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Option Pricing with Discrete Rebalancing

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  • Jean -Luc Prigent

    (Crest)

  • Olivier Renault

    (Crest)

  • Olivier Scaillet

    (Crest)

Abstract

We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after fixed relative variation of the stock price. The stock price follows a marked point process and the market is incomplete. We first characterize the equivalent martingale measures. An explicit formula based on the minimal martingale measure is then provided together with the hedging strategy underlying portfolio adjustments. Under adequate conditions on the stock price dynamics, the minimal pricing formula converges to the Black-Scholes formula when the triggering price increment shrinks to zero. This is shown theoretically and numerically on two examples : a marked Poisson process and a jump process driven by a latent geometric Brownian motion. For the empirical application we use IBM intraday transaction data and compare option prices given by the marked Poisson model and the Black-Scholes model.

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Bibliographic Info

Paper provided by Centre de Recherche en Economie et Statistique in its series Working Papers with number 99-61.

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Date of creation: 1999
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Handle: RePEc:crs:wpaper:99-61

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References

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  1. 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.
  2. Chesher, Andrew & Dhaene, Geert & Gouriéroux, Christian & Scaillet, Olivier, 1999. "Bartlett Identities Tests," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) 1999019, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
  3. Hua He William P. Keirstead and Joachim Rebholz., 1995. "Double Lookbacks," Research Program in Finance Working Papers RPF-248, University of California at Berkeley.
  4. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-64, May.
  5. Schweizer, Martin, 1992. "Martingale densities for general asset prices," Journal of Mathematical Economics, Elsevier, vol. 21(4), pages 363-378.
  6. Norbert Hofmann & Eckhard Platen & Martin Schweizer, 1992. "Option Pricing Under Incompleteness and Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 2(3), pages 153-187.
  7. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
  8. Chesher, Andrew & Spady, Richard, 1991. "Asymptotic Expansions of the Information Matrix Test Statistic," Econometrica, Econometric Society, vol. 59(3), pages 787-815, May.
  9. O. Scaillet & J.-L. Prigent & J.-P. Lesne, 2000. "Convergence of discrete time option pricing models under stochastic interest rates," Finance and Stochastics, Springer, vol. 4(1), pages 81-93.
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  11. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
  12. Bouleau, Nicolas & Lamberton, Damien, 1989. "Residual risks and hedging strategies in Markovian markets," Stochastic Processes and their Applications, Elsevier, vol. 33(1), pages 131-150, October.
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  17. repec:fth:inseep:9932 is not listed on IDEAS
  18. 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.
  19. 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.
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  22. repec:fth:louvco:9939 is not listed on IDEAS
  23. 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.
  24. Runggaldier, Wolfgang J. & Martin Schweizer, 1995. "Convergence of Option Values under Incompleteness," Discussion Paper Serie B 333, University of Bonn, Germany.
  25. 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.
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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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