A GMM approach to estimate the roughness of stochastic volatility. (English) Zbl 07704472
Summary: We develop a GMM approach for estimation of log-normal stochastic volatility models driven by a fractional Brownian motion with unrestricted Hurst exponent. We show that a parameter estimator based on the integrated variance is consistent and, under stronger conditions, asymptotically normally distributed. We inspect the behavior of our procedure when integrated variance is replaced with a noisy measure of volatility calculated from discrete high-frequency data. The realized estimator contains sampling error, which skews the fractal coefficient toward “illusive roughness.” We construct an analytical approach to control the impact of measurement error without introducing nuisance parameters. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show the bias correction attenuates any systematic deviance in the parameter estimates. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in stochastic volatility.
MSC:
| 62-XX | Statistics |
| 91-XX | Game theory, economics, finance, and other social and behavioral sciences |
Keywords:
fractional Brownian motion; GMM estimation; Hurst exponent; log-normal stochastic volatility; realized variance; roughnessReferences:
| [1] | Alizadeh, S.; Brandt, M. W.; Diebold, F. X., Range-based estimation of stochastic volatility models, J. Finance, 57, 3, 1047-1092 (2002) |
| [2] | Andersen, T. G.; Bollerslev, T., Intraday periodicity and volatility persistence in financial markets, J. Empir. Financ., 4, 2, 115-158 (1997) |
| [3] | Andersen, T. G.; Bollerslev, T., Answering the skeptics: Yes, standard volatility models do provide accurate forecasts, Int. Econ. Rev., 39, 4, 885-905 (1998) |
| [4] | Andersen, T. G.; Bollerslev, T., Deutsche Mark-Dollar volatility: Intraday activity patterns, macroeconomic announcements, and longer run dependencies, J. Finance, 53, 1, 219-265 (1998) |
| [5] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Ebens, H., The distribution of realized stock return volatility, J. Financ. Econ., 61, 1, 43-76 (2001) |
| [6] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P., Modeling and forecasting realized volatility, Econometrica, 71, 2, 579-625 (2003) · Zbl 1142.91712 |
| [7] | Andersen, T. G.; Sørensen, B. E., GMM estimation of a stochastic volatility model: A Monte Carlo study, J. Bus. Econom. Statist., 14, 3, 328-352 (1996) |
| [8] | Andrews, D. W.K., Heteroscedasticity and autocorrelation consistent covariance matrix estimation, Econometrica, 59, 3, 817-858 (1991) · Zbl 0732.62052 |
| [9] | Asmussen, S.; Glynn, P. W., Stochastic Simulation: Algorithms and Analysis (2007), Springer: Springer Berlin · Zbl 1126.65001 |
| [10] | Back, K., Asset prices for general processes, J. Math. Econom., 20, 4, 371-395 (1991) · Zbl 0727.90014 |
| [11] | Bandi, F. M.; Renò, R., Price and volatility co-jumps, J. Financ. Econ., 119, 1, 107-146 (2016) |
| [12] | Barboza, L. A.; Viens, F. G., Parameter estimation of Gaussian stationary processes using the generalized method of moments, Electron. J. Stat., 11, 1, 401-439 (2017) · Zbl 1473.62297 |
| [13] | Barndorff-Nielsen, O. E.; Basse-O’Connor, A., Quasi Ornstein-Uhlenbeck processes, Bernoulli, 17, 3, 916-941 (2011) · Zbl 1233.60020 |
