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Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing

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  • Devang Sinha
  • Siddhartha P. Chakrabarty

Abstract

In this paper, we propose and analyze a novel combination of multilevel Richardson-Romberg (ML2R) and importance sampling algorithm, with the aim of reducing the overall computational time, while achieving desired root-mean-squared error while pricing. We develop an idea to construct the Monte-Carlo estimator that deals with the parametric change of measure. We rely on the Robbins-Monro algorithm with projection, in order to approximate optimal change of measure parameter, for various levels of resolution in our multilevel algorithm. Furthermore, we propose incorporating discretization schemes with higher-order strong convergence, in order to simulate the underlying stochastic differential equations (SDEs) thereby achieving better accuracy. In order to do so, we study the Central Limit Theorem for the general multilevel algorithm. Further, we study the asymptotic behavior of our estimator, thereby proving the Strong Law of Large Numbers. Finally, we present numerical results to substantiate the efficacy of our developed algorithm.

Suggested Citation

  • Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Richardson-Romberg and Importance Sampling in Derivative Pricing," Papers 2209.00821, arXiv.org.
    • Handle: RePEc:arx:papers:2209.00821
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    References listed on IDEAS

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    1. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Methodology and Computing in Applied Probability, Springer, vol. 20(2), pages 611-641, June.
    2. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    3. Ahmed Kebaier & Jérôme Lelong, 2018. "Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation," Post-Print hal-01214840, HAL.
    4. Michael B. Giles & Kristian Debrabant & Andreas Ro{ss}ler, 2013. "Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation," Papers 1302.4676, arXiv.org, revised Jun 2019.
    5. Lelong, Jérôme, 2008. "Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2632-2636, November.
    6. Chen, Han-Fu & Guo, Lei & Gao, Ai-Jun, 1987. "Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds," Stochastic Processes and their Applications, Elsevier, vol. 27, pages 217-231.
    7. Chang-Han Rhee & Peter W. Glynn, 2015. "Unbiased Estimation with Square Root Convergence for SDE Models," Operations Research, INFORMS, vol. 63(5), pages 1026-1043, October.
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    Cited by:

    1. Devang Sinha & Siddhartha P. Chakrabarty, 2022. "Multilevel Monte Carlo and its Applications in Financial Engineering," Papers 2209.14549, arXiv.org.

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