Statistical inference for rough volatility: minimax theory. (English) Zbl 07928777
Summary: In recent years, rough volatility models have gained considerable attention in quantitative finance. In this paradigm, the stochastic volatility of the price of an asset has quantitative properties similar to that of a fractional Brownian motion with small Hurst index \(H < 1 / 2\). In this work, we provide the first rigorous statistical analysis of the problem of estimating \(H\) from historical observations of the underlying asset. We establish minimax lower bounds and design optimal procedures based on adaptive estimation of quadratic functionals based on wavelets. We prove in particular that the optimal rate of convergence for estimating \(H\) based on price observations at \(n\) time points is of order \(n^{- 1 / ( 4 H + 2 )}\) as \(n\) grows to infinity, extending results that were known only for \(H > 1 / 2\). Our study positively answers the question whether \(H\) can be inferred, although it is the regularity of a latent process (the volatility); in rough models, when \(H\) is close to 0, we even obtain an accuracy comparable to usual \(\sqrt{n} \)-consistent regular statistical models.
MSC:
| 60G22 | Fractional processes, including fractional Brownian motion |
| 62C20 | Minimax procedures in statistical decision theory |
| 62F12 | Asymptotic properties of parametric estimators |
| 62M09 | Non-Markovian processes: estimation |
| 62P20 | Applications of statistics to economics |
References:
| [1] | ABI JABER, E. (2022). The characteristic function of Gaussian stochastic volatility models: An analytic expression. Finance Stoch. 26 733-769. Digital Object Identifier: 10.1007/s00780-022-00489-4 Google Scholar: Lookup Link MathSciNet: MR4487683 · Zbl 1498.91443 · doi:10.1007/s00780-022-00489-4 |
| [2] | Abi Jaber, E. and El Euch, O. (2019). Multifactor approximation of rough volatility models. SIAM J. Financial Math. 10 309-349. Digital Object Identifier: 10.1137/18M1170236 Google Scholar: Lookup Link MathSciNet: MR3934104 · Zbl 1422.91765 · doi:10.1137/18M1170236 |
| [3] | Abi Jaber, E., Larsson, M. and Pulido, S. (2019). Affine Volterra processes. Ann. Appl. Probab. 29 3155-3200. Digital Object Identifier: 10.1214/19-AAP1477 Google Scholar: Lookup Link MathSciNet: MR4019885 · Zbl 1441.60052 · doi:10.1214/19-AAP1477 |
| [4] | AÏT-SAHALIA, Y. and JACOD, J. (2014). High-Frequency Financial Econometrics. Princeton Univ.Press, Princeton. · Zbl 1298.91018 |
| [5] | Alòs, E., León, J. A. and Vives, J. (2007). On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch. 11 571-589. Digital Object Identifier: 10.1007/s00780-007-0049-1 Google Scholar: Lookup Link MathSciNet: MR2335834 · Zbl 1145.91020 · doi:10.1007/s00780-007-0049-1 |
| [6] | ALÒS, E., ROLLOOS, F. and SHIRAYA, K. (2022). Forward start volatility swaps in rough volatility models. arXiv preprint, available at arXiv:2207.10370. |
| [7] | BARNDORFF-NIELSEN, O. E., CORCUERA, J. M. and PODOLSKIJ, M. (2013). Limit theorems for functionals of higher order differences of Brownian semi-stationary processes. In Prokhorov and Contemporary Probability Theory. Springer Proc. Math. Stat. 33 69-96. Springer, Heidelberg. Digital Object Identifier: 10.1007/978-3-642-33549-5_4 Google Scholar: Lookup Link MathSciNet: MR3070467 · Zbl 1273.60021 · doi:10.1007/978-3-642-33549-5_4 |
