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Sparse Grid Method for Highly Efficient Computation of Exposures for xVA

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  • Lech A. Grzelak

Abstract

Every "x"-adjustment in the so-called xVA financial risk management framework relies on the computation of exposures. Considering thousands of Monte Carlo paths and tens of simulation steps, a financial portfolio needs to be evaluated numerous times during the lifetime of the underlying assets. This is the bottleneck of every simulation of xVA. In this article, we explore numerical techniques for improving the simulation of exposures. We aim to decimate the number of portfolio evaluations, particularly for large portfolios involving multiple, correlated risk factors. The usage of the Stochastic Collocation (SC) method, together with Smolyak's sparse grid extension, allows for a significant reduction in the number of portfolio evaluations, even when dealing with many risk factors. The proposed model can be easily applied to any portfolio and size. We report that for a realistic portfolio comprising linear and non-linear derivatives, the expected reduction in the portfolio evaluations may exceed 6000 times, depending on the dimensionality and the required accuracy. We give illustrative examples and examine the method with realistic multi-currency portfolios consisting of interest rate swaps and swaptions.

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  • Lech A. Grzelak, 2021. "Sparse Grid Method for Highly Efficient Computation of Exposures for xVA," Papers 2104.14319, arXiv.org, revised May 2022.
    • Handle: RePEc:arx:papers:2104.14319
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    1. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, December.
    2. Mariano Zeron Medina Laris & Ignacio Ruiz, 2018. "Chebyshev Methods for Ultra-efficient Risk Calculations," Papers 1805.00898, arXiv.org.
    3. Judd, Kenneth L. & Maliar, Lilia & Maliar, Serguei & Valero, Rafael, 2014. "Smolyak method for solving dynamic economic models: Lagrange interpolation, anisotropic grid and adaptive domain," Journal of Economic Dynamics and Control, Elsevier, vol. 44(C), pages 92-123.
    4. Lech A. Grzelak & Cornelis W. Oosterlee, 2012. "On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 1-35, February.
    5. Shuaiqiang Liu & Lech A. Grzelak & Cornelis W. Oosterlee, 2022. "The Seven-League Scheme: Deep Learning for Large Time Step Monte Carlo Simulations of Stochastic Differential Equations," Risks, MDPI, vol. 10(3), pages 1-27, February.
    6. Lokman A. Abbas-Turki & Stéphane Crépey & Babacar Diallo, 2018. "Xva Principles, Nested Monte Carlo Strategies, And Gpu Optimizations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(06), pages 1-40, September.
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    Cited by:

    1. Griselda Deelstra & Lech A. Grzelak & Felix L. Wolf, 2022. "Accelerated Computations of Sensitivities for xVA," Papers 2211.17026, arXiv.org, revised Jan 2024.
    2. Lech A. Grzelak, 2022. "On Randomization of Affine Diffusion Processes with Application to Pricing of Options on VIX and S&P 500," Papers 2208.12518, arXiv.org.
    3. T. van der Zwaard & L. A. Grzelak & C. W. Oosterlee, 2022. "Efficient Wrong-Way Risk Modelling for Funding Valuation Adjustments," Papers 2209.12222, arXiv.org, revised Mar 2023.

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