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Arbitrage-Based Pricing When Volatility is Stochastic

  • Bossaerts, Peter
  • Ghysels, Eric
  • Gourieroux, Christian

One of the early examples of stochastic volatility models is Clark [1973]. He suggested that asset price movements should be tied to the rate at which transactions occur. To accomplish this, he made a distinction between transaction time and calendar time. This framework has hitherto been relatively unexploited to study derivative security pricing. This paper studies the implications of absence of arbitrage in economies where: (i) trade takes place in transaction time, (ii) there is a single state variable whose transaction-time price path is binomial, (iii) there are risk-free bonds with calendar-time maturities, and (iv) the relation between transaction time and calendar time is stochastic. The state variable could be interpreted in various ways. For example, it could be the price of a share of stock, as in Black and Scholes [1973], or a factor that summarizes changes in the investment opportunity set, as in Cox, Ingersoll and Ross [1985], or one that drives changes in the term structure of interest rates (Ho and Lee [1986], Heath, Jarrow and Morton [1992]). Property (iv) generally introduces stochastic volatility in the process of the state variable when recorded in calendar time. The paper investigates the pricing of derivative securities with calendar-time maturity. The restrictions obtained in Merton (1973) using simple buy-and-hold arbitrage portfolio arguments do not necessarily hold. Conditions are derived for all derivatives to be priced by dynamic arbitrage, i.e., for market completeness in the sense of Harrison and Pliska [1981]. A particular class of stationary economies where markets are indeed complete is characterized. Nous étudions la problématique de détermination de prix d'options lorsque la volatilité est stochastique. Normalement, la présence d'une volatilité stochastique entraîne une incomplétude des marchés. Nous proposons une approche par arbitrage, malgré cette apparente incomplétude. Elle consiste à exploiter une modélisation de la volatili

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Paper provided by California Institute of Technology, Division of the Humanities and Social Sciences in its series Working Papers with number 977.

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Date of creation: Jul 1996
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Publication status: Published:
Handle: RePEc:clt:sswopa:977
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  1. 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.
  2. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
  3. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
  4. Engle, Robert F. & Russell, Jeffrey R., 1997. "Forecasting the frequency of changes in quoted foreign exchange prices with the autoregressive conditional duration model," Journal of Empirical Finance, Elsevier, vol. 4(2-3), pages 187-212, June.
  5. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  6. Ghysels, E. & Jasiak, J., 1994. "Stochastic Volatility and time Deformation: An Application of trading Volume and Leverage Effects," Cahiers de recherche 9403, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  7. Kamara, Avraham & Miller, Thomas W., 1995. "Daily and Intradaily Tests of European Put-Call Parity," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 30(04), pages 519-539, December.
  8. Ho, Thomas S Y & Lee, Sang-bin, 1986. " Term Structure Movements and Pricing Interest Rate Contingent Claims," Journal of Finance, American Finance Association, vol. 41(5), pages 1011-29, December.
  9. Robert Jarrow & Dilip Madan, 1995. "Option Pricing Using The Term Structure Of Interest Rates To Hedge Systematic Discontinuities In Asset Returns," Mathematical Finance, Wiley Blackwell, vol. 5(4), pages 311-336.
  10. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-55, January.
  11. Harrison, J. Michael & Kreps, David M., 1979. "Martingales and arbitrage in multiperiod securities markets," Journal of Economic Theory, Elsevier, vol. 20(3), pages 381-408, June.
  12. Harrison, J. Michael & Pliska, Stanley R., 1981. "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and their Applications, Elsevier, vol. 11(3), pages 215-260, August.
  13. Cox, John C & Ingersoll, Jonathan E, Jr & Ross, Stephen A, 1985. "An Intertemporal General Equilibrium Model of Asset Prices," Econometrica, Econometric Society, vol. 53(2), pages 363-84, March.
  14. Olivier Scaillet & Boris Leblanc, 1998. "Path dependent options on yields in the affine term structure model," Finance and Stochastics, Springer, vol. 2(4), pages 349-367.
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