No-arbitrage semi-martingale restrictions for continuous-time volatility models subject to leverage effects, jumps and i.i.d. noise: theory and testable distributional implications. (English) Zbl 1418.62371
Summary: We develop a sequential procedure to test the adequacy of jump-diffusion models for return distributions. We rely on intraday data and nonparametric volatility measures, along with a new jump detection technique and appropriate conditional moment tests, for assessing the import of jumps and leverage effects. A novel robust-to-jumps approach is utilized to alleviate microstructure frictions for realized volatility estimation. Size and power of the procedure are explored through Monte Carlo methods. Our empirical findings support the jump-diffusive representation for S&P500 futures returns but reveal it is critical to account for leverage effects and jumps to maintain the underlying semi-martingale assumption.
MSC:
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91B84 | Economic time series analysis |
Keywords:
high-frequency data; realized volatility; jump detection; financial time sampling; normality testsReferences:
| [1] | Aı¨t-Sahalia, Y., Telling from discrete data whether the underlying continuous-time model is a diffusion, Journal of Finance, 57, 2075-2121 (2002) |
| [2] | Aı¨t-Sahalia, Y.; Mykland, P. A.; Zhang, L., How often to sample a continuous-time process in the presence of market microstructure noise, Review of Financial Studies, 18, 351-416 (2005) |
| [3] | Alizadeh, S.; Brandt, M.; Diebold, F. X., Range-based estimation of stochastic volatility models, Journal of Finance, 57, 1047-1091 (2002) |
| [4] | Andersen, T.G., 1992. Volatility. Manuscript, Northwestern University.; Andersen, T.G., 1992. Volatility. Manuscript, Northwestern University. |
| [5] | Andersen, T. G., Return volatility and trading volume: an information flow interpretation of stochastic volatility, Journal of Finance, 51, 169-204 (1996) |
| [6] | Andersen, T. G.; Bollerslev, T., Heterogeneous information arrivals and return volatility dynamics: Uncovering the long-run in high-frequency returns, Journal of Finance, 52, 975-1005 (1997) |
| [7] | Andersen, T. G.; Bollerslev, T., Intraday periodicity and volatility persistence in financial markets, Journal of Empirical Finance, 4, 115-158 (1997) |
| [8] | Andersen, T. G.; Bollerslev, T., Answering the skeptics: yes, standard volatility models do provide accurate forecasts, International Economic Review, 39, 885-905 (1998) |
| [9] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P., Great realizations, Risk, 13, 105-108 (2000) |
| [10] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Ebens, H., The distribution of realized stock return volatility, Journal of Financial Economics, 61, 43-76 (2001) |
| [11] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P., The distribution of realized exchange rate volatility, Journal of the American Statistical Association, 96, 42-55 (2001) · Zbl 1015.62107 |
| [12] | Andersen, T. G.; Benzoni, L.; Lund, J., Estimating jump-diffusions for equity returns, Journal of Finance, 57, 1239-1284 (2002) |
| [13] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Labys, P., Modeling and forecasting realized volatility, Econometrica, 71, 579-625 (2003) · Zbl 1142.91712 |
| [14] | Andersen, T. G.; Bollerslev, T.; Diebold, F. X.; Vega, C., Micro effects of macro announcements: real-time price discovery in foreign exchange, American Economic Review, 93, 38-62 (2003) |
