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October  2021, 26(10): 5567-5579. doi: 10.3934/dcdsb.2020367

Strong convergence rates for markovian representations of fractional processes

Philipp Harms

Department of Mathematical Stochastics, University of Freiburg, Germany

Received  February 2019 Revised  August 2020 Published  October 2021 Early access  December 2020

Fund Project: The author gratefully acknowledges support in the form of a Junior Fellowship of the Freiburg Institute of Advances Studies

Many fractional processes can be represented as an integral over a family of Ornstein–Uhlenbeck processes. This representation naturally lends itself to numerical discretizations, which are shown in this paper to have strong convergence rates of arbitrarily high polynomial order. This explains the potential, but also some limitations of such representations as the basis of Monte Carlo schemes for fractional volatility models such as the rough Bergomi model.

Keywords: Fractional processes, Markovian representation, Monte Carlo simulation, numerical discretization, strong convergence rates, rough Bergomi model.
Mathematics Subject Classification: Primary: 60G22; Secondary: 60G15, 65C05, 91G60.
Citation: Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5567-5579. doi: 10.3934/dcdsb.2020367
References:
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E. Abi Jaber, Lifting the Heston model, Quantitative Finance, 19 (2019), 1995-2013.  doi: 10.1080/14697688.2019.1615113.

[2]

E. Abi Jaber and O. El Euch, Markovian structure of the Volterra Heston model, Statistics & Probability Letters, 149 (2019), 63-72.  doi: 10.1016/j.spl.2019.01.024.

[3]

E. Abi Jaber and O. El Euch, Multifactor approximation of rough volatility models, SIAM Journal on Financial Mathematics, 10 (2019), 309-349.  doi: 10.1137/18M1170236.

[4]

E. Abi Jaber, M. Larsson and S. Pulido, Affine Volterra processes, The Annals of Applied Probability, 29 (2019), 3155-3200.  doi: 10.1214/19-AAP1477.

[5]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition

[6]

E. Alòs, J. A. León and J. Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance and Stochastics, 11 (2007), 571-589.  doi: 10.1007/s00780-007-0049-1.

[7]

C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904.  doi: 10.1080/14697688.2015.1099717.

[8]

C. Bayer, P. K. Friz, P. Gassiat, J. Martin and B. Stemper, A regularity structure for rough volatility, Mathematical Finance, 30 (2020), 782-832.  doi: 10.1111/mafi.12233.

[9]

C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath and B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), 779-798.  doi: 10.1080/14697688.2018.1529420.

[10]

M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21 (2015), 360-373.  doi: 10.3150/13-BEJ570.

[11]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Decoupling the short-and long-term behavior of stochastic volatility, arXiv: 1610.00332, 2016. doi: 10.2139/ssrn.2846756.

[12]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics, 21 (2017), 931-965.  doi: 10.1007/s00780-017-0335-5.

[13]

H. Brass and K. Petras, Quadrature Theory, vol. 178 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/178.

[14]

P. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electronic Communications in Probability, 3 (1998), 95-107.  doi: 10.1214/ECP.v3-998.

[15]

P. Carmona, L. Coutin and G. Montseny, Approximation of some Gaussian processes, Statistical Inference for Stochastic Processes, 3 (2000), 161-171.  doi: 10.1023/A:1009999518898.

[16]

C. Cuchiero and J. Teichmann, Generalized Feller processes and Markovian lifts of stochastic Volterra processes: The affine case, Journal of Evolution Equations, (2020), 1–48. doi: 10.1007/s00028-020-00557-2.

[17]

T. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, University of Twente, 2004.

[18]

C. R. Dietrich and G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM Journal on Scientific Computing, 18 (1997), 1088-1107.  doi: 10.1137/S1064827592240555.

[19]

M. Forde and H. Zhang, Asymptotics for rough stochastic volatility models, SIAM Journal on Financial Mathematics, 8 (2017), 114-145.  doi: 10.1137/15M1009330.

[20]

M. Fukasawa, Asymptotic analysis for stochastic volatility: Martingale expansion, Finance and Stochastics, 15 (2011), 635-654.  doi: 10.1007/s00780-010-0136-6.

