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Arbitrage Based Pricing When Volatility Is Stochastic

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  • Peter Bossaert
  • Eric Ghysels
  • Christian Gouriéroux

Abstract

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 volatilité, proposée par Clark (1973), fondée sur une distinction entre un temps calendaire et un temps de transaction. En faisant cette distinction et en supposant qu'il y a une simple variable d'état binomiale en temps de transaction et un taux sans risque en temps calendaire, nous discutons les conditions d'absence d'opportunités d'arbitrage. Nous caractérisons les conditions permettant la détermination des prix d'options par arbitrage dynamique dans le sens de Harrison et Pliska (1981) et nous montrons que les restrictions à la Merton (1973) ne s'appliquent plus.

Suggested Citation

  • Peter Bossaert & Eric Ghysels & Christian Gouriéroux, 1996. "Arbitrage Based Pricing When Volatility Is Stochastic," CIRANO Working Papers 96s-20, CIRANO.
  • Handle: RePEc:cir:cirwor:96s-20
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    References listed on IDEAS

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    1. Ghysels, E. & Jasiak, J., 1994. "Stochastic Volatility and time Deformation: an Application of trading Volume and Leverage Effects," Cahiers de recherche 9403, Universite de Montreal, Departement de sciences economiques.
    2. 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.
    3. 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.
    4. Ghysels, E. & Harvey, A. & Renault, E., 1995. "Stochastic Volatility," Papers 95.400, Toulouse - GREMAQ.
    5. 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.
    6. 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.
    7. 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 637-654, May-June.
    8. Clark, Peter K, 1973. "A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices," Econometrica, Econometric Society, vol. 41(1), pages 135-155, January.
    9. David Heath & Robert Jarrow & Andrew Morton, 2008. "Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation," World Scientific Book Chapters,in: Financial Derivatives Pricing Selected Works of Robert Jarrow, chapter 13, pages 277-305 World Scientific Publishing Co. Pte. Ltd..
    10. 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.
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    12. 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.
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    Cited by:

    1. Amigues, Jean-Pierre & Favard, Pascal & Gaudet, Gerard & Moreaux, Michel, 1998. "On the Optimal Order of Natural Resource Use When the Capacity of the Inexhaustible Substitute Is Limited," Journal of Economic Theory, Elsevier, vol. 80(1), pages 153-170, May.
    2. Álvaro Cartea & Thilo Meyer-Brandis, 2010. "How Duration Between Trades of Underlying Securities Affects Option Prices," Review of Finance, European Finance Association, vol. 14(4), pages 749-785.
    3. Bossaerts, Peter & Hillion, Pierre, 2003. "Local parametric analysis of derivatives pricing and hedging," Journal of Financial Markets, Elsevier, vol. 6(4), pages 573-605, August.
    4. Jean -Luc Prigent & Olivier Renault & Olivier Scaillet, 1999. "An Autoregressive Conditional Binomial Option Pricing Model," Working Papers 99-65, Center for Research in Economics and Statistics.

    More about this item

    Keywords

    Incomplete Markets; Transaction Time; Change of Time; Stochastic Volatility; Marchés incomplets; Temps de transaction; Changement de temps; Volatilité stochastique;

    JEL classification:

    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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