| [15] | Andersen, T.G., Bollerslev, T., Diebold, F.X., 2004a. Parametric and non-parametric volatility measurement. In: Hansen, L.P., Aı¨t-Sahalia, Y. (Eds.), Handbook of Financial Econometrics, Elsevier Science, New York, forthcoming.; Andersen, T.G., Bollerslev, T., Diebold, F.X., 2004a. Parametric and non-parametric volatility measurement. In: Hansen, L.P., Aı¨t-Sahalia, Y. (Eds.), Handbook of Financial Econometrics, Elsevier Science, New York, forthcoming. |
| [16] | Andersen, T. G.; Bollerslev, T.; Meddahi, N., Analytic evaluation of volatility forecasts, International Economic Review, 45, 1079-1110 (2004) |
| [17] | Andersen, T.G., Bollerslev, T., Diebold, F.X., 2005a. Roughing it up: including jump components in the measurement, modeling and forecasting of return volatility. Manuscript, Northwestern University, Duke University, and University of Pennsylvania.; Andersen, T.G., Bollerslev, T., Diebold, F.X., 2005a. Roughing it up: including jump components in the measurement, modeling and forecasting of return volatility. Manuscript, Northwestern University, Duke University, and University of Pennsylvania. |
| [18] | Andersen, T.G., Bollerslev, T., Diebold, F.X., Vega, C., 2005b. Real-time price discovery in stock, bond and foreign exchange markets. Manuscript, Northwestern University, Duke University, University of Pennsylvania, and University of Rochester.; Andersen, T.G., Bollerslev, T., Diebold, F.X., Vega, C., 2005b. Real-time price discovery in stock, bond and foreign exchange markets. Manuscript, Northwestern University, Duke University, University of Pennsylvania, and University of Rochester. |
| [19] | Andersen, T.G., Bollerslev, T., Frederiksen, P.H., Nielsen, M.Ø., 2005c. Jumps in common stock returns. Manuscript, Northwestern University, Duke University, and Cornell University.; Andersen, T.G., Bollerslev, T., Frederiksen, P.H., Nielsen, M.Ø., 2005c. Jumps in common stock returns. Manuscript, Northwestern University, Duke University, and Cornell University. |
| [20] | Andersen, T. G.; Bollerslev, T.; Meddahi, N., Correcting the errors: volatility forecast evaluation using high-frequency data and realized volatilities, Econometrica, 72, 279-296 (2005) · Zbl 1152.91720 |
| [21] | Andersen, T.G., Bollerslev, T., Frederiksen, P.H., Nielsen, M.Ø., 2006. Comment on realized variance and market microstructure noise, by P.R. Hansen and A. Lunde. Journal of Business and Economic Statistics 24, 173-179.; Andersen, T.G., Bollerslev, T., Frederiksen, P.H., Nielsen, M.Ø., 2006. Comment on realized variance and market microstructure noise, by P.R. Hansen and A. Lunde. Journal of Business and Economic Statistics 24, 173-179. |
| [22] | Ané, T.; Geman, H., Order flow, transaction clock, and normality of asset returns, Journal of Finance, 55, 2259-2284 (2000) |
| [23] | Back, K., Asset prices for general processes, Journal of Mathematical Economics, 20, 317-395 (1991) · Zbl 0727.90014 |
| [24] | Ball, C. A.; Torous, W. N., A simplified jump process for common stock returns, Journal of Financial and Quantitative Analysis, 18, 53-65 (1983) |
| [25] | Bandi, F.M., Russell, J.R., 2004a. Microstructure noise, realized volatility, and optimal sampling. Manuscript, University of Chicago.; Bandi, F.M., Russell, J.R., 2004a. Microstructure noise, realized volatility, and optimal sampling. Manuscript, University of Chicago. · Zbl 1138.91394 |
| [26] | Bandi, F.M., Russell, J.R., 2004b. Separating microstructure noise from volatility. Manuscript, University of Chicago.; Bandi, F.M., Russell, J.R., 2004b. Separating microstructure noise from volatility. Manuscript, University of Chicago. |
| [27] | Barndorff-Nielsen, O. E.; Shephard, N., Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics, Journal of the Royal Statistical Society Series B, 63, 167-241 (2001) · Zbl 0983.60028 |
