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Fast and Simple Method for Pricing Exotic Options using Gauss-Hermite Quadrature on a Cubic Spline Interpolation

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  • Xiaolin Luo
  • Pavel V. Shevchenko

Abstract

There is a vast literature on numerical valuation of exotic options using Monte Carlo, binomial and trinomial trees, and finite difference methods. When transition density of the underlying asset or its moments are known in closed form, it can be convenient and more efficient to utilize direct integration methods to calculate the required option price expectations in a backward time-stepping algorithm. This paper presents a simple, robust and efficient algorithm that can be applied for pricing many exotic options by computing the expectations using Gauss-Hermite integration quadrature applied on a cubic spline interpolation. The algorithm is fully explicit but does not suffer the inherent instability of the explicit finite difference counterpart. A `free' bonus of the algorithm is that it already contains the function for fast and accurate interpolation of multiple solutions required by many discretely monitored path dependent options. For illustrations, we present examples of pricing a series of American options with either Bermudan or continuous exercise features, and a series of exotic path-dependent options of target accumulation redemption note (TARN). Results of the new method are compared with Monte Carlo and finite difference methods, including some of the most advanced or best known finite difference algorithms in the literature. The comparison shows that, despite its simplicity, the new method can rival with some of the best finite difference algorithms in accuracy and at the same time it is significantly faster. Virtually the same algorithm can be applied to price other path-dependent financial contracts such as Asian options and variable annuities.

Suggested Citation

  • Xiaolin Luo & Pavel V. Shevchenko, 2014. "Fast and Simple Method for Pricing Exotic Options using Gauss-Hermite Quadrature on a Cubic Spline Interpolation," Papers 1408.6938, arXiv.org, revised Dec 2014.
    • Handle: RePEc:arx:papers:1408.6938
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    References listed on IDEAS

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    1. Dietmar Leisen & Matthias Reimer, 1996. "Binomial models for option valuation - examining and improving convergence," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(4), pages 319-346.
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    3. P. Forsyth & K. Vetzal & R. Zvan, 2002. "Convergence of numerical methods for valuing path-dependent options using interpolation," Review of Derivatives Research, Springer, vol. 5(3), pages 273-314, October.
    4. Xiaolin Luo & Pavel Shevchenko, 2013. "Pricing TARN Using a Finite Difference Method," Papers 1304.7563, arXiv.org, revised Aug 2014.
    5. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    7. Wilmott,Paul & Howison,Sam & Dewynne,Jeff, 1995. "The Mathematics of Financial Derivatives," Cambridge Books, Cambridge University Press, number 9780521497893, September.
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    Cited by:

    1. Xiaolin Luo & Pavel Shevchenko, 2015. "Variable Annuity with GMWB: surrender or not, that is the question," Papers 1507.08738, arXiv.org.
    2. Shevchenko, Pavel V. & Luo, Xiaolin, 2017. "Valuation of variable annuities with Guaranteed Minimum Withdrawal Benefit under stochastic interest rate," Insurance: Mathematics and Economics, Elsevier, vol. 76(C), pages 104-117.
    3. Luo, Xiaolin & Shevchenko, Pavel V., 2015. "Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization," Insurance: Mathematics and Economics, Elsevier, vol. 62(C), pages 5-15.
    4. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A unified pricing of variable annuity guarantees under the optimal stochastic control framework," Papers 1605.00339, arXiv.org.
    5. Pavel V. Shevchenko & Xiaolin Luo, 2016. "Valuation of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Stochastic Interest Rate," Papers 1602.03238, arXiv.org, revised Jan 2017.
    6. Xiaolin Luo & Pavel Shevchenko, 2014. "Fast Numerical Method for Pricing of Variable Annuities with Guaranteed Minimum Withdrawal Benefit under Optimal Withdrawal Strategy," Papers 1410.8609, arXiv.org.
    7. Pavel V. Shevchenko & Xiaolin Luo, 2016. "A Unified Pricing of Variable Annuity Guarantees under the Optimal Stochastic Control Framework," Risks, MDPI, vol. 4(3), pages 1-31, July.

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