Estimation of the instantaneous volatility

  • Alexander Alvarez
  • Fabien Panloup
  • Monique Pontier
  • Nicolas Savy


This paper is concerned with the estimation of the volatility process in a stochastic volatility model of the following form: dX t a t dt + σ t dW t , where X denotes the log-price and σ is a càdlàg semi-martingale. In the spirit of a series of recent works on the estimation of the cumulated volatility, we here focus on the instantaneous volatility for which we study estimators built as finite differences of the power variations of the log-price. We provide central limit theorems with an optimal rate depending on the local behavior of σ. In particular, these theorems yield some confidence intervals for σ t .


Central limit theorem Power variation Semimartingale 

Mathematics Subject Classification (2000)

Primary 60F05 Secondary 91B70 91B82 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aït-Sahalia Y (2004) Disentangling diffusion from jumps. J Financ Econ 74: 487–528CrossRefGoogle Scholar
  2. Aït-Sahalia Y, Jacod J (2007) Volatility estimators for discretely sampled Lévy processes. Ann Stat 35(1): 355–392MATHCrossRefGoogle Scholar
  3. Aït-Sahalia Y, Jacod J (2009) Testing for jumps in a discretely observed process. Ann Stat 37(1): 184–222MATHCrossRefGoogle Scholar
  4. Alvarez A (2007) Modélisation de séries financières, estimations, ajustement de modèles et tests d’hypothèses. PhD thesis, Université de ToulouseGoogle Scholar
  5. Barndorff-Nielsen OE (2001) Modelling by Lévy processes. In: Selected proceedings of the symposium on inference for stochastic processes, Athens, GA, 2000. IMS lecture notes monograph series, vol 37. Inst Math Statist, Beachwood, OH, pp 25–31Google Scholar
  6. Barndorff-Nielsen OE, Shephard N (2002) Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J R Stat Soc B Stat Methodol 64(2): 253–280MathSciNetMATHCrossRefGoogle Scholar
  7. Barndorff-Nielsen OE, Shephard N (2006) Econometrics of testing for jumps in financial economics using bipower variation. J Financ Econometr 4: 1–30MathSciNetCrossRefGoogle Scholar
  8. Barndorff-Nielsen OE, Graversen SE, Jacod J, Podolskij M, Shephard N (2006) A central limit theorem for realised power and bipower variations of continuous semimartingales. In: Kabanov Y, Liptser R, Stoyanov J (eds) From stochastic analysis to mathematical finance, the Shiryaev Festschrift. Springer, Berlin,, pp 33–68CrossRefGoogle Scholar
  9. Cont R, Tankov P (2004) Financial modelling with jump processes. Financial mathematics series. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  10. Corsi F, Pirini D, Roberto R (2008) Volatility forecasting: the jumps do matter. PreprintGoogle Scholar
  11. Eagleson GK (1975) Martingale convergence to mixtures of infinitely divisible laws. Ann Probab 3(3): 557–562MathSciNetMATHCrossRefGoogle Scholar
  12. Espinosa F, Vives J (2006) A volatility-varying and jump-diffusion Merton type model of interest rate risk. Insur Math Econ 38(1): 157–166MathSciNetMATHCrossRefGoogle Scholar
  13. Fouque J-P, Papanicolaou G, Sircar KR (2000) Derivatives in financial markets with stochastic volatility. Cambridge University Press, CambridgeMATHGoogle Scholar
  14. Hall P, Heyde CC (1980) Martingale limit theory and its application. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New YorkMATHGoogle Scholar
  15. Jacod J (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stoch Process Appl 118(4): 517–559MathSciNetMATHCrossRefGoogle Scholar
  16. Jacod J, Podolskij M, Vetter M Limit theorems for moving averages of discretized processes plus noise-madpn-3 (preprint)Google Scholar
  17. Lépingle D (1976) La variation d’ordre p des semi-martingales. Z. Wahrscheinlichkeitstheorie und Verw Gebiete 36(4): 295–316MathSciNetMATHCrossRefGoogle Scholar
  18. Mancini C (2001) Disentangling the jumps of the diffusion in a geometric jumping Brownian. Giornale dell’Istituto Italiano degli Attuari LXIV:19–47Google Scholar
  19. Mancini C (2004) Estimating the integrated volatility in stochastic volatility models with Lévy type jumps. Technical report, Universita di FirenzeGoogle Scholar
  20. Robert CY, Rosenbaum M (2009) Volatility and covariation estimation when microstructure noise and trading times are endogenous. To appear in Math FinancGoogle Scholar
  21. Todorov V, Tauchen G (2008) Volatility jumps. PreprintGoogle Scholar
  22. Veraart AED (2010) Inference for the jump part of quadratic variation of Ito semimartingales. Econ Theory (26-02): 331–368MathSciNetCrossRefGoogle Scholar
  23. Woerner JHC (2005) Estimation of integrated volatility in stochastic volatility models. Appl Stoch Models Bus Indus 21(1): 27–44MathSciNetMATHCrossRefGoogle Scholar
  24. Woerner JHC (2006) Power and multipower variation: inference for high frequency data. In: Stochastic finance. Springer, New York, pp 343–364Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Alexander Alvarez
    • 1
  • Fabien Panloup
    • 2
  • Monique Pontier
    • 3
  • Nicolas Savy
    • 3
  1. 1.La Habana UniversityHavanaCuba
  2. 2.Institut de Mathématiques de Toulouse et INSA ToulouseToulouseFrance
  3. 3.Institut de Mathétiques de Toulouse et Université de ToulouseToulouseFrance

Personalised recommendations