| [28] | Barndorff-Nielsen, O. E.; Shephard, N., Econometric analysis of realised volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society, 64, 253-280 (2002) · Zbl 1059.62107 |
| [29] | Barndorff-Nielsen, O. E.; Shephard, N., Estimating quadratic variation using realized variance, Journal of Applied Econometrics, 17, 457-478 (2002) |
| [30] | Barndorff-Nielsen, O. E.; Shephard, N., Power and bipower variation with stochastic volatility and jumps, Journal of Financial Econometrics, 2, 1-37 (2004) · Zbl 1095.60023 |
| [31] | Barndorff-Nielsen, O. E.; Shephard, N., Econometrics of testing for jumps in financial economics using bipower variation, Journal of Financial Econometrics, 4, 1-60 (2006) · Zbl 1095.60023 |
| [32] | Bates, D. S., Post-‘87 Crash fears in the S&P500 futures option market, Journal of Econometrics, 94, 181-238 (2000) · Zbl 0942.62118 |
| [33] | Bekaert, G.; Wu, G., Asymmetric volatility and risk in equity markets, Review of Financial Studies, 13, 1-42 (2000) |
| [34] | Bollen, B.; Inder, B., Estimating daily volatility in financial markets utilizing intraday data, Journal of Empirical Finance, 9, 551-562 (2002) |
| [35] | Bollerslev, T., A conditionally heteroskedastic time series model for speculative prices and rates of return, Review of Economics and Statistics, 69, 542-547 (1987) |
| [36] | Bollerslev, T., Litvinova, J., Tauchen, G., 2005. Leverage and volatility feedback effects in high-frequency data. Manuscript, Duke University.; Bollerslev, T., Litvinova, J., Tauchen, G., 2005. Leverage and volatility feedback effects in high-frequency data. Manuscript, Duke University. |
| [37] | Bontemps, C., Meddahi, N., 2002. Testing normality: a GMM approach. Manuscript, CIRANO, Montrèal.; Bontemps, C., Meddahi, N., 2002. Testing normality: a GMM approach. Manuscript, CIRANO, Montrèal. · Zbl 1336.62056 |
| [38] | Bontemps, C.; Meddahi, N., Testing normality: a GMM approach, Journal of Econometrics, 124, 149-186 (2005) · Zbl 1336.62056 |
| [39] | Campbell, J. Y.; Hentschel, L., No news is good news: an asymmetric model of changing volatility in stock returns, Journal of Financial Economics, 31, 282-331 (1992) |
| [40] | Carr, P.; Geman, H.; Madan, D.; Yor, M., The fine structure of asset returns: an empirical investigation, Journal of Business, 75, 305-332 (2002) |
| [41] | Carr, P.; Geman, H.; Madan, D.; Yor, M., Stochastic volatility for Lévy processes, Mathematical Finance, 13, 345-382 (2003) · Zbl 1092.91022 |
| [42] | Chan, W. H.; Maheu, J. M., Conditional jump dynamics in stock market returns, Journal of Business and Economic Statistics, 20, 377-389 (2002) |
| [43] | Chang, Y., Park, J.Y., 2004. Taking a new contour: a novel view on unit root tests. Manuscript, Rice University.; Chang, Y., Park, J.Y., 2004. Taking a new contour: a novel view on unit root tests. Manuscript, Rice University. |
| [44] | Chernov, M.; Gallant, A. R.; Ghysels, E.; Tauchen, G., Alternative models for stock price dynamics, Journal of Econometrics, 116, 225-257 (2003) · Zbl 1043.62087 |
| [45] | Chib, S.; Nardari, F.; Shephard, N., Markov chain Monte Carlo methods for stochastic volatility models, Journal of Econometrics, 108, 281-316 (2002) · Zbl 1099.62539 |
| [46] | Clark, P. K., A subordinate stochastic process model with finite variance for speculative prices, Econometrica, 41, 135-155 (1973) · Zbl 0308.90011 |
| [47] | Comte, F.; Renault, E., Long memory in continuous time stochastic volatility models, Mathematical Finance, 8, 291-323 (1998) · Zbl 1020.91021 |
| [48] | Corsi, F.; Zumbach; Müller, U. A.; Dacorogna, M., Consistent high-precision volatility from high-frequency data, Economic Notes, 30, 183-204 (2001) |
| [49] | Dacorogna, M. M.; Gencay, R.; Müller, U. A.; Pictet, O. V.; Olsen, R. B., An Introduction to High-Frequency Finance (2001), Academic Press: Academic Press San Diego |
| [50] | Dambis, K. E., On the decomposition of continuous submartingales, Theory of Probability Applications, 10, 401-410 (1965) · Zbl 0141.15102 |
| [51] | Davidson, J., Stochastic Limit Theory (1994), Oxford University Press: Oxford University Press New York |
| [52] | Drost, F. C.; Nijman, T. E.; Werker, B. J.M., Estimation and testing in models containing both jumps and conditional heteroskedasticity, Journal of Business and Economic Statistics, 16, 237-243 (1998) |
| [53] | Dubins, L. E.; Schwartz, G., On continuous martingales, Proceedings of the National Academy of Sciences, 53, 913-916 (1965) · Zbl 0203.17504 |
| [54] | Epps, T. W.; Epps, M. L., The stochastic dependence of security price changes and transaction volumes: implications for the mixture-of-distributions hypothesis, Econometrica, 44, 305-321 (1976) · Zbl 0319.90014 |
| [55] | Eraker, B., Do stock prices and volatility jump?. Reconciling evidence from spot and option prices, Journal of Finance, 59, 1367-1403 (2004) |
| [56] | Eraker, B.; Johannes, M. S.; Polson, N. G., The impact of jumps in volatility, Journal of Finance, 58, 1269-1300 (2003) |
| [57] | Forsberg, L.; Bollerslev, T., Bridging the gap between the distribution of realized (ECU) volatility and ARCH modeling (of the Euro): the GARCH-NIG model, Journal of Applied Econometrics, 17, 535-548 (2002) |
| [58] | Gallant, A. R.; Hsu, C. T.; Tauchen, G., Using daily range data to calibrate volatility diffusions and extract the forward integrated variance, Review of Economics and Statistics, 81, 617-631 (1999) |
| [59] | Garman, M. B.; Klass, M. J., On the estimation of price volatility from historical data, Journal of Business, 53, 67-78 (1980) |
| [60] | Gençay, R., Selçuk, F., 2004. Volatility-return dynamics across different timescales. Manuscript, Simon Fraser University.; Gençay, R., Selçuk, F., 2004. Volatility-return dynamics across different timescales. Manuscript, Simon Fraser University. |
| [61] | Gillemot, L., Farmer, J.D., Lillo, F., 2005. There’s more to volatility than volume. Manuscript, Santa Fe Institute.; Gillemot, L., Farmer, J.D., Lillo, F., 2005. There’s more to volatility than volume. Manuscript, Santa Fe Institute. · Zbl 1134.91547 |
| [62] | Hansen, P. R.; Lunde, A., Realized variance and market microstructure noise, Journal of Business and Economic Statistics, 24, 127-161 (2006) |
| [63] | Harris, L., A transaction data study of weekly and intradaily patterns in stock returns, Journal of Financial Economics, 16, 99-117 (1986) |
| [64] | Huang, X.; Tauchen, G., The relative contribution of jumps to total price variance, Journal of Financial Econometrics, 3, 456-499 (2005) |
| [65] | Jarque, C.; Bera, A., Efficient tests for normality, heteroskedasticity, and serial independence or regression residuals, Economic Letters, 6, 255-259 (1980) |
| [66] | Johannes, M., The statistical and economic role of jumps in continuous-time interest rate models, Journal of Finance, 59, 227-260 (2004) |
| [67] | Karatzas, I.; Shreve, S. E., Brownian Motion and Stochastic Calculus (1991), Springer: Springer New York · Zbl 0734.60060 |
| [68] | Lai, T. L.; Siegmund, D., Fixed accuracy estimation of an autoregressive parameter, Annals of Statistics, 11, 478-485 (1983) · Zbl 0519.62076 |
| [69] | Maheu, J. M.; McCurdy, T. H., News arrival, jump dynamics and volatility components for individual stock returns, Journal of Finance, 59, 755-793 (2004) |
| [70] | Mancini, C., Estimation of the parameters of jump of a general poisson-diffusion model, Scandinavian Actuarial Journal, 104, 42-52 (2004) · Zbl 1114.62081 |
| [71] | Mancini, C., 2005a. Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Manuscript, Universita di Firenze.; Mancini, C., 2005a. Disentangling the jumps of the diffusion in a geometric jumping Brownian motion. Manuscript, Universita di Firenze. |
| [72] | Mancini, C., 2005b. Estimating the integrated volatility in stochastic volatility models with levy typw jumps. Manuscript, Universita di Firenze.; Mancini, C., 2005b. Estimating the integrated volatility in stochastic volatility models with levy typw jumps. Manuscript, Universita di Firenze. |
| [73] | Meddahi, N., A theoretical comparison between integrated and realized volatility, Journal of Applied Econometrics, 17, 479-508 (2002) |
| [74] | Merton, R. C., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 125-144 (1976) · Zbl 1131.91344 |
| [75] | Nelson, D. B., ARCH models as diffusion approximations, Journal of Econometrics, 45, 7-38 (1990) · Zbl 0719.60089 |
| [76] | Nelson, D. B., Conditional heteroskedasticity in asset returns: a new approach, Econometrica, 59, 347-370 (1991) · Zbl 0722.62069 |
| [77] | Nelson, D. B., Filtering and forecasting with misspecified ARCH models I: getting the right variance with the wrong model, Journal of Econometrics, 52, 61-90 (1992) · Zbl 0761.62169 |
| [78] | Oomen, R. C.A., Properties of biased-corrected realized variance under alternative sampling schemes, Journal of Financial Econometrics, 4, 555-577 (2005) |
| [79] | Oomen, R. C.A., Properties of realized variance under alternative sampling schemes, Journal of Business & Economic Statistics, 24, 219-237 (2006) |
| [80] | Pan, J., The jump-risk premia implicit in options: evidence from an integrated time series study, Journal of Financial Economics, 63, 3-50 (2002) |
| [81] | Parkinson, M., The extreme value method for estimating the variance of the rate of return, Journal of Business, 53, 61-65 (1980) |
| [82] | Peters, R.T., de Vilder, R.G., 2004. Testing the diffusion model for the S&P500. Manuscript, University of Amsterdam.; Peters, R.T., de Vilder, R.G., 2004. Testing the diffusion model for the S&P500. Manuscript, University of Amsterdam. |
| [83] | Tauchen, G.; Pitts, M., The price variability-volume relationship on speculative markets, Econometrica, 51, 485-505 (1983) · Zbl 0495.90026 |
| [84] | Todorov, V., Tauchen, G., 2005. Simulation methods for Lévy-driven CARMA stochastic volatility models. Manuscript, Duke University.; Todorov, V., Tauchen, G., 2005. Simulation methods for Lévy-driven CARMA stochastic volatility models. Manuscript, Duke University. |
| [85] | Wood, R. A.; McInish, T. H.; Ord, J. K., An investigation of transaction data for NYSE stocks, Journal of Finance, 25, 723-739 (1985) |
| [86] | Zhang, L.; Aı¨t-Sahalia, Y.; Mykland, P. A., A tale of two time scales: determining integrated volatility with noisy high-frequency data, Journal of the American Statistical Association, 100, 1394-1411 (2005) · Zbl 1117.62461 |
| [87] | Zhou, B., High-frequency data and volatility in foreign exchange rates, Journal of Business and Economic Statistics, 14, 45-52 (1996) |
| [88] | Zhou, B., F-consistency, De-volatilization and normalization of high-frequency financial data, (Dunis, C.; Zhou, B., Nonlinear Modelling of High Frequency Financial Time Series (1998), Wiley: Wiley London) |
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