| [8] | Bayer, C., Friz, P. and Gatheral, J. (2016). Pricing under rough volatility. Quant. Finance 16 887-904. Digital Object Identifier: 10.1080/14697688.2015.1099717 Google Scholar: Lookup Link MathSciNet: MR3494612 · Zbl 1465.91108 · doi:10.1080/14697688.2015.1099717 |
| [9] | Bayer, C., Friz, P. K., Gassiat, P., Martin, J. and Stemper, B. (2020). A regularity structure for rough volatility. Math. Finance 30 782-832. Digital Object Identifier: 10.1111/mafi.12233 Google Scholar: Lookup Link MathSciNet: MR4116451 · Zbl 1508.91548 · doi:10.1111/mafi.12233 |
| [10] | BENNEDSEN, M., LUNDE, A. and PAKKANEN, M. S. (2017). Hybrid scheme for Brownian semistationary processes. Finance Stoch. 21 931-965. Digital Object Identifier: 10.1007/s00780-017-0335-5 Google Scholar: Lookup Link MathSciNet: MR3723378 · Zbl 1385.65010 · doi:10.1007/s00780-017-0335-5 |
| [11] | BENNEDSEN, M., LUNDE, A. and PAKKANEN, M. S. (2022). Decoupling the short-and long-term behavior of stochastic volatility. J. Financ. Econom. 25 961-1006. Digital Object Identifier: 10.1093/jjfinec/nbaa049 Google Scholar: Lookup Link · doi:10.1093/jjfinec/nbaa049 |
| [12] | BIBINGER, M., JIRAK, M. and VETTER, M. (2017). Nonparametric change-point analysis of volatility. Ann. Statist. 45 1542-1578. Digital Object Identifier: 10.1214/16-AOS1499 Google Scholar: Lookup Link MathSciNet: MR3670188 · Zbl 1421.62163 · doi:10.1214/16-AOS1499 |
| [13] | BOLKO, A. E., CHRISTENSEN, K., PAKKANEN, M. S. and VELIYEV, B. (2023). A GMM approach to estimate the roughness of stochastic volatility. J. Econometrics 235 745-778. Digital Object Identifier: 10.1016/j.jeconom.2022.06.009 Google Scholar: Lookup Link MathSciNet: MR4602890 · Zbl 07704472 · doi:10.1016/j.jeconom.2022.06.009 |
| [14] | BROUSTE, A. and FUKASAWA, M. (2018). Local asymptotic normality property for fractional Gaussian noise under high-frequency observations. Ann. Statist. 46 2045-2061. Digital Object Identifier: 10.1214/17-AOS1611 Google Scholar: Lookup Link MathSciNet: MR3845010 · Zbl 1411.62045 · doi:10.1214/17-AOS1611 |
| [15] | CALLEGARO, G., GRASSELLI, M. and PAGÈS, G. (2021). Fast hybrid schemes for fractional Riccati equations (rough is not so tough). Math. Oper. Res. 46 221-254. Digital Object Identifier: 10.1287/moor.2020.1054 Google Scholar: Lookup Link MathSciNet: MR4224426 · Zbl 1481.60079 · doi:10.1287/moor.2020.1054 |
| [16] | CHONG, C. H., HOFFMANN, M., LIU, Y., ROSENBAUM, M. and SZYMANSKI, G. (2024). Statistical inference for rough volatility: Central limit theorems. Ann. Appl. Probab. 34 2600-2649. Digital Object Identifier: 10.1214/23-aap2002 Google Scholar: Lookup Link MathSciNet: MR4757488 · doi:10.1214/23-aap2002 |
| [17] | CHONG, C. H. and TODOROV, V. (2022). Short-time expansion of characteristic functions in a rough volatility setting with applications. Bernoulli. To appear. |
| [18] | CHONG, C. H., HOFFMANN, M., LIU, Y., ROSENBAUM, M. and SZYMANSKY, G. (2024). Supplement to “Statistical inference for rough volatility: Minimax theory.” https://doi.org/10.1214/23-AOS2343SUPP |
| [19] | CIESIELSKI, Z., KERKYACHARIAN, G. and ROYNETTE, B. (1993). Quelques espaces fonctionnels associés à des processus gaussiens. Studia Math. 107 171-204. Digital Object Identifier: 10.4064/sm-107-2-171-204 Google Scholar: Lookup Link MathSciNet: MR1244574 · Zbl 0809.60004 · doi:10.4064/sm-107-2-171-204 |
