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Enhancing Quasi-Monte Carlo Simulation by Minimizing Effective Dimension for Derivative Pricing

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  • Ye Xiao

    (Tsinghua University)

  • Xiaoqun Wang

    (Tsinghua University)

Abstract

Many problems in derivative pricing can be formulated as high-dimensional integrals. Many of them do not have closed-form solutions and have to be estimated by numerical integrations such as Monte Carlo or quasi-Monte Carlo (QMC) methods. Since the quasi-random points used for QMC simulation have perfect projections at the first few dimensions, reducing the effective dimension of the integrands can improve the efficiency of QMC. In this paper, based on the first-order Taylor approximations of the functions at Gaussian sample points, we propose a new general method based on principal component analysis (PCA) to reduce the effective dimensions of the functions. Rather than aiming at decomposing the covariance matrix of the Brownian motions as in the traditional PCA, the new method implements PCA on the gradients of the functions at sample points and then an orthogonal transformation is found to reduce the effective dimensions. Numerical experiments show that by using the new dimension reduction method, a significant efficient improvement of QMC can be achieved on pricing exotic options and mortgage-backed securities.

Suggested Citation

  • Ye Xiao & Xiaoqun Wang, 2019. "Enhancing Quasi-Monte Carlo Simulation by Minimizing Effective Dimension for Derivative Pricing," Computational Economics, Springer;Society for Computational Economics, vol. 54(1), pages 343-366, June.
    • Handle: RePEc:kap:compec:v:54:y:2019:i:1:d:10.1007_s10614-017-9732-2
    DOI: 10.1007/s10614-017-9732-2
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    References listed on IDEAS

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    1. Xiaoqun Wang & Ken Seng Tan, 2013. "Pricing and Hedging with Discontinuous Functions: Quasi-Monte Carlo Methods and Dimension Reduction," Management Science, INFORMS, vol. 59(2), pages 376-389, July.
    2. Paul Glasserman & Philip Heidelberger & Perwez Shahabuddin, 1999. "Asymptotically Optimal Importance Sampling and Stratification for Pricing Path‐Dependent Options," Mathematical Finance, Wiley Blackwell, vol. 9(2), pages 117-152, April.
    3. Xiaoqun Wang, 2009. "Dimension Reduction Techniques in Quasi-Monte Carlo Methods for Option Pricing," INFORMS Journal on Computing, INFORMS, vol. 21(3), pages 488-504, August.
    4. Xiaoqun Wang & Ian H. Sloan, 2011. "Quasi-Monte Carlo Methods in Financial Engineering: An Equivalence Principle and Dimension Reduction," Operations Research, INFORMS, vol. 59(1), pages 80-95, February.
    5. Spassimir H. Paskov & Joseph F. Traub, 1995. "Faster Valuation of Financial Derivatives," Working Papers 95-03-034, Santa Fe Institute.
    6. Xiaoqun Wang, 2016. "Handling Discontinuities in Financial Engineering: Good Path Simulation and Smoothing," Operations Research, INFORMS, vol. 64(2), pages 297-314, April.
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    Cited by:

    1. Paul Bilokon & Sergei Kucherenko & Casey Williams, 2022. "Quasi-Monte Carlo methods for calculating derivatives sensitivities on the GPU," Papers 2209.11337, arXiv.org.
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    3. Zhijian He & Xiaoqun Wang, 2021. "An Integrated Quasi-Monte Carlo Method for Handling High Dimensional Problems with Discontinuities in Financial Engineering," Computational Economics, Springer;Society for Computational Economics, vol. 57(2), pages 693-718, February.

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