Pricing American Options by Simulation Using a Stochastic Mesh with Optimized Weights

  • Mark Broadie
  • Paul Glasserman
  • Zachary Ha
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 49)


This paper develops a simulation method for pricing path-dependent American options, and American options on a large number of underlying assets, such as basket options. Standard numerical procedures (lattice methods and finite difference methods) are generally inapplicable to such high-dimensional problems, and this has motivated research into simulation-based methods. The optimal stopping problem embedded in the pricing of American options makes this a nonstandard problem for simulation.

This paper extends the stochastic mesh introduced in Broadie and Glasserman [5]. In its original form, the stochastic mesh method required knowledge of the transition density of the underlying process of asset prices and other state variables. This paper extends the method to settings in which the transition density is either unknown or fails to exist. We avoid the need for a transition density by choosing mesh weights through a constrained optimization problem. If the weights are constrained to correctly price sufficiently many simple instruments, they can be expected to work well in pricing a more complex American option. We investigate two criteria for use in the optimization — maximum entropy and least squares. The methods are illustrated through numerical examples.


Maximum Entropy Transition Density American Option Maximum Entropy Method Underlying Asset 
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  1. [1]
    Andersen, L., 1999, “A Simple Approach to the Pricing of Bermudan Swaptions in the Multi-Factor Libor Market Model,” working paper, General Re Financial Products, NY.Google Scholar
  2. [2]
    Avellaneda, M., C. Friedman, R. Holmes, D. Samperi, 1997, “Calibrating Volatility Surfaces via Relative-Entropy Minimization,” Applied Mathematical Finance, Vol.4, No.l, 37–64.MATHCrossRefGoogle Scholar
  3. [3]
    Barraquand, J., and D. Martineau, 1995, “Numerical Valuation of High Dimensional Multivariate American Securities,” Journal of Financial and Quantitative Analysis, Vol. 30, No. 3, 383–405.CrossRefGoogle Scholar
  4. [4]
    Broadie, M., and P. Glasserman, 1997, “Pricing American-Style Securities Using Simulation,” Journal of Economic Dynamics and Control, Vol. 21, Nos. 8–9, 1323–1352.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Broadie, M., and P. Glasserman, 1997, “A Stochastic Mesh Method for Pricing High-Dimensional American Options,” working paper, Columbia University.Google Scholar
  6. [6]
    Broadie, M., P. Glasserman, and G. Jain, 1997, “Enhanced Monte Carlo Estimates for American Option Prices,” Journal of Derivatives, Vol. 5, No. 1 (Fall), 25–44.CrossRefGoogle Scholar
  7. [7]
    Carrière, J.F., 1996, “Valuation of the Early-Exercise Price for Derivative Securities using Simulations and Splines,” Insurance: Mathematics and Economics, Vol. 19, 19–30.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Golan, A., G. Judge and D. Miller, 1996, Maximum Entropy Econometrics: Robust Estimation with Limited Data, Wiley, New York.MATHGoogle Scholar
  9. [9]
    Longstaff, F., and E. Schwartz, 1998, “Valuing American Options By Simulation: A Simple Least Squares Approach,” working paper, UCLA Anderson Graduate School of Management.Google Scholar
  10. [10]
    Pedersen, M., 1999, “Bermudan Swaptions in the LIBOR Market Model,” working paper, SimCorp A/S, Copenhagen, Denmark.Google Scholar
  11. [11]
    Tilley, J.A., 1993, “Valuing American Options in a Path Simulation Model,” Transactions of the Society of Actuaries, Vol. 45, 83–104.Google Scholar
  12. [12]
    Tsitsiklis, J., and B. Van Roy, 1997, “Optimal Stopping of Markov Processes: Hubert Space Theory, Approximation Algorithms, and an Application to Pricing High-Dimensional Financial Derivatives,” working paper, Laboratory for Information and Decision Sciences, MIT, Cambridge, MA.Google Scholar
  13. [13]
    Wilmott, P., Derivatives, Wiley, Chichester, England, 1998.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Mark Broadie
    • 1
  • Paul Glasserman
    • 1
  • Zachary Ha
    • 2
  1. 1.Columbia Business SchoolNew YorkUSA
  2. 2.Goldman Sachs & Co.New YorkUSA

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