[21]

M. Gaß, K. Glau, M. Mahlstedt and M. Mair, Chebyshev interpolation for parametric option pricing, Finance and Stochastics, 22 (2018), 701-731.  doi: 10.1007/s00780-018-0361-y.

[22]

M. Gaß, K. Glau and M. Mair, Magic points in finance: Empirical integration for parametric option pricing, SIAM Journal on Financial Mathematics, 8 (2017), 766-803.  doi: 10.1137/16M1101301.

[23]

P. Gassiat, On the martingale property in the rough Bergomi model, Electronic Communications in Probability, 24 (2019), Paper No. 33, 9 pp. doi: 10.1214/19-ECP239.

[24]

J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949.  doi: 10.1080/14697688.2017.1393551.

[25]

S. E. Graversen and G. Peskir, Maximal inequalities for the Ornstein–Uhlenbeck process, Proceedings of the American Mathematical Society, 128 (2000), 3035-3041.  doi: 10.1090/S0002-9939-00-05345-4.

[26]

P. Harms and D. Stefanovits, Affine representations of fractional processes with applications in mathematical finance, Stochastic Processes and their Applications, 129 (2019), 1185-1228.  doi: 10.1016/j.spa.2018.04.010.

[27]

B. Horvath, A. Jacquier and A. Muguruza, Functional central limit theorems for rough volatility, arXiv: 1711.03078, 2017. doi: 10.2139/ssrn.3078743.

[28]

J. R. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20 (1984), 1898-1908.  doi: 10.1029/WR020i012p01898.

[29]

T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces, vol. 67, Springer, Cham, 2017. doi: 10.1007/978-3-319-69808-3.

[30]

R. McCrickerd and M. S. Pakkanen, Turbocharging Monte Carlo pricing for the rough Bergomi model, Quantitative Finance, 18 (2018), 1877–1886. doi: 10.1080/14697688.2018.1459812.

[31]

A. A. Muravlev, Representation of a fractional Brownian motion in terms of an infinite-dimensional Ornstein–Uhlenbeck process, Russian Mathematical Surveys, 66 (2011), 439-441.  doi: 10.1070/RM2011v066n02ABEH004746.

[32]

L. Mytnik and T. S. Salisbury, Uniqueness for Volterra-type stochastic integral equations, arXiv: 1502.05513, 2015.

[33]

N. N. Vakhania, V. I. Tarieladze and S. A.Chobanyan, Probability distributions on Banach spaces, vol. 14, Springer Science & Business Media, 1987. doi: 10.1007/978-94-009-3873-1.

[34]

M. Veraar, The stochastic Fubini theorem revisited, Stochastics An International Journal of Probability and Stochastic Processes, 84 (2012), 543-551.  doi: 10.1080/17442508.2011.618883.

show all references

References:
[1]

E. Abi Jaber, Lifting the Heston model, Quantitative Finance, 19 (2019), 1995-2013.  doi: 10.1080/14697688.2019.1615113.

[2]

E. Abi Jaber and O. El Euch, Markovian structure of the Volterra Heston model, Statistics & Probability Letters, 149 (2019), 63-72.  doi: 10.1016/j.spl.2019.01.024.

[3]

E. Abi Jaber and O. El Euch, Multifactor approximation of rough volatility models, SIAM Journal on Financial Mathematics, 10 (2019), 309-349.  doi: 10.1137/18M1170236.

[4]

E. Abi Jaber, M. Larsson and S. Pulido, Affine Volterra processes, The Annals of Applied Probability, 29 (2019), 3155-3200.  doi: 10.1214/19-AAP1477.

[5]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide, 3rd edition

[6]

E. Alòs, J. A. León and J. Vives, On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance and Stochastics, 11 (2007), 571-589.  doi: 10.1007/s00780-007-0049-1.

[7]

C. Bayer, P. Friz and J. Gatheral, Pricing under rough volatility, Quantitative Finance, 16 (2016), 887-904.  doi: 10.1080/14697688.2015.1099717.

[8]

C. Bayer, P. K. Friz, P. Gassiat, J. Martin and B. Stemper, A regularity structure for rough volatility, Mathematical Finance, 30 (2020), 782-832.  doi: 10.1111/mafi.12233.

