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Topics in Numerical Methods for Finance pp 57–94Cite as

American Option Pricing Using Simulation and Regression: Numerical Convergence Results

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Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 19))

Abstract

Recently, simulation methods combined with regression techniques have gained importance when it comes to American option pricing. In this paper, we consider such methods and we examine numerically their convergence properties. We first consider the Least Squares Monte-Carlo (LSM) method of (Longstaff and Schwartz, Rev. Financ. Stud., 14:113–147, 2001) and report convergence rates for the cross-sectional regressions as well as for the estimated price. The results show that the method converges fast, and this holds even with multiple early exercises and with multiple stochastic factors as long as the payoff function is smooth. We also compare the convergence rates to those obtained when using the related methods proposed by (Carriere, Insur. Math. Econ., 19:19–30, 1996; Tsitsiklis JN, Van Roy, IEEE Trans. Neural Network, 12(4):694–703, 2001). The results show that the price estimates from the latter methods converge significantly slower in the multi-period situation.

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Notes

  1. 1.

    Though the methods proposed in [5, 6] also use simulation, they do not directly rely on regression techniques for the approximation. Instead, the first method uses additional subsampling whereas the latter requires information about the transition densities to approximate the continuation value.

  2. 2.

    See [22] for the proof and the necessary assumptions.

  3. 3.

    See [22] for the proof and the necessary assumptions.

  4. 4.

    The exception to this is option number 9, which is a deep out of the money option with low volatility. For this option, very few paths are in the money and hence used in the cross-sectional regression, and as a result the price estimate is biased upwards due to overfitting. In fact, when using 10 times the number of paths the estimated value for βBIAS 2, M is significantly negative for this option also.

  5. 5.

    To be precise, what should be increased is the number of in-the-money paths, Ñ, used in the cross-sectional regressions. Unfortunately, it is not possible to control directly this number by the nature of the Monte Carlo study. However, the proportion of the paths that are in the money should be approximately constant and we will therefore have that Ñ ∝ M 4.

  6. 6.

    The total number of regressors with a maximum order of at most m in r dimensions is given by \(\left (m + r\right )!/\left (m!r!\right )\) (see also [11]).

  7. 7.

    To achieve this we set C = 0. 10928 and round the number of paths.

  8. 8.

    Also note that the estimated order of convergence in Table 7 are very close to those in Table 4 in which the true conditional expectation is used.

  9. 9.

    In fact, the cross-sectional regressions break down in more than 80% of the cases when using eight Laguerre polynomials and in all the cases when using more than eight polynomials. The regressions also break down in all cases when using ten Hermite polynomials.

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Authors and Affiliations

  1. Department of Finance, HEC Montreal, 3000 chemin de la Cote-Sainte-Catherine, Montreal, Quebec, Canada, H3T 2A7

    Lars Stentoft

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  1. Lars Stentoft
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Correspondence to Lars Stentoft .

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Editors and Affiliations

  1. , Dublin City Univ. Business School, Dublin City University, Ballymun Rd, Dublin 9, Dublin 9, Ireland

    Mark Cummins

  2. , Kemmy Business School, University of Limerick, Limerick, Ireland

    Finbarr Murphy

  3. Inst. for Numerical Computat. & Analy., 7 - 9 Dame Court, Dublin 2, Dublin 2, Ireland

    John J.H. Miller

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Stentoft, L. (2012). American Option Pricing Using Simulation and Regression: Numerical Convergence Results. In: Cummins, M., Murphy, F., Miller, J. (eds) Topics in Numerical Methods for Finance. Springer Proceedings in Mathematics & Statistics, vol 19. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-3433-7_5

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