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Efficient volatility estimation in a two‐factor model

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  • Olivier Féron
  • Pierre Gruet
  • Marc Hoffmann

Abstract

We statistically analyze a multivariate Heath‐Jarrow‐Morton diffusion model with stochastic volatility. The volatility process of the first factor is left totally unspecified while the volatility of the second factor is the product of an unknown process and an exponential function of time to maturity. This exponential term includes some real parameter measuring the rate of increase of the second factor as time goes to maturity. From historical data, we efficiently estimate the time to maturity parameter in the sense of constructing an estimator that achieves an optimal information bound in a semiparametric setting. We also nonparametrically identify the paths of the volatility processes and achieve minimax bounds. We address the problem of degeneracy that occurs when the dimension of the process is greater than two, and give in particular optimal limit theorems under suitable regularity assumptions on the drift process. We consistently analyze the numerical behavior of our estimators on simulated and real datasets of prices of forward contracts on electricity markets.

Suggested Citation

  • Olivier Féron & Pierre Gruet & Marc Hoffmann, 2020. "Efficient volatility estimation in a two‐factor model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 47(3), pages 862-898, September.
    • Handle: RePEc:bla:scjsta:v:47:y:2020:i:3:p:862-898
    DOI: 10.1111/sjos.12431
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    References listed on IDEAS

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    2. Hoffmann, Marc, 1999. "Adaptive estimation in diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 135-163, January.
    3. Hoffmann, Marc, 1997. "Minimax estimation of the diffusion coefficient through irregular samplings," Statistics & Probability Letters, Elsevier, vol. 32(1), pages 11-24, February.
    4. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    5. Juri Hinz & Lutz Von Grafenstein & Michel Verschuere & Martina Wilhelm, 2005. "Pricing electricity risk by interest rate methods," Quantitative Finance, Taylor & Francis Journals, vol. 5(1), pages 49-60.
    6. Benth, Fred Espen & Koekebakker, Steen, 2008. "Stochastic modeling of financial electricity contracts," Energy Economics, Elsevier, vol. 30(3), pages 1116-1157, May.
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