Source: Adv. in Appl. Probab.
Volume 39, Number 2
Based on the concept of multipower variation we establish a class of easily
computable and robust estimators for the integrated volatility, especially
squared integrated volatility, in Lévy-type stochastic volatility
models. We derive consistency and feasible distributional results for the estimators.
Furthermore, we discuss the applications to time-changed CGMY, normal
inverse Gaussian, and hyperbolic models with and without leverage, where
the time-changes are based on integrated Cox-Ingersoll-Ross or
Ornstein-Uhlenbeck-type processes. We deduce which type of market
microstructure does not affect the estimates.
Ait-Sahalia, Y. and Jacod, J. (2007). Volatility estimators for discretely sampled Lévy processes. To appear in Ann. Statist.
Barndorff-Nielsen, O. E. et al. (2006a). A central limit theorem for realised power and bipower variations of continuous semimartingales. In From Stochastic Analysis to Mathematical Finance, eds Y. Kabanov, R. Lipster and J. Stoyanov, Springer, Berlin, pp. 33--68.
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein--Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. Roy. Statist. Soc. Ser. B 63, 167--241.
Barndorff-Nielsen, O. E. and Shephard, N. (2003). Realised power variation and stochastic volatility models. Bernoulli 9, 243--265.
Barndorff-Nielsen, O. E. and Shephard, N. (2004). Power and bipower variation with stochastic volatility and jumps. J. Financial Econometrics 2, 1--48.
Barndorff-Nielsen, O. E. and Shephard, N. (2005). Power variation and time change. Theory Prob. Appl. 50, 1--15.
Barndorff-Nielsen, O. E. and Shephard, N. (2006). Econometrics of testing for jumps in financial econometrics using bipower variation. J. Financial Econometrics 4, 1--30.
Barndorff-Nielsen, O. E. and Shephard, N. (2007). Variation, jumps, market frictions and high frequency data in financial econometrics. To appear in Advances in Economics and Econometrics, Theory and Applications, Ninth World Congress, Cambridge University Press.
Barndorff-Nielsen, O. E., Shephard, N. and Winkel, M. (2006b). Limit theorems for multipower variation in the presence of jumps. Stoch. Process. Appl. 116, 796--806.
Berman, S. M. (1965). Sign-invariant random variables and stochastic processes with sign invariant increments. Trans. Amer. Math. Soc. 119, 216--243.
Mathematical Reviews (MathSciNet): MR185651
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305--332.
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345--382.
Comte, F. and Renault, E. (1998). Long memory in continuous-time stochastic valatility models. Math. Finance 8, 291--323.
Corcuera, J. M., Nualart, D. and Woerner, J. H. C. (2006). Power variation of some integral fractional processes. Bernoulli 12, 713--735.
Corcuera, J. M., Nualart, D. and Woerner, J. H. C. (2007). A functional central limit theorem for the realized power variation of integrated stable processes. Stoch. Anal. Appl. 25, 169--186.
Eberlein, E. and von Hammerstein, E. A. (2004). Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. Seminar on Stochastic Analysis, Random Fields and Applications IV (Progress. Prob. 58), Birkhäuser, Basel, pp. 221--264.
Geman, H., Madan, D. B. and Yor, M. (2001). Time changes for Lévy processes. Math. Finance 11, 79--96.
Gnedenko, B. V. and Kolmogorov, A. N. (1968). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading, MA.
Mathematical Reviews (MathSciNet): MR233400
Howison, S., Rafailidis, A. and Rasmussen, H. O. (2002). A note on the pricing and hedging of volatility derivatives. Tech. Rep. 2001-MF-09, OCIAM, University of Oxford.
Hudson, W. N. and Mason, J. D. (1976). Variational sums for additive processes. Proc. Amer. Math. Soc. 55, 395--399.
Mathematical Reviews (MathSciNet): MR405593
Lepingle, D. (1976). La variation d'ordre p des semi-martingales. Z. Wahrscheinlichkeitsth. 36, 295--316.
Mathematical Reviews (MathSciNet): MR420837
Raible, S. (1999). Lévy processes in finance: Theory, numerics, and empirical facts. PhD thesis, University of Freiburg, 2000.
Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
Woerner, J. H. C. (2003a). Purely discontinuous Lévy processes and power variation: inference for integrated volatility and the scale parameter. 2003-MF-07, Working Paper, University of Oxford.
Woerner, J. H. C. (2003b). Variational sums and power variation: a unifying approach to model selection and estimation in semimartingale models. Statist. Decisions 21, 47--68.
Woerner, J. H. C. (2005). Estimation of integrated volatility in stochastic volatility models. Appl. Stochastic Models Bus. Ind., 21:27--44.
Woerner, J. H. C. (2006). Power and multipower variation: inference for high frequency data. In Stochastic Finance, eds A. N. Shiryaev, M. do Rosário Grossihno, P. Oliviera, and M. Esquivel. Springer, New York, pp. 343--364.