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Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model

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  • Masuda, H.
  • Yoshida, N.

Abstract

With the help of a general methodology of asymptotic expansions for mixing processes, we obtain the Edgeworth expansion for log-returns of a stock price process in Barndorff-Nielsen and Shephard's stochastic volatility model, in which the latent volatility process is described by a stationary non-Gaussian Ornstein-Uhlenbeck process (OU process) with invariant selfdecomposable distribution on . The present result enables us to simultaneously explain non-Gaussianity for short time-lags as well as approximate Gaussianity for long time-lags. The Malliavin calculus formulated by Bichteler, Gravereaux and Jacod for processes with jumps and the exponential mixing property of the OU process play substantial roles in order to ensure a conditional type Cramér condition under a certain truncation. Owing to several inherent properties of OU processes, the regularity conditions for the expansions can be verified without any difficulty, and the coefficients of the expansions up to any order can be explicitly computed.

Suggested Citation

  • Masuda, H. & Yoshida, N., 2005. "Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model," Stochastic Processes and their Applications, Elsevier, vol. 115(7), pages 1167-1186, July.
    • Handle: RePEc:eee:spapps:v:115:y:2005:i:7:p:1167-1186
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    References listed on IDEAS

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    1. Ole Barndorff-Nielsen & Elisa Nicolato & Neil Shephard, 2002. "Some recent developments in stochastic volatility modelling," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 11-23.
    2. Elisa Nicolato & Emmanouil Venardos, 2003. "Option Pricing in Stochastic Volatility Models of the Ornstein‐Uhlenbeck type," Mathematical Finance, Wiley Blackwell, vol. 13(4), pages 445-466, October.
    3. Dassios, Angelos & Jang, Jiwook, 2003. "Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity," LSE Research Online Documents on Economics 2849, London School of Economics and Political Science, LSE Library.
    4. Yuji Sakamoto & Nakahiro Yoshida, 2004. "Asymptotic expansion formulas for functionals of ε-Markov processes with a mixing property," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(3), pages 545-597, September.
    5. Ole E. Barndorff‐Nielsen & Neil Shephard, 2003. "Integrated OU Processes and Non‐Gaussian OU‐based Stochastic Volatility Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 30(2), pages 277-295, June.
    6. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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    Cited by:

    1. Keller-Ressel, Martin & Mijatović, Aleksandar, 2012. "On the limit distributions of continuous-state branching processes with immigration," Stochastic Processes and their Applications, Elsevier, vol. 122(6), pages 2329-2345.
    2. Yoshida, Nakahiro, 2023. "Asymptotic expansion and estimates of Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 176-248.
    3. Yoshida, Nakahiro, 2013. "Martingale expansion in mixed normal limit," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 887-933.
    4. Tudor, Ciprian A. & Yoshida, Nakahiro, 2023. "High order asymptotic expansion for Wiener functionals," Stochastic Processes and their Applications, Elsevier, vol. 164(C), pages 443-492.
    5. Masuda, Hiroki, 2007. "Ergodicity and exponential [beta]-mixing bounds for multidimensional diffusions with jumps," Stochastic Processes and their Applications, Elsevier, vol. 117(1), pages 35-56, January.
    6. Vyacheslav Abramov & Fima Klebaner, 2007. "Estimation and Prediction of a Non-Constant Volatility," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 14(1), pages 1-23, March.
    7. Honda, Tetsuhiro & Tamaki, Kenichiro & Shiohama, Takayuki, 2010. "Higher order asymptotic bond price valuation for interest rates with non-Gaussian dependent innovations," Finance Research Letters, Elsevier, vol. 7(1), pages 60-69, March.

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