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Option pricing with discrete rebalancing

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  • Prigent, Jean-Luc
  • Renault, Olivier
  • Scaillet, Olivier

Abstract

We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after ¯xed relative changes in the stock price. The stock price follows a marked point process and the market is incomplete. We first characterisethe equivalent martingale measures. An explicit pricing formula based on the minimal martingale measure is then provided together with the hedging strategy underlying port-folio adjustments. Two examples illustrate our pricing framework : a jump process driven by a latent geometric Brownian motion and a marked Poisson process. We establish the convergence to the Black-Scholes model when the triggering price increment shrinks to zero. For the empirical application we use IBM, France Telecom and CAC 40 intraday transaction data, and compare option prices given by the marked Poisson model, the Black-Scholes model and observed option prices.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Empirical Finance.

Volume (Year): 11 (2004)
Issue (Month): 1 (January)
Pages: 133-161

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Handle: RePEc:eee:empfin:v:11:y:2004:i:1:p:133-161

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Web page: http://www.elsevier.com/locate/jempfin

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References

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  1. Naoto Kunitomo & Masayuki Ikeda, 1992. "Pricing Options With Curved Boundaries," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 275-298.
  2. Laurent, J.P. & Scaillet, O., 1997. "Variance Optimal Cap Pricing Models," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 1999002, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES), revised 01 Jan 1999.
  3. Jean-Philippe Lesne & Jean-Luc Prigent & Olivier Scaillet, 1998. "Convergence of Discrete Time Option Pricing Models Under Stochastic Interest Rates," Working Papers 98-51, Centre de Recherche en Economie et Statistique.
  4. Andrew Chesher & Geert Dhaene & Christian Gourieroux & Olivier Scaillet, 1999. "Bartlett Identities Tests," Working Papers 99-32, Centre de Recherche en Economie et Statistique.
  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. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1998. "Implied Volatility Functions: Empirical Tests," Journal of Finance, American Finance Association, vol. 53(6), pages 2059-2106, December.
  7. 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.
  8. 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.
  9. Runggaldier, Wolfgang J. & Martin Schweizer, 1995. "Convergence of Option Values under Incompleteness," Discussion Paper Serie B 333, University of Bonn, Germany.
  10. Hua He William P. Keirstead and Joachim Rebholz., 1995. "Double Lookbacks," Research Program in Finance Working Papers RPF-248, University of California at Berkeley.
  11. Fabio Mercurio & Ton Vorst, 1996. "Option pricing with hedging at fixed trading dates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(2), pages 135-158.
  12. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-64, May.
  13. Schweizer, Martin, 1992. "Martingale densities for general asset prices," Journal of Mathematical Economics, Elsevier, vol. 21(4), pages 363-378.
  14. Cox, John C. & Ross, Stephen A. & Rubinstein, Mark, 1979. "Option pricing: A simplified approach," Journal of Financial Economics, Elsevier, vol. 7(3), pages 229-263, September.
  15. N. Hofmann & E. Platen & M. Schweizer, 1992. "Option Pricing under Incompleteness and Stochastic Volatility," Discussion Paper Serie B 209, University of Bonn, Germany.
  16. Chernov, Mikhail & Ghysels, Eric, 2000. "A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation," Journal of Financial Economics, Elsevier, vol. 56(3), pages 407-458, June.
  17. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
  18. repec:fth:louvco:9939 is not listed on IDEAS
  19. 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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  22. David B. Colwell & Robert J. Elliott, 1993. "Discontinuous Asset Prices And Non-Attainable Contingent Claims," Mathematical Finance, Wiley Blackwell, vol. 3(3), pages 295-308.
  23. Chesher, Andrew & Spady, Richard, 1991. "Asymptotic Expansions of the Information Matrix Test Statistic," Econometrica, Econometric Society, vol. 59(3), pages 787-815, May.
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Cited by:
  1. J.L. Prigent & O. Scaillet, 2000. "Weak Convergence of Hedging Strategies of Contingent Claims," THEMA Working Papers 2000-50, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.

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