Finance and Stochastics

, Volume 12, Issue 1, pp 1–19 | Cite as

Optimal importance sampling with explicit formulas in continuous time

Article

Abstract

In the Black–Scholes model, consider the problem of selecting a change of drift which minimizes the variance of Monte Carlo estimators for prices of path-dependent options.

Employing large deviations techniques, the asymptotically optimal change of drift is identified as the solution to a one-dimensional variational problem, which may be reduced to the associated Euler–Lagrange differential equation.

Closed-form solutions for geometric and arithmetic average Asian options are provided.

Keywords

Monte Carlo methods Variance reduction Importance sampling Large deviations 

Mathematics Subject Classification (2000)

91B28 60F10 65C05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dembo, A., Zeitouni, O.: Large Deviations Techniques and Applications, 2nd edn. Applications of Mathematics (New York), vol. 38. Springer, New York (1998) MATHGoogle Scholar
  2. 2.
    Deuschel, J.-D., Stroock, D.W.: Large Deviations. Pure and Applied Mathematics, vol. 137. Academic, Boston (1989) MATHGoogle Scholar
  3. 3.
    Dufresne, D.: The integral of geometric Brownian motion. Adv. Appl. Probab. 33, 223–241 (2001) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Dupuis, P., Ellis, R.S.: A Weak Convergence Approach to the Theory of Large Deviations. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1997) MATHGoogle Scholar
  5. 5.
    Dupuis, P., Wang, H.: Importance sampling, large deviations, and differential games. Stoch. Stoch. Rep. 76, 481–508 (2004) MATHMathSciNetGoogle Scholar
  6. 6.
    Dupuis, P., Wang, H.: Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab. 15, 1–38 (2005) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Geman, H., Yor, M.: Bessel processes, Asian options and perpetuities. Math. Finance 3, 349–375 (1993) MATHCrossRefGoogle Scholar
  8. 8.
    Glasserman, P., Wang, Y.: Counterexamples in importance sampling for large deviations probabilities. Ann. Appl. Probab. 7, 731–746 (1997) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Glasserman, P., Heidelberger, P., Shahabuddin, P.: Asymptotically optimal importance sampling and stratification for pricing path-dependent options. Math. Finance 9, 117–152 (1999) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kemna, A., Vorst, A.: A pricing method for options based on average values. J. Bank. Finance 14, 113–129 (1990) CrossRefGoogle Scholar
  11. 11.
    Levy, E.: Pricing European average rate currency options. J. Int. Money Finance 11, 474–491 (1992) CrossRefGoogle Scholar
  12. 12.
    Schilder, M.: Some asymptotic formulas for Wiener integrals. Trans. Am. Math. Soc. 125, 63–85 (1966) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Siegmund, D.: Importance sampling in the Monte Carlo study of sequential tests. Ann. Stat. 4, 673–684 (1976) MATHMathSciNetGoogle Scholar
  14. 14.
    Stroock, D.W.: Probability Theory, an Analytic View. Cambridge University Press, Cambridge (1993) MATHGoogle Scholar
  15. 15.
    Turnbull, S.M., Wakeman, L.M.D.: A quick algorithm for pricing European average options. J. Financ. Quant. Anal. 26, 377–389 (1991) CrossRefGoogle Scholar
  16. 16.
    Varadhan, S.R.S.: Asymptotic probabilities and differential equations. Commun. Pure Appl. Math. 19, 261–286 (1966) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA

Personalised recommendations