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A Comparison of Monte Carlo, Lattice Rules and Other Low-Discrepancy Point Sets

  • Christiane Lemieux
  • Pierre L’Ecuyer
Conference paper

Abstract

We explore how lattice rules can reduce the variance of the estimators for simulation problems, in comparison with the Monte Carlo method. To do this, we compare these two methods on option valuation problems in finance, along with two types of (t, s)-sequences. We also look at the effect of combining variance reduction techniques with the preceding approaches. Our numerical results seem to indicate that lattice rules are less affected by the “curse of dimensionality” than the other types of quasi-Monte Carlo methods and provide more precise estimators than Monte Carlo does.

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References

  1. 1.
    P. Acworth, M. Broadie, and P. Glasserman. A comparison of some Monte Carlo and quasi-Monte Carlo techniques for option pricing. In P. Hellekalek and H. Niederreiter, editors,Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, number 127 in Lecture Notes in Statistics, pages 1–18. Springer-Verlag, 1997.Google Scholar
  2. 2.
    A.N. Avramidis and J.R. Wilson. Integrated variance reduction strategies for simulation.Operations Research, 44: 327–346, 1996.CrossRefzbMATHGoogle Scholar
  3. 3.
    F. Black and M. Scholes. The pricing of options and corporate liabilities.Journal of Political Economy, 81: 637–659, 1973.CrossRefGoogle Scholar
  4. 4.
    P. Bratley, B.L. Fox, and H. Niederreiter. Implementation and tests of low- discrepancy sequences.ACM Transactions on Modeling and Computer Simulation, 2: 195–213, 1992.CrossRefzbMATHGoogle Scholar
  5. 5.
    P. Bratley, B.L. Fox, and L.E. Schrage.A Guide to Simulation. Springer- Verlag, New York, second edition, 1987.CrossRefGoogle Scholar
  6. 6.
    R.E. Caflish, W. Morokoff, and A. Owen. Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension.The Journal of Computational Finance, l(l): 27–46, 1997.Google Scholar
  7. 7.
    W.G. Cochran.Sampling Techniques. John Wiley and Sons, New York, second edition, 1977.zbMATHGoogle Scholar
  8. 8.
    R. Cranley and T.N.L. Patterson. Randomization of number theoretic methods for multiple integration.SIAM Journal on Numerical Analysis, 13 (6): 904–914, 1976.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    K Entacher, P Hellekalek, and P L’Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. Submitted, 1998.Google Scholar
  10. 10.
    H. Faure. Discrépance des suites associées à un système de numération.Acta Arithmetica, 61: 337–351, 1982.MathSciNetGoogle Scholar
  11. 11.
    G.S. Fishman.Monte Carlo: Concepts, Algorithms, and Applications. Springer Series in Operations Research. Springer-Verlag, New York, 1996.CrossRefzbMATHGoogle Scholar
  12. 12.
    G.S. Fishman and B.D. Wang. Antithetic variates revisited.Communications of the ACM, 26: 964–971, 1983.CrossRefzbMATHGoogle Scholar
  13. 13.
    B.L. Fox. Implementation and relative efficiency of quasirandom sequence generators.ACM Transactions on Mathematical Software, 12: 362–376, 1986.CrossRefzbMATHGoogle Scholar
  14. 14.
    F.J. Hickernell. Lattice rules: How well do they measure up? In P Hellekalek and G Larcher, editors,Random and Quasi-Random Point Sets, volume 138 ofLecture Notes in Statistics, pages 109–166. Springer, New York, 1998.CrossRefGoogle Scholar
  15. 15.
    AGZ Kemna and CF Vorst. A pricing method for options based on average asset values.Journal of Banking and Finance, 14: 113–129, 1990.CrossRefGoogle Scholar
  16. 16.
    D.E. Knuth.The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Addison-Wesley, Reading, Mass., third edition, 1997.Google Scholar
  17. 17.
    P. L’Ecuyer. Efficiency improvement via variance reduction. InProceedings of the 1994 Winter Simulation Conference, pages 122–132. IEEE Press, 1994.Google Scholar
  18. 18.
    P. L’Ecuyer. Good parameters and implementations for combined multiple recursive random number generators.Operations Research, 47(1), 1999. To appear.Google Scholar
  19. 19.
    P. L’Ecuyer. Tables of linear congruential generators of different sizes and good lattice structure.Mathematics of Computation, 68 (225): 249–260, 1999.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    C. Lemieux Léutilisation de méthodes quasi-Monte Carlo randomisées pour améliorer léefficacité des estimateurs en simulation. PhD thesis, Université de Montréal, 1999. In preparation.Google Scholar
  21. 21.
    C. Lemieux and P. L’Ecuyer. Efficiency improvement by lattice rules for pricing asian options. InProceedings of the 1998 Winter Simulation Conference, pages 579–586. IEEE Press, 1998.Google Scholar
  22. 22.
    H. Niederreiter. Multidimensional numerical integration using pseudorandom numbers.Mathematical Programming Study, 27: 17–38, 1986.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    H. Niederreiter.Random Number Generation and Quasi-Monte Carlo Methods, volume 63 ofSIAM CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1992.CrossRefzbMATHGoogle Scholar
  24. 24.
    AB Owen. Randomly permuted(t, m, s)-nets and(t, s)-sequences. In H Niederreiter and PJ-S Shiue, editors,Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, number 106 in Lecture Notes in Statistics, pages 299–317. Springer-Verlag, 1995.Google Scholar
  25. 25.
    AB Owen. Scrambled net variance for integrals of smooth functions.Annals of Statistics, 25 (4): 1541–1562, 1997.zbMATHMathSciNetGoogle Scholar
  26. 26.
    A Papageorgiou and J Traub. Beating Monte Carlo.Risk, 9: 63–65, June 1996.Google Scholar
  27. 27.
    S Paskov and J Traub. Faster valuation of financial derivatives.Journal of Portfolio Management, 22: 113 - 120, 1995.CrossRefGoogle Scholar
  28. 28.
    IH Sloan and S Joe.Lattice Methods for Multiple Integration. Clarendon Press, Oxford, 1994.zbMATHGoogle Scholar
  29. 29.
    IH Sloan and L Walsh. A computer search of rank 2 lattice rules for multi-dimensional quadrature.Mathematics of Computation, 54: 281–302, 1990.zbMATHMathSciNetGoogle Scholar
  30. 30.
    IM Sobol’. The distribution of points in a cube and the approximate evaluation of integrals.U.S.S.R. Comput. Math, and Math. Phys., 7: 86–112, 1967.CrossRefMathSciNetGoogle Scholar
  31. 31.
    J Spanier and EH Maize. Quasi-random methods for estimating integrals using relatively small samples.SIAM Review, 36: 18–44, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    S Tezuka.Uniform Random Numbers: Theory and Practice. Kluwer Academic Publishers, Norwell, Mass., 1995.CrossRefzbMATHGoogle Scholar
  33. 33.
    B Tuffin. On the use of low-discrepancy sequences in Monte Carlo methods. Technical Report No. 1060, I.R.I.S.A., Rennes, Prance, 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Christiane Lemieux
    • 1
  • Pierre L’Ecuyer
    • 1
  1. 1.Département d’Informatique et de Recherche OpérationnelleUniversité de MontréalCanada

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