| [20] | COEURJOLLY, J.-F. (2001). Estimating the parameters of a fractional Brownian motion by discrete variations of its sample paths. Stat. Inference Stoch. Process. 4 199-227. Digital Object Identifier: 10.1023/A:1017507306245 Google Scholar: Lookup Link MathSciNet: MR1856174 · Zbl 0984.62058 · doi:10.1023/A:1017507306245 |
| [21] | Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic volatility models. Math. Finance 8 291-323. Digital Object Identifier: 10.1111/1467-9965.00057 Google Scholar: Lookup Link MathSciNet: MR1645101 · Zbl 1020.91021 · doi:10.1111/1467-9965.00057 |
| [22] | Cuchiero, C. and Teichmann, J. (2019). Markovian lifts of positive semidefinite affine Volterra-type processes. Decis. Econ. Finance 42 407-448. Digital Object Identifier: 10.1007/s10203-019-00268-5 Google Scholar: Lookup Link MathSciNet: MR4031333 · Zbl 1432.91110 · doi:10.1007/s10203-019-00268-5 |
| [23] | El Euch, O., Fukasawa, M., Gatheral, J. and Rosenbaum, M. (2019). Short-term at-the-money asymptotics under stochastic volatility models. SIAM J. Financial Math. 10 491-511. Digital Object Identifier: 10.1137/18M1167565 Google Scholar: Lookup Link MathSciNet: MR3945237 · Zbl 1417.91495 · doi:10.1137/18M1167565 |
| [24] | EL EUCH, O., FUKASAWA, M. and ROSENBAUM, M. (2018). The microstructural foundations of leverage effect and rough volatility. Finance Stoch. 22 241-280. Digital Object Identifier: 10.1007/s00780-018-0360-z Google Scholar: Lookup Link MathSciNet: MR3778355 zbMATH: 1410.91491 · Zbl 1410.91491 · doi:10.1007/s00780-018-0360-z |
| [25] | El Euch, O. and Rosenbaum, M. (2018). Perfect hedging in rough Heston models. Ann. Appl. Probab. 28 3813-3856. Digital Object Identifier: 10.1214/18-AAP1408 Google Scholar: Lookup Link MathSciNet: MR3861827 zbMATH: 1418.91467 · Zbl 1418.91467 · doi:10.1214/18-AAP1408 |
| [26] | El Euch, O. and Rosenbaum, M. (2019). The characteristic function of rough Heston models. Math. Finance 29 3-38. Digital Object Identifier: 10.1111/mafi.12173 Google Scholar: Lookup Link MathSciNet: MR3905737 · Zbl 1411.91553 · doi:10.1111/mafi.12173 |
| [27] | FORDE, M., FUKASAWA, M., GERHOLD, S. and SMITH, B. (2020). The Rough Bergomi model as \(\mathit{H} \to 0\) - skew flattening/blow up and non-Gaussian rough volatility. Preprint. |
| [28] | Forde, M. and Zhang, H. (2017). Asymptotics for rough stochastic volatility models. SIAM J. Financial Math. 8 114-145. Digital Object Identifier: 10.1137/15M1009330 Google Scholar: Lookup Link MathSciNet: MR3608743 · Zbl 1422.91693 · doi:10.1137/15M1009330 |
| [29] | FRIZ, P. K., GASSIAT, P. and PIGATO, P. (2022). Short-dated smile under rough volatility: Asymptotics and numerics. Quant. Finance 22 463-480. Digital Object Identifier: 10.1080/14697688.2021.1999486 Google Scholar: Lookup Link MathSciNet: MR4401811 · Zbl 1487.91137 · doi:10.1080/14697688.2021.1999486 |
| [30] | FRIZ, P. K., GATHERAL, J. and RADOIČIĆ, R. (2022). Forests, cumulants, martingales. Ann. Probab. 50 1418-1445. Digital Object Identifier: 10.1214/21-aop1560 Google Scholar: Lookup Link MathSciNet: MR4420423 · Zbl 1534.60053 · doi:10.1214/21-aop1560 |
| [31] | FUKASAWA, M. (2021). Volatility has to be rough. Quant. Finance 21 1-8. Digital Object Identifier: 10.1080/14697688.2020.1825781 Google Scholar: Lookup Link MathSciNet: MR4188876 · Zbl 1484.91474 · doi:10.1080/14697688.2020.1825781 |
