Approximation of Stochastic Volterra Equations with kernels of completely monotone type

Alfonsi, Aurélien; Kebaier, Ahmed

doi:10.1090/mcom/3911

Approximation of Stochastic Volterra Equations with kernels of completely monotone type
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by Aurélien Alfonsi and Ahmed Kebaier

Math. Comp. 93 (2024), 643-677

DOI: https://doi.org/10.1090/mcom/3911

Published electronically: November 2, 2023

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Abstract:

In this work, we develop a multifactor approximation for $d$-dimensional Stochastic Volterra Equations (SVE) with Lipschitz coefficients and kernels of completely monotone type that may be singular. First, we prove an $L^2$-estimation between two SVEs with different kernels, which provides a quantification of the error between the SVE and any multifactor Stochastic Differential Equation (SDE) approximation. For the particular rough kernel case with Hurst parameter lying in $(0,1/2)$, we propose various approximating multifactor kernels, state their rates of convergence and illustrate their efficiency for the rough Bergomi model. Second, we study a Euler discretization of the multifactor SDE and establish a convergence result towards the SVE that is uniform with respect to the approximating multifactor kernels. These obtained results lead us to build a new multifactor Euler scheme that reduces significantly the computational cost in an asymptotic way compared to the Euler scheme for SVEs. Finally, we show that our multifactor Euler scheme outperforms the Euler scheme for SVEs for option pricing in the rough Heston model.