[9]

C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath and B. Stemper, Short-time near-the-money skew in rough fractional volatility models, Quantitative Finance, 19 (2019), 779-798.  doi: 10.1080/14697688.2018.1529420.

[10]

M. Beiglböck and P. Siorpaes, Pathwise versions of the Burkholder–Davis–Gundy inequality, Bernoulli, 21 (2015), 360-373.  doi: 10.3150/13-BEJ570.

[11]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Decoupling the short-and long-term behavior of stochastic volatility, arXiv: 1610.00332, 2016. doi: 10.2139/ssrn.2846756.

[12]

M. Bennedsen, A. Lunde and M. S. Pakkanen, Hybrid scheme for Brownian semistationary processes, Finance and Stochastics, 21 (2017), 931-965.  doi: 10.1007/s00780-017-0335-5.

[13]

H. Brass and K. Petras, Quadrature Theory, vol. 178 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011. doi: 10.1090/surv/178.

[14]

P. Carmona and L. Coutin, Fractional Brownian motion and the Markov property, Electronic Communications in Probability, 3 (1998), 95-107.  doi: 10.1214/ECP.v3-998.

[15]

P. Carmona, L. Coutin and G. Montseny, Approximation of some Gaussian processes, Statistical Inference for Stochastic Processes, 3 (2000), 161-171.  doi: 10.1023/A:1009999518898.

[16]

C. Cuchiero and J. Teichmann, Generalized Feller processes and Markovian lifts of stochastic Volterra processes: The affine case, Journal of Evolution Equations, (2020), 1–48. doi: 10.1007/s00028-020-00557-2.

[17]

T. Dieker, Simulation of Fractional Brownian Motion, Master's thesis, University of Twente, 2004.

[18]

C. R. Dietrich and G. N. Newsam, Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM Journal on Scientific Computing, 18 (1997), 1088-1107.  doi: 10.1137/S1064827592240555.

[19]

M. Forde and H. Zhang, Asymptotics for rough stochastic volatility models, SIAM Journal on Financial Mathematics, 8 (2017), 114-145.  doi: 10.1137/15M1009330.

[20]

M. Fukasawa, Asymptotic analysis for stochastic volatility: Martingale expansion, Finance and Stochastics, 15 (2011), 635-654.  doi: 10.1007/s00780-010-0136-6.

[21]

M. Gaß, K. Glau, M. Mahlstedt and M. Mair, Chebyshev interpolation for parametric option pricing, Finance and Stochastics, 22 (2018), 701-731.  doi: 10.1007/s00780-018-0361-y.

[22]

M. Gaß, K. Glau and M. Mair, Magic points in finance: Empirical integration for parametric option pricing, SIAM Journal on Financial Mathematics, 8 (2017), 766-803.  doi: 10.1137/16M1101301.

[23]

P. Gassiat, On the martingale property in the rough Bergomi model, Electronic Communications in Probability, 24 (2019), Paper No. 33, 9 pp. doi: 10.1214/19-ECP239.

[24]

J. Gatheral, T. Jaisson and M. Rosenbaum, Volatility is rough, Quantitative Finance, 18 (2018), 933-949.  doi: 10.1080/14697688.2017.1393551.

[25]

S. E. Graversen and G. Peskir, Maximal inequalities for the Ornstein–Uhlenbeck process, Proceedings of the American Mathematical Society, 128 (2000), 3035-3041.  doi: 10.1090/S0002-9939-00-05345-4.

[26]

P. Harms and D. Stefanovits, Affine representations of fractional processes with applications in mathematical finance, Stochastic Processes and their Applications, 129 (2019), 1185-1228.  doi: 10.1016/j.spa.2018.04.010.

[27]

B. Horvath, A. Jacquier and A. Muguruza, Functional central limit theorems for rough volatility, arXiv: 1711.03078, 2017. doi: 10.2139/ssrn.3078743.

[28]

J. R. Hosking, Modeling persistence in hydrological time series using fractional differencing, Water Resources Research, 20 (1984), 1898-1908.  doi: 10.1029/WR020i012p01898.