| [14] | Barndorff-Nielsen, O. E.; Hansen, P. R.; Shephard, N., Designing realized kernels to measure the ex post variation of equity prices in the presence of noise, Econometrica, 76, 6, 1481-1536 (2008) · Zbl 1153.91416 |
| [15] | Barndorff-Nielsen, O. E.; Shephard, N., Non-Gaussian Orstein-Uhlenbeck-based models and some of their uses in financial economics, J. R. Stat. Soc. Ser. B Stat. Methodol., 63, 2, 167-241 (2001) · Zbl 0983.60028 |
| [16] | Barndorff-Nielsen, O. E.; Shephard, N., Econometric analysis of realized volatility and its use in estimating stochastic volatility models, J. R. Stat. Soc. Ser. B Stat. Methodol., 64, 2, 253-280 (2002) · Zbl 1059.62107 |
| [17] | Barndorff-Nielsen, O. E.; Shephard, N., Power and bipower variation with stochastic volatility and jumps, J. Financ. Econom., 2, 1, 1-48 (2004) |
| [18] | Bayer, C.; Friz, P.; Gatheral, J., Pricing under rough volatility, Quant. Finance, 16, 6, 887-904 (2016) · Zbl 1465.91108 |
| [19] | Belyaev, Y. K., Continuity and Hölder’s conditions for sample functions of stationary Gaussian processes, (Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability. Vol. II (1961), University of California Press: University of California Press Berkeley), 23-33 · Zbl 0111.33003 |
| [20] | Bennedsen, M.; Lunde, A.; Pakkanen, M. S., Hybrid scheme for Brownian semistationary processes, Finance Stoch., 21, 4, 931-965 (2017) · Zbl 1385.65010 |
| [21] | Bennedsen, M.; Lunde, A.; Pakkanen, M. S., Decoupling the short- and long-term behavior of stochastic volatility, J. Financ. Econom. (2022), (Forthcoming) |
| [22] | Bingham, N. H.; Goldie, C. M.; Teugels, J. L., Regular Variation (1989), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0667.26003 |
| [23] | Bollerslev, T.; Zhou, H., Estimating stochastic volatility diffusion using conditional moments of integrated volatility, J. Econometrics, 109, 1, 33-65 (2002) · Zbl 1020.62096 |
| [24] | Breusch, T.; Qian, H.; Wyhowski, D., Redundancy of moment conditions, J. Econometrics, 91, 1, 89-111 (1999) · Zbl 1041.62504 |
| [25] | Carrasco, M.; Florens, J.-P., Generalization of GMM to a continuum of moment conditions, Econom. Theory, 16, 6, 797-834 (2000) · Zbl 0968.62028 |
| [26] | Cheridito, P.; Kawaguchi, H.; Maejima, M., Fractional Ornstein-Uhlenbeck processes, Electron. J. Probab., 8, 3, 1-14 (2003) · Zbl 1065.60033 |
| [27] | Christensen, K.; Oomen, R. C.A.; Renò, R., The drift burst hypothesis, J. Econometrics, 227, 2, 461-497 (2022) · Zbl 07491168 |
| [28] | Christensen, K.; Thyrsgaard, M.; Veliyev, B., The realized empirical distribution function of stochastic variance with application to goodness-of-fit testing, J. Econometrics, 212, 2, 556-583 (2019) · Zbl 1452.62753 |
| [29] | Christie, A. A., The stochastic behavior of common stock variances: Value, leverage and interest rate effects, J. Financial Econ., 10, 4, 407-432 (1982) |
| [30] | Christoffersen, P. F.; Jacobs, K.; Mimouni, K., Volatility dynamics for the S & P500: Evidence from realized volatility, daily returns, and option prices, Rev. Financial Stud., 23, 8, 3141-3189 (2010) |
| [31] | Comte, F.; Renault, E., Long memory in continuous-time stochastic volatility models, Math. Finance, 8, 4, 291-323 (1998) · Zbl 1020.91021 |
| [32] | Corradi, V.; Distaso, W., Semi-parametric comparison of stochastic volatility models using realized measures, Rev. Econom. Stud., 73, 3, 635-667 (2006) · Zbl 1145.91345 |
