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Empirical Martingale Simulation for Asset Prices

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  • Jin-Chuan Duan
  • Jean-Guy Simonato

Abstract

This paper proposes a simple modification to the standard Monte Carlo simulation procedure for computing the prices of derivative securities. The modification imposes the martingale property on the simulated sample paths of the underlying asset price. This procedure is referred to as the empirical martingale simulation (EMS). The EMS ensures that the price estimated by simulation satisfies rational option pricing bounds. The EMS also yields a substantial error reduction for the price estimate. The EMS can be easily coupled with the standard variance reduction methods to obtain greater computational efficiency. Simulation studies are conducted for European and Asian call options using both the Black and Scholes and GARCH option pricing frameworks. The results indicate that the EMS yields substantial variance reduction particularly for in- and at-the-money options. Cette étude propose une modification simple aux procédures traditionnelles de calcul de prix des produits dérivés par simulation de Monte Carlo. La modification impose la propriété de martingale aux trajectoires simulées de la variable d'état sous-jacente. L'utilisation de cette procédure assure que l'estimé de prix respecte les bornes rationnelles d'option tout en diminuant de façon substantielle la variance des estimés de prix. La procédure peut aisément être jumelée aux méthodes traditionnelles de réduction de variance afin d'obtenir une plus grande efficacité. Une étude de simulation est présentée pour des options d'achat Européennes et Asiatiques. Les résultats indiquent que la méthode obtient des réductions substantielle de la variance des estimés de prix et ce, particulièrement pour les options in et at the money .

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Bibliographic Info

Paper provided by CIRANO in its series CIRANO Working Papers with number 95s-43.

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Date of creation: 01 Oct 1995
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Handle: RePEc:cir:cirwor:95s-43

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Keywords: Martingale; Option Pricing; Monte Carlo Simulation; GARCH; Asian Options; Martingale; Evaluation des Options; Simulation de Monte Carlo; GARCH; Options Asiatiques;

References

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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. Robert C. Merton, 1973. "Theory of Rational Option Pricing," Bell Journal of Economics, The RAND Corporation, vol. 4(1), pages 141-183, Spring.
  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. 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.
  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. Peter Ritchken & L. Sankarasubramanian & Anand M. Vijh, 1993. "The Valuation of Path Dependent Contracts on the Average," Management Science, INFORMS, vol. 39(10), pages 1202-1213, October.
  7. Kemna, A. G. Z. & Vorst, A. C. F., 1990. "A pricing method for options based on average asset values," Journal of Banking & Finance, Elsevier, vol. 14(1), pages 113-129, March.
  8. Lucas, Robert E, Jr, 1978. "Asset Prices in an Exchange Economy," Econometrica, Econometric Society, vol. 46(6), pages 1429-45, November.
  9. Boyle, Phelim P., 1977. "Options: A Monte Carlo approach," Journal of Financial Economics, Elsevier, vol. 4(3), pages 323-338, May.
  10. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
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