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Analyzing the Fine Structure of Continuous Time Stochastic Processes

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Seminar on Stochastic Analysis, Random Fields and Applications VI

Part of the book series: Progress in Probability ((PRPR,volume 63))

Abstract

In the recent years especially in finance many different models either based on semimartingales, purely continuous, pure jump and a mixture of both, or fractional Brownian motion have been proposed in the literature. We provide a class of easily computable estimators which allows to infer the fine structure of the underlying process in terms of the Blumenthal-Getoor index or the Hurst exponent based on high frequency data. This method makes it possible not only to detect jumps, but also determine their activity and the regularity of continuous components, which can be used for model selection or to analyze the market microstructure by taking into account different time scales. Furthermore, our method provides a simple graphical tool for detecting jumps.

Mathematics Subject Classification (2000). Primary: 62M99; Secondary: 91B84.

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References

  1. Y. Ait-Sahalia, Telling from discrete data whether the underlying continuous-time model is a diffusion, Journal of Finance, 57 (2002), 2075–2112.

    Article  Google Scholar 

  2. Y. Ait-Sahalia and J. Jacod, Estimating the degree of activity of jumps in high frequency data, Annals of Statistics, to appear, 2008.

    Google Scholar 

  3. D.J. Aldous and G.K. Eagleson, On mixing and stability of limit theorems, Ann. Probab., 6 (1978), 325–331.

    Article  MathSciNet  MATH  Google Scholar 

  4. O.E. Barndorff-Nielsen, J.M. Corcuera, M. Podolskij, and J.H.C. Woerner, Bipower variation for Gaussian processes with stationary increments, J. Appl. Probab., 46 (1) (2009), 132–150.

    Article  MathSciNet  MATH  Google Scholar 

  5. O.E. Barndorff-Nielsen and N. Shephard, Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion), Journal of the Royal Statistical Society, Series B, 63 (2001), 167–241.

    Article  MathSciNet  MATH  Google Scholar 

  6. O.E. Barndorff-Nielsen and N. Shephard, Realised power variation and stochastic volatility models, Bernoulli, 9 (2003), 243–265.

    Article  MathSciNet  MATH  Google Scholar 

  7. O.E. Barndorff-Nielsen and N. Shephard, Econometrics of testing for jumps in financial econometrics using bipower variation, Journal of Financial Econometrics, 4 (2006), 1–30.

    Article  MathSciNet  Google Scholar 

  8. O.E. Barndorff-Nielsen, E. Graversen, J. Jacod, M. Podolskij, and N. Shephard, A central limit theorem for realised power and bipower variation of continuous semimartingales, From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev, 2005.

    Google Scholar 

  9. S.M. Berman, Sign-invariant random variables and stochastic processes with sign invariant increments, Trans. Amer. Math. Soc., 119 (1965), 216–243.

    Article  MathSciNet  MATH  Google Scholar 

  10. D.C. Brody, J. Syroka, and M. Zervos, Dynamical pricing of weather derivatives, Quantitative Finance, 2 (2002), 189–198.

    Article  MathSciNet  Google Scholar 

  11. P. Carr, H. Geman, D.B. Madan, and M. Yor, Stochastic volatility for L´evy processes, Math. Finance, 13 (2003), 345–382.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Cheridito, Arbitrage in fractional Brownian motion models, Finance and Stochastics,7 (2003), 533–553.

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Cont and C. Mancini, Nonparametric tests of analyzing the fine structure of price fluctuations, preprint, 2007.

    Google Scholar 

  14. J.M. Corcuera, Power variation analysis of some integral long-memory processes. Proceedings of the Abel Symposium 2005, 2006.

    Google Scholar 

  15. J.M. Corcuera, D. Nualart, and J.H.C. Woerner, Power Variation of some integral fractional processes, Bernoulli, 12 (2006), 713–735.

    Article  MathSciNet  MATH  Google Scholar 

  16. T. Di Matteo, T. Aste, and M.M. Dacorogna, Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development, Journal of Banking & Finance, 29 (2005), 827–851.

    Article  Google Scholar 

  17. P.D. Ditlevsen, Observation of α-stable noise induced millenal climate changes from a ice record, Geophysical Research Letters, 26 (1999), 1441–1444.

    Article  Google Scholar 

  18. E. Eberlein, J. Kallsen, and J. Kristen, Riskmanagement Based on Stochastic Volatility, Journal of Risk, 2 (2003), 19–44.

    Google Scholar 

  19. P. Guasoni, M. Rasonyi, and W. Schachermayer, Consistent price systems and facelifting pricing under transaction costs, Annals of Applied Probability, 18 (2008), 491–520.

    Article  MathSciNet  MATH  Google Scholar 

  20. Y. Hu and D. Nualart, Renormalized self-intersection local time for fractional Brownian motion, Ann. Probab., 33 (2005), 948–983.