| [33] | Davidson, J., Stochastic Limit Theory (1994), Oxford University Press: Oxford University Press Oxford |
| [34] | Davidson, J., A new consistency proof for HAC variance estimators, Econom. Lett., 186, 1, Article 108811 pp. (2020) · Zbl 1435.62133 |
| [35] | Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. Ann., 300, 1, 463-520 (1994) · Zbl 0865.90014 |
| [36] | Duffie, D.; Singleton, K. J., Simulated moments estimation of markov models of asset prices, Econometrica, 61, 4, 929-952 (1993) · Zbl 0783.62099 |
| [37] | Dym, H.; McKean, H. P., Gaussian Processes, Function Theory, and the Inverse Spectral Problem (1976), Academic Press: Academic Press Massachusetts · Zbl 0327.60029 |
| [38] | El Euch, O.; Rosenbaum, M., The characteristic function of rough Heston models, Math. Finance, 29, 1, 3-38 (2018) · Zbl 1411.91553 |
| [39] | Eraker, B.; Johannes, M.; Polson, N., The impact of jumps in volatility and returns, J. Finance, 58, 3, 1269-1300 (2003) |
| [40] | Fridman, M.; Harris, L., A maximum likelihood approach for non-Gaussian stochastic volatility models, J. Bus. Econom. Statist., 16, 3, 284-291 (1998) |
| [41] | Fukasawa, M.; Takabatake, T.; Westphal, R., Consistent estimation for fractional stochastic volatility model under high-frequency asymptotics, Math. Finance (2022), (Forthcoming) · Zbl 1522.91272 |
| [42] | Gallant, A. R.; Hsieh, D. A.; Tauchen, G. E., Estimation of stochastic volatility with diagnostics, J. Econometrics, 81, 1, 159-192 (1997) · Zbl 0904.62134 |
| [43] | Garnier, J.; Sølna, K., Option pricing under fast-varying and rough stochastic volatility, Ann. Finance, 14, 4, 489-516 (2018) · Zbl 1418.91514 |
| [44] | Gatheral, J.; Jaisson, T.; Rosenbaum, M., Volatility is rough, Quant. Finance, 18, 6, 933-949 (2018) · Zbl 1400.91590 |
| [45] | Gradshteyn, I. S.; Ryzhik, I. M., Table of Integrals, Series, and Products (2007), Academic Press: Academic Press Massachusetts · Zbl 1208.65001 |
| [46] | Hall, A. R.; Inoue, A.; Jana, K.; Shin, C., Information in generalized method of moments estimation and entropy-based moment selection, J. Econometrics, 138, 2, 488-512 (2007) · Zbl 1418.62459 |
| [47] | Hansen, L. P., Large sample properties of generalized method of moments estimators, Econometrica, 50, 4, 1029-1054 (1982) · Zbl 0502.62098 |
| [48] | Hansen, P. R.; Lunde, A., Realized variance and market microstructure noise, J. Bus. Econom. Statist., 24, 2, 127-161 (2006) |
| [49] | Hansen, P. R.; Lunde, A., Estimating the persistence and the autocorrelation function of a time series that is measured with error, Econom. Theory, 30, 1, 60-93 (2014) · Zbl 1290.91132 |
| [50] | Harvey, A. C.; Ruiz, E.; Shephard, N., Multivariate stochastic variance models, Rev. Financ. Stud., 61, 2, 247-264 (1994) · Zbl 0805.90026 |
| [51] | Heston, S. L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6, 2, 327-343 (1993) · Zbl 1384.35131 |
| [52] | Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, J. Finance, 42, 2, 281-300 (1987) |
| [53] | Hurst, H. E., Long-term storage capacity of reservoirs, Trans. Am. Soc. Civ. Eng., 116, 1, 770-799 (1951) |
| [54] | Jacod, J.; Li, Y.; Mykland, P. A.; Podolskij, M.; Vetter, M., Microstructure noise in the continuous case: The pre-averaging approach, Stochastic Process. Appl., 119, 7, 2249-2276 (2009) · Zbl 1166.62078 |