[29]

T. Hytönen, J. van Neerven, M. Veraar and L. Weis, Analysis in Banach Spaces, vol. 67, Springer, Cham, 2017. doi: 10.1007/978-3-319-69808-3.

[30]

R. McCrickerd and M. S. Pakkanen, Turbocharging Monte Carlo pricing for the rough Bergomi model, Quantitative Finance, 18 (2018), 1877–1886. doi: 10.1080/14697688.2018.1459812.

[31]

A. A. Muravlev, Representation of a fractional Brownian motion in terms of an infinite-dimensional Ornstein–Uhlenbeck process, Russian Mathematical Surveys, 66 (2011), 439-441.  doi: 10.1070/RM2011v066n02ABEH004746.

[32]

L. Mytnik and T. S. Salisbury, Uniqueness for Volterra-type stochastic integral equations, arXiv: 1502.05513, 2015.

[33]

N. N. Vakhania, V. I. Tarieladze and S. A.Chobanyan, Probability distributions on Banach spaces, vol. 14, Springer Science & Business Media, 1987. doi: 10.1007/978-94-009-3873-1.

[34]

M. Veraar, The stochastic Fubini theorem revisited, Stochastics An International Journal of Probability and Stochastic Processes, 84 (2012), 543-551.  doi: 10.1080/17442508.2011.618883.

Figure 1.  Volterra Brownian motion of Hurst index $ H\in(0,1/2) $ can be represented as an integral $ W^H_t = \int_0^\infty Y_t(x) x^{-1/2-H}dx $ over a Gaussian random field $ Y_t(x) $. The smoothness of the random field in the spatial dimension $ x $ allows one to approximate this integral efficiently using high order quadrature rules
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Figure 2.  Dependence of the approximations on the number $ n $ of quadrature intervals and the Hurst index $ H $. Left: varying the number $ n\in\{2,5,10,20,40\}$ = of quadrature intervals with fixed parameters $ H = 0.1 $, $ m = 5 $. Right: varying the Hurst index $ H\in\{0.1,0.2,0.3,0.4\}$ = with fixed parameters $ n = 40 $, $ m = 5 $
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Figure 3.  The upper bound $ 2Hm/3 $ on the convergence rate established in Remark 6.2 for $ m $-point interpolatory quadrature closely matches the numerically observed one (here: at $ t = 1 $, computed analytically from the covariance functions of the Gaussian processes $ W^H $ and $ W^{H,n} $). Left: relative error $ e = \|W^H_1-W^{H,n}_1\|_{L^2(\Omega)}/\|W^H_1\|_{L^2(\Omega)} $ for $ m\in\{2,3,\dots,20\}$ = with $ H = 0.1 $. Right: slopes of the lines in the left plot (dots) and predicted convergence rate (line)
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Table 1.  Complexity of several numerical methods for sampling a fractional process $ (W^H_{i/k})_{i\in\{1,\dots,k\}} $ with Hurst index $ H\in(0,1/2) $ at $ k $ equidistant time points
Method Structure Error Complexity
Cholesky Static 0 $ k^3 $
Hosking, Dieker [28,17] Recursive 0 $ k^2 $
Dietrich, Newsam [18] Static 0 $ k\log k $
Bennedsen, Lunde, Pakkanen [12] Recursive $ k^{-H} $ $ k \log k $
Carmona, Coutin, Montseny [15] Recursive $ \epsilon $ $ k\epsilon^{-3/(4H)} $
This paper Recursive $ \epsilon $ $ k\epsilon^{-1/r} $ for $ r\in(0,\infty) $
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Method Structure Error Complexity
Cholesky Static 0 $ k^3 $
Hosking, Dieker [28,17] Recursive 0 $ k^2 $
Dietrich, Newsam [18] Static 0 $ k\log k $
Bennedsen, Lunde, Pakkanen [12] Recursive $ k^{-H} $ $ k \log k $
Carmona, Coutin, Montseny [15] Recursive $ \epsilon $ $ k\epsilon^{-3/(4H)} $
This paper Recursive $ \epsilon $ $ k\epsilon^{-1/r} $ for $ r\in(0,\infty) $
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