    Article  MathSciNet  MATH  Google Scholar 

  21. W.N. Hudson and J.D. Mason, Variational sums for additive processes, Proc. Amer. Math. Soc, 55 (1976), 395–399.

    Article  MathSciNet  MATH  Google Scholar 

  22. D. Hull and A. White, The pricing of options on assets with stochastic volatilities, Journal of Finance, 42 (1987), 281–300.

    Article  Google Scholar 

  23. R.J. Hyndman, (n.d.) Time Series Data Library, FVD1.DAT (Accessed November 2008), http://www.robhyndman.info/TSDL.

  24. P. Imkeller and I. Pavlyukevich, First exit times of SDEs driven by stable L´evy processes, Stochastic Processes and their Applications, 116 (2006), 611–642.

    Article  MathSciNet  MATH  Google Scholar 

  25. D. Lepingle, La variation d’ordre p des semi-martingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 36 (1976), 295–316.

    Article  MathSciNet  MATH  Google Scholar 

  26. C. Mancini, Estimating the integrated volatility in stochastic volatility models with L´evy type jumps, University of Firenze, preprint, 2005.

    Google Scholar 

  27. D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab., 33 (2005), 177–193.

    Article  MathSciNet  MATH  Google Scholar 

  28. G. Peccati and C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, Lecture Notes in Math. S´eminaire de Probabilit´es XXXVIII, (2005), 247–262.

    Google Scholar 

  29. R.F. Peltier and J. Levy Vehel, A new method for estimating the parameter of a fractional Brownian motion, Technical Report No 2396, INRIA Rocquencourt, 1994.

    Google Scholar 

  30. K. Sato, L´evy Processes and Infinitely Divisible Distributions, Cambridge University Press, 1999.

    Google Scholar 

  31. L. Scott, Option pricing when the variance changes randomly: theory, estimation and an application, Journal of Financial and Quantitative Analysis, 22 (1987), 419–438.

    Article  Google Scholar 

  32. A.N. Shiryaev, Essentials of Stochastic Finance: Facts, Models and Theory, World Scientific, Singapore, 1999.

    Google Scholar 

  33. E. Stein and C. Stein, Stock price distributions with stochastic volatility: an analytic approach, Review of Financial Studies, 4 (1991), 727–752.

    Article  Google Scholar 

  34. V. Todorov and G. Tauchen, Activity signature plots and the generalized Blumenthal-Getoor index, working paper, 2007.

    Google Scholar 

  35. J.H.C.Woerner, Purely discontinuous L´evy Processes and Power Variation: inference for integrated volatility and the scale parameter, 2003-MF-07, Working Paper Series in Mathematical Finance, University of Oxford.

    Google Scholar 

  36. J.H.C. Woerner, Estimation of integrated volatility in stochastic volatility models, Appl. Stochastic Models Bus. Ind., 21 (2005), 27–44.

    Article  MathSciNet  MATH  Google Scholar 

  37. J.H.C. Woerner, Power and multipower variation: inference for high frequency data, In: Stochastic Finance, A.N. Shiryaev, M. do Ros´ario Grossihno, P. Oliviera, and M. Esquivel, Springer, (2006), 343–364.

    Google Scholar 

  38. J.H.C. Woerner, Analyzing the fine structure of continuous time stochastic processes, working paper, 2006.

    Google Scholar 

  39. J.H.C. Woerner, Inference in L´evy type stochastic volatility models, Advances in Applied Probability, 39 (2007), 531–549.

    Article  MathSciNet  MATH  Google Scholar 

  40. J.H.C. Woerner, Volatility estimates for high frequency data: market microstucture noise versus fractional Brownian motion models, preprint, 2007.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227, Dortmund, Germany

    Jeannette H. C. Woerner

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  1. Jeannette H. C. Woerner
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Correspondence to Jeannette H. C. Woerner .

Editor information

Editors and Affiliations

  1. Fac. Sciences de Base (FSB), Institut de Mathématiques (IMA), Ecole Polytechnique Fédérale de Lausanne, Lausanne, 1015, Switzerland

    Robert Dalang

  2. Institut Élie Cartan IECN, Nancy Université, Vandoeuvre-les-Nancy CX, France

    Marco Dozzi

  3. , Unité de Mathématiques appliquées, Ecole Nationale Supérieure des Technique, Boulevard Victor 32, Paris Cedex 15, 75739, France

    Francesco Russo

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Woerner, J.H.C. (2011). Analyzing the Fine Structure of Continuous Time Stochastic Processes. In: Dalang, R., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability, vol 63. Springer, Basel. https://doi.org/10.1007/978-3-0348-0021-1_26

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