| [55] | Kaarakka, T.; Salminen, P., On fractional Ornstein-Uhlenbeck processes, Commun. Stoch. Anal., 5, 1, 121-133 (2011) · Zbl 1331.60065 |
| [56] | Karhunen, K., Über die struktur stationärer zufälliger Funktionen, Ark. Mat., 1, 2, 141-160 (1950) · Zbl 0038.08603 |
| [57] | Li, J.; Todorov, V.; Tauchen, G., Adaptive estimation of continuous-time regression models using high-frequency data, J. Econometrics, 200, 1, 36-47 (2017) · Zbl 1388.62050 |
| [58] | Lindgren, G., Lectures on Stationary Stochastic Processes, Lecture Notes (2006), Lund University |
| [59] | Mancini, C., Non-parametric threshold estimation for models with stochastic diffusion coefficient and jumps, Scand. J. Stat., 36, 2, 270-296 (2009) · Zbl 1198.62079 |
| [60] | Mandelbrot, B. B.; Van Ness, J. W., Fractional brownian motions, fractional noises and applications, SIAM Rev., 10, 4, 422-437 (1968) · Zbl 0179.47801 |
| [61] | Maruyama, G., The harmonic analysis of stationary stochastic processes, (Memoirs of the Faculty of Science. Vol. 4 (1949), Kyūshū University), 45-106 · Zbl 0045.40602 |
| [62] | Meddahi, N., A theoretical comparison between integrated and realized volatility, J. Appl. Econometrics, 17, 5, 479-508 (2002) |
| [63] | Meddahi, N., ARMA representation of integrated and realized variances, Econom. J., 6, 2, 335-356 (2003) · Zbl 1036.62085 |
| [64] | Melino, A.; Turnbull, S. M., Pricing foreign currency options with stochastic volatility, J. Econometrics, 45, 1-2, 239-265 (1990) |
| [65] | Merlevède, F.; Peligrad, M.; Utev, S., Functional Gaussian Approximation for Dependent Structures (2019), Oxford University Press: Oxford University Press Oxford · Zbl 1447.60003 |
| [66] | Newey, W. K.; McFadden, D., Large sample estimation and hypothesis, (Engle, R. F.; McFadden, D., Handbook of Econometrics: Vol. IV (1994), North-Holland: North-Holland Amsterdam), 2112-2245 |
| [67] | Newey, W. K.; West, K. D., A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix, Econometrica, 55, 3, 703-708 (1987) · Zbl 0658.62139 |
| [68] | Patton, A. J., Volatility forecast comparison using imperfect volatility proxies, J. Econometrics, 160, 1, 246-256 (2011) · Zbl 1441.62830 |
| [69] | Peccati, G.; Taqqu, M. S., Wiener Chaos: Moments, Cumulants and Diagrams (2011), Springer: Springer Berlin · Zbl 1231.60003 |
| [70] | Taqqu, M. S., Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 31, 4, 287-302 (1975) · Zbl 0303.60033 |
| [71] | Taylor, S. J., Modelling Financial Time Series (1986), John Wiley and Sons: John Wiley and Sons Chichester · Zbl 1130.91345 |
| [72] | Todorov, V., Estimation of continuous-time stochastic volatility models with jumps using high-frequency data, J. Econometrics, 148, 2, 131-148 (2009) · Zbl 1429.62480 |
| [73] | Todorov, V.; Tauchen, G., Volatility jumps, J. Bus. Econom. Statist., 29, 3, 356-371 (2011) · Zbl 1219.91156 |
| [74] | Vetter, M., Limit theorems for bipower variation of semimartingales, Stochastic Process. Appl., 120, 1, 22-38 (2010) · Zbl 1183.60010 |
| [75] | Wright, J. H., Detecting lack of identification in GMM, Econom. Theory, 19, 2, 322-330 (2003) · Zbl 1441.62905 |
| [76] | Zhang, L.; Mykland, P. A.; Aït-Sahalia, Y., A tale of two time scales: determining integrated volatility with noisy high-frequency data, J. Amer. Statist. Assoc., 100, 472, 1394-1411 (2005) · Zbl 1117.62461 |
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