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Multilevel Monte Carlo For Exponential L\'{e}vy Models

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  • Mike Giles
  • Yuan Xia

Abstract

We apply multilevel Monte Carlo for option pricing problems using exponential L\'{e}vy models with a uniform timestep discretisation to monitor the running maximum required for lookback and barrier options. The numerical results demonstrate the computational efficiency of this approach. We derive estimates of the convergence rate for the error introduced by the discrete monitoring of the running supremum of a broad class of L\'{e}vy processes. We use these to obtain upper bounds on the multilevel Monte Carlo variance convergence rate for the Variance Gamma, NIG and $\alpha$-stable processes used in the numerical experiments. We also show numerical results and analysis of a trapezoidal approximation for Asian options.

Suggested Citation

  • Mike Giles & Yuan Xia, 2014. "Multilevel Monte Carlo For Exponential L\'{e}vy Models," Papers 1403.5309, arXiv.org, revised May 2017.
    • Handle: RePEc:arx:papers:1403.5309
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    References listed on IDEAS

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    1. Peter Carr & Liuren Wu, 2003. "The Finite Moment Log Stable Process and Option Pricing," Journal of Finance, American Finance Association, vol. 58(2), pages 753-777, April.
    2. Jos'e E. Figueroa-L'opez & Peter Tankov, 2012. "Small-time asymptotics of stopped L\'evy bridges and simulation schemes with controlled bias," Papers 1203.2355, arXiv.org, revised Jul 2014.
    3. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    4. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    5. Dereich, Steffen & Heidenreich, Felix, 2011. "A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1565-1587, July.
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    Cited by:

    1. Jorge González Cázares & Aleksandar Mijatović, 2022. "Simulation of the drawdown and its duration in Lévy models via stick-breaking Gaussian approximation," Finance and Stochastics, Springer, vol. 26(4), pages 671-732, October.
    2. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c & Ger'onimo Uribe Bravo, 2018. "Geometrically Convergent Simulation of the Extrema of L\'{e}vy Processes," Papers 1810.11039, arXiv.org, revised Jun 2021.
    3. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.
    4. Nabil Kahale, 2018. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," Papers 1805.09427, arXiv.org, revised Sep 2018.
    5. Søren Asmussen & Jevgenijs Ivanovs, 2018. "Discretization error for a two-sided reflected Lévy process," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 199-212, June.
    6. Kathrin Glau & Daniel Kressner & Francesco Statti, 2019. "Low-rank tensor approximation for Chebyshev interpolation in parametric option pricing," Papers 1902.04367, arXiv.org.
    7. Fomichov, Vladimir & González Cázares, Jorge & Ivanovs, Jevgenijs, 2021. "Implementable coupling of Lévy process and Brownian motion," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 407-431.
    8. Alaya, Mohamed Ben & Hajji, Kaouther & Kebaier, Ahmed, 2016. "Importance sampling and statistical Romberg method for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1901-1931.
    9. Kahalé, Nabil, 2020. "General multilevel Monte Carlo methods for pricing discretely monitored Asian options," European Journal of Operational Research, Elsevier, vol. 287(2), pages 739-748.
    10. Jorge Gonz'alez C'azares & Aleksandar Mijatovi'c, 2020. "Simulation of the drawdown and its duration in L\'{e}vy models via stick-breaking Gaussian approximation," Papers 2011.06618, arXiv.org, revised Mar 2021.
    11. Jie Chen & Liaoyuan Fan & Lingfei Li & Gongqiu Zhang, 2022. "A multidimensional Hilbert transform approach for barrier option pricing and survival probability calculation," Review of Derivatives Research, Springer, vol. 25(2), pages 189-232, July.

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