References

Eduardo Abi Jaber, Weak existence and uniqueness for affine stochastic Volterra equations with $L^1$-kernels, Bernoulli 27 (2021), no. 3, 1583–1615. MR 4260503, DOI 10.3150/20-bej1284
Eduardo Abi Jaber and Omar El Euch, Multifactor approximation of rough volatility models, SIAM J. Financial Math. 10 (2019), no. 2, 309–349. MR 3934104, DOI 10.1137/18M1170236
Aurélien Alfonsi, High order discretization schemes for the CIR process: application to affine term structure and Heston models, Math. Comp. 79 (2010), no. 269, 209–237. MR 2552224, DOI 10.1090/S0025-5718-09-02252-2
Christian Bayer, Peter Friz, and Jim Gatheral, Pricing under rough volatility, Quant. Finance 16 (2016), no. 6, 887–904. MR 3494612, DOI 10.1080/14697688.2015.1099717
C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath, and B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quant. Finance 19 (2019), no. 5, 779–798. MR 3939657, DOI 10.1080/14697688.2018.1529420
Denis Belomestny, Stefan Häfner, Tigran Nagapetyan, and Mikhail Urusov, Variance reduction for discretised diffusions via regression, J. Math. Anal. Appl. 458 (2018), no. 1, 393–418. MR 3711910, DOI 10.1016/j.jmaa.2017.09.002
Mohamed Ben Alaya and Ahmed Kebaier, Central limit theorem for the multilevel Monte Carlo Euler method, Ann. Appl. Probab. 25 (2015), no. 1, 211–234. MR 3297771, DOI 10.1214/13-AAP993
Mikkel Bennedsen, Asger Lunde, and Mikko S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance Stoch. 21 (2017), no. 4, 931–965. MR 3723378, DOI 10.1007/s00780-017-0335-5
Marc A. Berger and Victor J. Mizel, Volterra equations with Itô integrals. I, J. Integral Equations 2 (1980), no. 3, 187–245. MR 581430
Marc A. Berger and Victor J. Mizel, Volterra equations with Itô integrals. II, J. Integral Equations 2 (1980), no. 4, 319–337. MR 596459
Philippe Carmona and Laure Coutin, Fractional Brownian motion and the Markov property, Electron. Comm. Probab. 3 (1998), 95–107. MR 1658690, DOI 10.1214/ECP.v3-998
Philippe Carmona, Laure Coutin, and G. Montseny, Approximation of some Gaussian processes, Stat. Inference Stoch. Process. 3 (2000), no. 1-2, 161–171. 19th “Rencontres Franco-Belges de Statisticiens” (Marseille, 1998). MR 1819293, DOI 10.1023/A:1009999518898
W. George Cochran, Jung-Soon Lee, and Jürgen Potthoff, Stochastic Volterra equations with singular kernels, Stochastic Process. Appl. 56 (1995), no. 2, 337–349. MR 1325227, DOI 10.1016/0304-4149(94)00072-2
L. Coutin and L. Decreusefond, Stochastic Volterra equations with singular kernels, Stochastic analysis and mathematical physics, Progr. Probab., vol. 50, Birkhäuser Boston, Boston, MA, 2001, pp. 39–50. MR 1886561
L. Decreusefond, Regularity properties of some stochastic Volterra integrals with singular kernel, Potential Anal. 16 (2002), no. 2, 139–149. MR 1881594, DOI 10.1023/A:1012628013041
Omar El Euch and Mathieu Rosenbaum, The characteristic function of rough Heston models, Math. Finance 29 (2019), no. 1, 3–38. MR 3905737, DOI 10.1111/mafi.12173
Peter K. Friz, Paul Gassiat, and Paolo Pigato, Short-dated smile under rough volatility: asymptotics and numerics, Quant. Finance 22 (2022), no. 3, 463–480. MR 4401811, DOI 10.1080/14697688.2021.1999486
Masaaki Fukasawa, Short-time at-the-money skew and rough fractional volatility, Quant. Finance 17 (2017), no. 2, 189–198. MR 3592946, DOI 10.1080/14697688.2016.1197410
Masaaki Fukasawa, Volatility has to be rough, Quant. Finance 21 (2021), no. 1, 1–8. MR 4188876, DOI 10.1080/14697688.2020.1825781
Jim Gatheral, Thibault Jaisson, and Mathieu Rosenbaum, Volatility is rough, Quant. Finance 18 (2018), no. 6, 933–949. MR 3805308, DOI 10.1080/14697688.2017.1393551
Michael B. Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56 (2008), no. 3, 607–617. MR 2436856, DOI 10.1287/opre.1070.0496
G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra integral and functional equations, Encyclopedia of Mathematics and its Applications, vol. 34, Cambridge University Press, Cambridge, 1990. MR 1050319, DOI 10.1017/CBO9780511662805
Philipp Harms, Strong convergence rates for Markovian representations of fractional processes, Discrete Contin. Dyn. Syst. Ser. B 26 (2021), no. 10, 5567–5579. MR 4271188, DOI 10.3934/dcdsb.2020367
Philipp Harms and David Stefanovits, Affine representations of fractional processes with applications in mathematical finance, Stochastic Process. Appl. 129 (2019), no. 4, 1185–1228. MR 3926553, DOI 10.1016/j.spa.2018.04.010
Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, Dover Publications, Inc., New York, 1994. Corrected reprint of the 1966 original [Wiley, New York; MR0201039 (34 #924)]. MR 1280462
Benjamin Jourdain and Jérôme Lelong, Robust adaptive importance sampling for normal random vectors, Ann. Appl. Probab. 19 (2009), no. 5, 1687–1718. MR 2569805, DOI 10.1214/09-AAP595
Shigeo Kusuoka, Approximation of expectation of diffusion processes based on Lie algebra and Malliavin calculus, Advances in mathematical economics. Vol. 6, Adv. Math. Econ., vol. 6, Springer, Tokyo, 2004, pp. 69–83. MR 2079333, DOI 10.1007/978-4-431-68450-3_{4}
Vincent Lemaire and Gilles Pagès, Unconstrained recursive importance sampling, Ann. Appl. Probab. 20 (2010), no. 3, 1029–1067. MR 2680557, DOI 10.1214/09-AAP650
Vincent Lemaire and Gilles Pagès, Multilevel Richardson-Romberg extrapolation, Bernoulli 23 (2017), no. 4A, 2643–2692. MR 3648041, DOI 10.3150/16-BEJ822
Nigel J. Newton, Variance reduction for simulated diffusions, SIAM J. Appl. Math. 54 (1994), no. 6, 1780–1805. MR 1301282, DOI 10.1137/S0036139992236220
Syoiti Ninomiya and Nicolas Victoir, Weak approximation of stochastic differential equations and application to derivative pricing, Appl. Math. Finance 15 (2008), no. 1-2, 107–121. MR 2409419, DOI 10.1080/13504860701413958
Étienne Pardoux and Philip Protter, Stochastic Volterra equations with anticipating coefficients, Ann. Probab. 18 (1990), no. 4, 1635–1655. MR 1071815
Philip Protter, Volterra equations driven by semimartingales, Ann. Probab. 13 (1985), no. 2, 519–530. MR 781420
Alexandre Richard, Xiaolu Tan, and Fan Yang, Discrete-time simulation of stochastic Volterra equations, Stochastic Process. Appl. 141 (2021), 109–138. MR 4293770, DOI 10.1016/j.spa.2021.07.003
Alexandre Richard, Xiaolu Tan, and Fan Yang, On the discrete-time simulation of the rough Heston model, SIAM J. Financial Math. 14 (2023), no. 1, 223–249. MR 4544542, DOI 10.1137/21M1443807
S. E. Rømer, Hybrid multifactor scheme for stochastic Volterra equations with completely monotone kernels, preprint, 2022, DOI 10.2139/ssrn.3706253.
Yuji Shinozaki, Construction of a third-order K-scheme and its application to financial models, SIAM J. Financial Math. 8 (2017), no. 1, 901–932. MR 3725272, DOI 10.1137/16M1067986
Denis Talay and Luciano Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990), no. 4, 483–509 (1991). MR 1091544, DOI 10.1080/07362999008809220
Zhidong Wang, Existence and uniqueness of solutions to stochastic Volterra equations with singular kernels and non-Lipschitz coefficients, Statist. Probab. Lett. 78 (2008), no. 9, 1062–1071. MR 2422961, DOI 10.1016/j.spl.2007.10.007
David Vernon Widder, The Laplace Transform, Princeton Mathematical Series, vol. 6, Princeton University Press, Princeton, NJ, 1941. MR 5923
Xicheng Zhang, Euler schemes and large deviations for stochastic Volterra equations with singular kernels, J. Differential Equations 244 (2008), no. 9, 2226–2250. MR 2413840, DOI 10.1016/j.jde.2008.02.019
Xicheng Zhang, Stochastic Volterra equations in Banach spaces and stochastic partial differential equation, J. Funct. Anal. 258 (2010), no. 4, 1361–1425. MR 2565842, DOI 10.1016/j.jfa.2009.11.006