| [32] | FUKASAWA, M., HORVATH, B. and TANKOV, P. (2024). Hedging under rough volatility. In Rough Volatility. Financ. Math. 115-125. Society for Industrial and Applied Mathematics, Philadelphia, PA. Digital Object Identifier: 10.1137/1.9781611977783.ch6 Google Scholar: Lookup Link · doi:10.1137/1.9781611977783.ch6 |
| [33] | FUKASAWA, M. and TAKABATAKE, T. (2019). Asymptotically efficient estimators for self-similar stationary Gaussian noises under high frequency observations. Bernoulli 25 1870-1900. Digital Object Identifier: 10.3150/18-BEJ1039 Google Scholar: Lookup Link MathSciNet: MR3961234 · Zbl 1466.62382 · doi:10.3150/18-BEJ1039 |
| [34] | GASSIAT, P. (2019). On the martingale property in the rough Bergomi model. Electron. Commun. Probab. 24 Paper No. 33, 9. Digital Object Identifier: 10.1214/19-ECP239 Google Scholar: Lookup Link MathSciNet: MR3962483 · Zbl 1488.60109 · doi:10.1214/19-ECP239 |
| [35] | GASSIAT, P. (2023). Weak error rates of numerical schemes for rough volatility. SIAM J. Financial Math. 14 475-496. Digital Object Identifier: 10.1137/22M1485760 Google Scholar: Lookup Link MathSciNet: MR4589567 · Zbl 1517.91280 · doi:10.1137/22M1485760 |
| [36] | Gatheral, J., Jaisson, T. and Rosenbaum, M. (2018). Volatility is rough. Quant. Finance 18 933-949. Digital Object Identifier: 10.1080/14697688.2017.1393551 Google Scholar: Lookup Link MathSciNet: MR3805308 · Zbl 1400.91590 · doi:10.1080/14697688.2017.1393551 |
| [37] | GATHERAL, J., JUSSELIN, P. and ROSENBAUM, M. (2020). The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem. Risk. |
| [38] | Gatheral, J. and Keller-Ressel, M. (2019). Affine forward variance models. Finance Stoch. 23 501-533. Digital Object Identifier: 10.1007/s00780-019-00392-5 Google Scholar: Lookup Link MathSciNet: MR3968277 · Zbl 1430.91110 · doi:10.1007/s00780-019-00392-5 |
| [39] | GERHOLD, S., GERSTENECKER, C. and PINTER, A. (2019). Moment explosions in the rough Heston model. Decis. Econ. Finance 42 575-608. Digital Object Identifier: 10.1007/s10203-019-00267-6 Google Scholar: Lookup Link MathSciNet: MR4031338 · Zbl 1432.91123 · doi:10.1007/s10203-019-00267-6 |
| [40] | GLOTER, A. and HOFFMANN, M. (2004). Stochastic volatility and fractional Brownian motion. Stochastic Process. Appl. 113 143-172. Digital Object Identifier: 10.1016/j.spa.2004.03.008 Google Scholar: Lookup Link MathSciNet: MR2078541 · Zbl 1065.62179 · doi:10.1016/j.spa.2004.03.008 |
| [41] | GLOTER, A. and HOFFMANN, M. (2007). Estimation of the Hurst parameter from discrete noisy data. Ann. Statist. 35 1947-1974. Digital Object Identifier: 10.1214/009053607000000316 Google Scholar: Lookup Link MathSciNet: MR2363959 · Zbl 1126.62073 · doi:10.1214/009053607000000316 |
| [42] | HOFFMANN, M. (2002). Rate of convergence for parametric estimation in a stochastic volatility model. Stochastic Process. Appl. 97 147-170. Digital Object Identifier: 10.1016/S0304-4149(01)00130-2 Google Scholar: Lookup Link MathSciNet: MR1870964 · Zbl 1058.62024 · doi:10.1016/S0304-4149(01)00130-2 |
| [43] | HORVATH, B., TEICHMANN, J. and ŽURIČ, Ž. (2021). Deep hedging under rough volatility. Risks 9 138. |
