TY - JOUR
T1 - Asymptotic expansion for Barndorff-Nielsen and Shephard's stochastic volatility model
AU - Masuda, Hiroki
AU - Yoshida, Nakahiro
N1 - Funding Information:
This work was in part supported by Cooperative Research Program 16-2002 of the Institute of Statistical Mathematics, and 21st Century COE Program “Development of Dynamic Mathematics with High Functionality” of Kyushu University.
PY - 2005/7
Y1 - 2005/7
N2 - 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 ℝ+. 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.
AB - 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 ℝ+. 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.
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U2 - 10.1016/j.spa.2005.02.007
DO - 10.1016/j.spa.2005.02.007
M3 - Article
AN - SCOPUS:20344381297
VL - 115
SP - 1167
EP - 1186
JO - Stochastic Processes and their Applications
JF - Stochastic Processes and their Applications
SN - 0304-4149
IS - 7
ER -