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Option Pricing with Discrete Rebalancing

  • Prigent, J.-L.

    (Université de Cergy-Pontoise, THEMA)

  • Renault, O.

    (UNIVERSITE CATHOLIQUE DE LOUVAIN, Institut de Recherches Economiques et Sociales (IRES))

  • Scaillet, O.

    (UNIVERSITE CATHOLIQUE DE LOUVAIN, Institut de Recherches Economiques et Sociales (IRES); UNIVERSITE CATHOLIQUE DE LOUVAIN, Institut d’Administration et de Gestion (IAG))

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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Paper provided by Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES) in its series Discussion Papers (IRES - Institut de Recherches Economiques et Sociales) with number 1999029.

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Length: 18
Date of creation: 01 Feb 1999
Date of revision: 00 Oct 1999
Handle: RePEc:ctl:louvir:1999029
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  1. 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.
  2. 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.
  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. CHESHER, Andrew & DHAENE, Geert & GOURIEROUX, Christian & SCAILLET, Olivier, 1999. "Bartlett identities tests," CORE Discussion Papers 1999039, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  5. N. Hofmann & E. Platen & M. Schweizer, 1992. "Option Pricing under Incompleteness and Stochastic Volatility," Discussion Paper Serie B 209, University of Bonn, Germany.
  6. Duffie, Darrell & Lando, David, 2001. "Term Structures of Credit Spreads with Incomplete Accounting Information," Econometrica, Econometric Society, vol. 69(3), pages 633-64, May.
  7. Hua He William P. Keirstead and Joachim Rebholz., 1995. "Double Lookbacks," Research Program in Finance Working Papers RPF-248, University of California at Berkeley.
  8. Schweizer, Martin, 1992. "Martingale densities for general asset prices," Journal of Mathematical Economics, Elsevier, vol. 21(4), pages 363-378.
  9. Jean-Paul Laurent & Olivier Scaillet, 1999. "Variance Optimal Cap Pricing Models," Working Papers 99-07, Centre de Recherche en Economie et Statistique.
  10. 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.
  11. 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.
  12. 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.
  13. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  14. Naoto Kunitomo & Masayuki Ikeda, 1992. "Pricing Options With Curved Boundaries," Mathematical Finance, Wiley Blackwell, vol. 2(4), pages 275-298.
  15. 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.
  16. 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.
  17. 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.
  18. Schweizer, Martin, 1991. "Option hedging for semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 37(2), pages 339-363, April.
  19. 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.
  20. Runggaldier, Wolfgang J. & Martin Schweizer, 1995. "Convergence of Option Values under Incompleteness," Discussion Paper Serie B 333, University of Bonn, Germany.
  21. 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.
  22. repec:fth:inseep:9932 is not listed on IDEAS
  23. 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.
  24. 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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