| [44] | Istas, J. and Lang, G. (1997). Quadratic variations and estimation of the local Hölder index of a Gaussian process. Ann. Inst. Henri Poincaré Probab. Stat. 33 407-436. Digital Object Identifier: 10.1016/S0246-0203(97)80099-4 Google Scholar: Lookup Link MathSciNet: MR1465796 · Zbl 0882.60032 · doi:10.1016/S0246-0203(97)80099-4 |
| [45] | JACQUIER, A., MUGURUZA, A. and PANNIER, A. (2021). Rough multifactor volatility for SPX and VIX options. arXiv preprint, available at arXiv:2112.14310. |
| [46] | JACQUIER, A. and PANNIER, A. (2022). Large and moderate deviations for stochastic Volterra systems. Stochastic Process. Appl. 149 142-187. Digital Object Identifier: 10.1016/j.spa.2022.03.017 Google Scholar: Lookup Link MathSciNet: MR4410038 · Zbl 1489.60044 · doi:10.1016/j.spa.2022.03.017 |
| [47] | Jaisson, T. and Rosenbaum, M. (2015). Limit theorems for nearly unstable Hawkes processes. Ann. Appl. Probab. 25 600-631. Digital Object Identifier: 10.1214/14-AAP1005 Google Scholar: Lookup Link MathSciNet: MR3313750 · Zbl 1319.60101 · doi:10.1214/14-AAP1005 |
| [48] | JUSSELIN, P. and ROSENBAUM, M. (2020). No-arbitrage implies power-law market impact and rough volatility. Math. Finance 30 1309-1336. Digital Object Identifier: 10.1111/mafi.12254 Google Scholar: Lookup Link MathSciNet: MR4154771 · Zbl 1508.91536 · doi:10.1111/mafi.12254 |
| [49] | KAWAI, R. (2013). Fisher information for fractional Brownian motion under high-frequency discrete sampling. Comm. Statist. Theory Methods 42 1628-1636. Digital Object Identifier: 10.1080/03610926.2011.594540 Google Scholar: Lookup Link MathSciNet: MR3041490 · Zbl 1411.60059 · doi:10.1080/03610926.2011.594540 |
| [50] | LIVIERI, G., MOUTI, S., PALLAVICINI, A. and ROSENBAUM, M. (2018). Rough volatility: Evidence from option prices. IISE Trans. 50 767-776. |
| [51] | MCCRICKERD, R. and PAKKANEN, M. S. (2018). Turbocharging Monte Carlo pricing for the rough Bergomi model. Quant. Finance 18 1877-1886. Digital Object Identifier: 10.1080/14697688.2018.1459812 Google Scholar: Lookup Link MathSciNet: MR3867719 · Zbl 1406.91486 · doi:10.1080/14697688.2018.1459812 |
| [52] | MEYER, Y., SELLAN, F. and TAQQU, M. S. (1999). Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion. J. Fourier Anal. Appl. 5 465-494. Digital Object Identifier: 10.1007/BF01261639 Google Scholar: Lookup Link MathSciNet: MR1755100 · Zbl 0948.60026 · doi:10.1007/BF01261639 |
| [53] | ROSENBAUM, M. (2008). Estimation of the volatility persistence in a discretely observed diffusion model. Stochastic Process. Appl. 118 1434-1462. Digital Object Identifier: 10.1016/j.spa.2007.09.004 Google Scholar: Lookup Link MathSciNet: MR2427046 · Zbl 1142.62055 · doi:10.1016/j.spa.2007.09.004 |
| [54] | ROSENBAUM, M. (2009). First order \(p\)-variations and Besov spaces. Statist. Probab. Lett. 79 55-62. Digital Object Identifier: 10.1016/j.spl.2008.07.019 Google Scholar: Lookup Link MathSciNet: MR2483397 · Zbl 1172.60008 · doi:10.1016/j.spl.2008.07.019 |
| [55] | ROSENBAUM, M. and ZHANG, J. (2021). Deep calibration of the quadratic rough Heston model. arXiv preprint, available at arXiv:2107.01611. |
| [56] | SZYMANSKI, G. (2024). Optimal estimation of the rough Hurst parameter in additive noise. Stochastic Process. Appl. 170 Paper no. 104302. Digital Object Identifier: 10.1016/j.spa.2024.104302 Google Scholar: Lookup Link MathSciNet: MR4700236 · Zbl 07812501 · doi:10.1016/j.spa.2024.104302 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
