility model which has received considerable empirical support. In this context we examine the properties and hedging behavior of volatility options. Unlike American options, European call options on volatility are found to display concavity at high levels of volatility. Cet article examine l'évaluation des options sur volatilité, de type européen ou américain, dans le cadre d'un modèle d'équilibre général avec volatilité stochastique. Certaines propriétés de la région d'exercise optimal et du prix de l'option sont établies lorsque la volatilité suit un processus général de diffusion. Des formules d'évaluation explicites sont ensuite dérivées dans quatre cas particuliers. Nous étudions en détail le cas d'un processus de volatilité de type MRLP (mean-reverting in the log) qui semble être conforme à l'évidence empirique. Les propriétés et le comportement de couverture des options sur volatilité sont examinées dans ce cadre. ¸ l'opposeé d'une option d'achat américaine, le prix d'une option d'achat européenne sur volatilité s'avère être une fonction concave lorsque le niveau de volatilité s'élève.

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    Bibliographic Info

    Paper provided by CIRANO in its series CIRANO Working Papers with number 99s-43.

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    Date of creation: 01 Nov 1999
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    Handle: RePEc:cir:cirwor:99s-43

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    Keywords: Stochastic volatility; European options; American options; optimal exercise; early exercise premium; hedging; viability; Volatilité stochastique; options européennes; options américaines; exercice optimal; prime d'exercice anticipé; couverture de risque; viabilité;

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    1. Yaacov Z. Bergman & Bruce D. Grundy & Zvi Wiener, . "General Properties of Option Prices (Revision of 11-95) (Reprint 058)," Rodney L. White Center for Financial Research Working Papers 1-96, Wharton School Rodney L. White Center for Financial Research.
    2. Johnson, Herb & Shanno, David, 1987. "Option Pricing when the Variance Is Changing," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(02), pages 143-151, June.
    3. Jagannathan, Ravi, 1984. "Call options and the risk of underlying securities," Journal of Financial Economics, Elsevier, vol. 13(3), pages 425-434, September.
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    5. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    6. Broadie, Mark & Detemple, Jerome, 1996. "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," Review of Financial Studies, Society for Financial Studies, vol. 9(4), pages 1211-50.
    7. Hentschel, Ludger, 1995. "All in the family Nesting symmetric and asymmetric GARCH models," Journal of Financial Economics, Elsevier, vol. 39(1), pages 71-104, September.
    8. Melino, Angelo & Turnbull, Stuart M., 1990. "Pricing foreign currency options with stochastic volatility," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 239-265.
    9. Yacine Aït-Sahalia & Andrew W. Lo, 1998. "Nonparametric Estimation of State-Price Densities Implicit in Financial Asset Prices," Journal of Finance, American Finance Association, vol. 53(2), pages 499-547, 04.
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    11. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
    12. Broadie, Mark & Detemple, Jerome & Ghysels, Eric & Torres, Olivier, 2000. "American options with stochastic dividends and volatility: A nonparametric investigation," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 53-92.
    13. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
    14. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. " General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
    15. Grunbichler, Andreas & Longstaff, Francis A., 1996. "Valuing futures and options on volatility," Journal of Banking & Finance, Elsevier, vol. 20(6), pages 985-1001, July.
    16. Cox, John C & Ross, Stephen A, 1976. "A Survey of Some New Results in Financial Option Pricing Theory," Journal of Finance, American Finance Association, vol. 31(2), pages 383-402, May.
    17. Ju, Nengjiu, 1998. "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," Review of Financial Studies, Society for Financial Studies, vol. 11(3), pages 627-46.
    18. Danielsson, Jon, 1994. "Stochastic volatility in asset prices estimation with simulated maximum likelihood," Journal of Econometrics, Elsevier, vol. 64(1-2), pages 375-400.
    19. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    20. S. D. Jacka, 1991. "Optimal Stopping and the American Put," Mathematical Finance, Wiley Blackwell, vol. 1(2), pages 1-14.
    21. Peter Carr & Robert Jarrow & Ravi Myneni, 1992. "Alternative Characterizations Of American Put Options," Mathematical Finance, Wiley Blackwell, vol. 2(2), pages 87-106.
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