Skip to main content
SpringerLink
Log in

Estimation of the instantaneous volatility

  • Published:
Statistical Inference for Stochastic Processes Aims and scope Submit manuscript

Abstract

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 .

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aït-Sahalia Y (2004) Disentangling diffusion from jumps. J Financ Econ 74: 487–528

    Article  Google Scholar 

  • Aït-Sahalia Y, Jacod J (2007) Volatility estimators for discretely sampled Lévy processes. Ann Stat 35(1): 355–392

    Article  MATH  Google Scholar 

  • Aït-Sahalia Y, Jacod J (2009) Testing for jumps in a discretely observed process. Ann Stat 37(1): 184–222

    Article  MATH  Google Scholar 

  • Alvarez A (2007) Modélisation de séries financières, estimations, ajustement de modèles et tests d’hypothèses. PhD thesis, Université de Toulouse

  • 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 25–31

  • 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): 253–280

    Article  MathSciNet  MATH  Google Scholar 

  • Barndorff-Nielsen OE, Shephard N (2006) Econometrics of testing for jumps in financial economics using bipower variation. J Financ Econometr 4: 1–30

    Article  MathSciNet  Google Scholar 

  • 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 33–68

    Chapter  Google Scholar 

  • Cont R, Tankov P (2004) Financial modelling with jump processes. Financial mathematics series. Chapman & Hall/CRC, Boca Raton

    Google Scholar 

  • Corsi F, Pirini D, Roberto R (2008) Volatility forecasting: the jumps do matter. Preprint

  • Eagleson GK (1975) Martingale convergence to mixtures of infinitely divisible laws. Ann Probab 3(3): 557–562

    Article  MathSciNet  MATH  Google Scholar 

  • Espinosa F, Vives J (2006) A volatility-varying and jump-diffusion Merton type model of interest rate risk. Insur Math Econ 38(1): 157–166

    Article  MathSciNet  MATH  Google Scholar 

  • Fouque J-P, Papanicolaou G, Sircar KR (2000) Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Hall P, Heyde CC (1980) Martingale limit theory and its application. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York

    MATH  Google Scholar 

  • Jacod J (2008) Asymptotic properties of realized power variations and related functionals of semimartingales. Stoch Process Appl 118(4): 517–559

    Article  MathSciNet  MATH  Google Scholar 

  • Jacod J, Podolskij M, Vetter M Limit theorems for moving averages of discretized processes plus noise-madpn-3 (preprint)

  • Lépingle D (1976) La variation d’ordre p des semi-martingales. Z. Wahrscheinlichkeitstheorie und Verw Gebiete 36(4): 295–316

    Article  MathSciNet  MATH  Google Scholar 

  • Mancini C (2001) Disentangling the jumps of the diffusion in a geometric jumping Brownian. Giornale dell’Istituto Italiano degli Attuari LXIV:19–47

  • Mancini C (2004) Estimating the integrated volatility in stochastic volatility models with Lévy type jumps. Technical report, Universita di Firenze

  • Robert CY, Rosenbaum M (2009) Volatility and covariation estimation when microstructure noise and trading times are endogenous. To appear in Math Financ

  • Todorov V, Tauchen G (2008) Volatility jumps. Preprint

  • Veraart AED (2010) Inference for the jump part of quadratic variation of Ito semimartingales. Econ Theory (26-02): 331–368

    Article  MathSciNet  Google Scholar 

  • Woerner JHC (2005) Estimation of integrated volatility in stochastic volatility models. Appl Stoch Models Bus Indus 21(1): 27–44

    Article  MathSciNet  MATH  Google Scholar 

  • Woerner JHC (2006) Power and multipower variation: inference for high frequency data. In: Stochastic finance. Springer, New York, pp 343–364

Download references

Author information

Authors and Affiliations

  1. La Habana University, Havana, Cuba

    Alexander Alvarez

  2. Institut de Mathématiques de Toulouse et INSA Toulouse, Toulouse, France

    Fabien Panloup

  3. Institut de Mathétiques de Toulouse et Université de Toulouse, Toulouse, France

    Monique Pontier & Nicolas Savy

Authors
  1. Alexander Alvarez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fabien Panloup
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Monique Pontier
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Nicolas Savy
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabien Panloup.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alvarez, A., Panloup, F., Pontier, M. et al. Estimation of the instantaneous volatility. Stat Inference Stoch Process 15, 27–59 (2012). https://doi.org/10.1007/s11203-011-9062-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11203-011-9062-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation