Advertisement

Applications of Monte Carlo/Quasi-Monte Carlo Methods in Finance: Option Pricing

  • Yongzeng Lai
  • Jerome Spanier

Abstract

The pricing of options is a very important problem encountered in financial markets today. The famous Black-Scholes model provides explicit closed form solutions for the values of certain (European style) call and put options. But for many other options, either there are no closed form solutions, or if such closed form solutions exist, the formulas exhibiting them are complicated and difficult to evaluate accurately by conventional methods. In this case, Monte Carlo methods may prove to be valuable.

In this paper, we illustrate two separate applications of Monte Carlo and/or quasi-Monte Carlo methods to the pricing of options: first, the method is used to estimate multiple integrals related to the evaluation of European style options; second, an adaptive Monte Carlo method is applied to a finite difference approximation of a partial differential equation formulation of a class of finance problems. Some of the advantages in using the Monte Carlo method for such problems are discussed.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    . Barrett, J., Moore, G., Wilmott, P.: Inelegant Efficiency.Risk Magazine 5(1995) 82–84Google Scholar
  2. 2.
    . Black, F., Scholes, M.: The Pricing of Options and Corporate Liabilities.J. Pol and Econ. 81(1973) 637–659CrossRefGoogle Scholar
  3. 3.
    . Boyle, P.P.: Options: a Monte Carlo Approach.J. Finan. Econ. 4(1977) 323–338CrossRefGoogle Scholar
  4. 4.
    . Chance, D. M.:An Introduction to Derivatives, (third edition) The Dryden Press 1995Google Scholar
  5. 5.
    . Cox, J. C., Rubinstein, M.:Options Markets. Prentice Hall 1985Google Scholar
  6. 6.
    . Cox, J. C. Ross, S. A., Rubinstein, M.: Option Pricing: a Simplified Approach.J. Fin. Econ. 71979 229–263CrossRefzbMATHGoogle Scholar
  7. 7.
    . Duffie, D.:Dynamic Asset Pricing Theory. Princeton 1992Google Scholar
  8. 8.
    . Duffie, D.:Security Markets: Stochastic Models. Academic Press, Inc. 1988zbMATHGoogle Scholar
  9. 9.
    . Gemmili, G.:Options Pricing. McGraw-Hill 1992Google Scholar
  10. 10.
    . Hull, J. C.:Options, Futures, and other Derivative Securities. Prentice-Hall, Inc. 1993Google Scholar
  11. 11.
    . Paskov, S.: Computing High Dimensional Integrals with Applications to Finance. preprint Columbia Univ. (1994)Google Scholar
  12. 12.
    . Wilmott, P., Dewynne, J., Howison, S.:Option Pricing: Mathematical Models and Computation. Oxford University Press 1995zbMATHGoogle Scholar
  13. 13.
    . Harrison, J. M., Kresps, D.: Martingales and Arbitrage in Multiperiod Securities Markets.J. Econ. Theory 20(1979) 381–408CrossRefzbMATHGoogle Scholar
  14. 14.
    . Niederreiter, H.:Random Number Generation and Quasi-Monte Carlo Methods. SIAM 1992CrossRefzbMATHGoogle Scholar
  15. 15.
    . Harrison, J. M., Pliska, S. R.: Martingales and Stochastic Integrals in the Theory of Continuous Trading.Stoch. Proc. Appi 11(1981) 261–271CrossRefMathSciNetGoogle Scholar
  16. 16.
    . Halton, J.: Sequential Monte Carlo. Proc. Camb. Phil. Soc.58(1962) 57–73CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    . Halton, J.: Sequential Monte Carlo Techniques for the Solution of Linear Systems.J. Sei. Comp. 9(1994) 213–257zbMATHMathSciNetGoogle Scholar
  18. 18.
    . Li, L., Spanier, J.: Approximation of Transport Equations by Matrix Equations and Sequential Sampling.Monte Carlo Methods and Appi 3(1997) 171–198zbMATHMathSciNetGoogle Scholar
  19. 19.
    . Lai, Y.: Monte Carlo and Quasi-Monte Carlo Methods and Their Applications. Ph.D. dissertation Claremont Graduate University 1998Google Scholar
  20. 20.
    . Broadie, M., Glasserman, P.: Pricing American-style Securities Using Simulation.J. of Economic Dynamics and Control 21(1997) 1323–1352CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    . Spanier, J., Gelbard, E. M.:Monte Carlo Principles and Neutron Transport Problems. Addison-Wesley 1969zbMATHGoogle Scholar
  22. 22.
    . Karatzas, I.:Lectures on the Mathematics of Finance. American Mathematical Society 1998Google Scholar
  23. 23.
    . Joy, C., Boyle, P. P., Tan, K. S.: Quasi-Monte Carlo Methods in Numerical Finance.Management ScienceVol. 42 No. 6 June 1996Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Yongzeng Lai
    • 1
  • Jerome Spanier
    • 1
  1. 1.Department of MathematicsClaremont Graduate UniversityClaremontUSA

Personalised recommendations