4. Empirical Results
4.2 Data and Model Parameter Estimation
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changes to the Brownian motions (with drift), the proposed asset return model emphasizes that stochastic asymmetric volatility is driven by multiple sources of random shock.
According to the general model specifications, the extended model of Carr et al. (2002) with a diffusion is adopted as a special case when the stochastic time change is constant, while remains random as proposed by Madan and Yor (2008), and asymmetric volatili-ty(e.g. leverage effect and volatility feedback effect) is ignored. On the other hand, VG model is also my special case similar to those of Carr et al. (2002), but evolves like Gamma process.
Formally, this work permits the separate treatment of the pure-continuous and infinite-jump time-changed Lévy processes to enable the generation of stochastic asymmetric volatility via both components.
4.2 Data and Model Parameter Estimation
This study uses the S&P 500 index data from January 1, 1980 to June 30, 2008 at two different frequencies, daily and weekly, to estimate the proposed general stochastic asymmetric volatility asset return model, based on the characteristic function with spectral GMM estimation proposed by Chacko and Viceira (2003).
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Figure12: Daily S&P index log-return, from January 1, 1980 to June 30, 2008.
Numerous studies have estimated the parameters of the stochastic volatility model through the characteristic function methods, such as Singleton (2001), Jiang and Knight (2002) and Chacko and Viceira (2003). Among these studies, spectral GMM estimation is particularly suitable for time-changed Lévy processes because of its direct use of characteristic function without inver-sion to recover the density function and closed form of the return moment. Generally, few esti-mation methods using the traditional maximum likelihood are easy to be use because no analyt-ical expression is known for the density function of the time-changed Lévy process, as the mod-el presented here. Therefore, to respond to the above difficulty, this study employs spectral GMM to estimate the parameters of the general stochastic asymmetric volatility asset return
1000 2000 3000 4000 5000 6000 7000
-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05
days
log-return
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model.
First, the conditional characteristic function , ; , log must be defined as follows:
, ; , log , log
cos log , log sin log , log (22)
where is a vector of parameters of the general asset return model, 0. The Eqn. (22) de-notes the Euler expansion of exponential of complex variable. The definition of the conditional characteristic function implies
, ; , log 0 (23) for all . The Eqn. (23) defines an infinite set of complex-valued moment conditions. The following pair of real-valued moment conditions can be induced:
Re exp log , ; , log 0
Im exp log , ; , log 0 (24)
where Re · and Im · are real-valued operators that extract the real and imaginary parts of the complex number. The r-dimensional vector of instrument log , orthogonal to , ; is given as follows:
log , , ; 0
where , ; exp log , ; , log . Furthermore, the above
ex-pression can be considered the complex-valued moment conditions implying the following pair of real-valued moment conditions:
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Re log , , ; 0
Im log , , ; 0
Choosing a set of appropriate points for , , … , , the pair of real-valued moment conditions can be obtained as
; log 0
where ; log , denotes a 2 1 vector of moment condition satisfying
; Re log , , ;
Im log , , ;
and , ; , ; , , ; , … , , ; is an n-dimensional vector orthogonal to log , .
Based on the above discussion, Chacko and Viceira (2003) treat above transformation as the standard procedure involved in the GMM estimation. Given a set of index data observed at a discrete time, the correspondingsample moment conditions is denoted as
; log , 1
; log
Spectral GMM estimation identifies values of that make the sample moment condition as close to zero as possible, based on the following quadratic form:
SGMM arg min ; log , ; log , ; log ,
where ; log , denotes a positive-definite, symmetric weighting matrix. Similar to the properties of GMM, the asymptotic variance of the spectral GMM estimator is minimized when
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choosing the following weighting matrix:
; log ,
where ; log , ; log , . In fact, I can find any consistent es-timator to replace the ; log , .
The conditional characteristic function is defined as follows:
, ; , log , , log , 1,2.
Next, this study integrates the unobservable state variable in the asset return model out of the above conditional characteristic function to yield the characteristic function based only on cur-rent asset price. The conditional characteristic function should be provided to estimate the pa-rameters of the proposed general asset return model via the Kolmogorov backward equation11. The conditional characteristic functions of the pure-continuous and infinite-jump stochastic time-changed asset price models are prioritized in calibrating the general stochastic asymmetric volatility asset return model.
Proposition 4: If the asset percentage return is introduced as Eqn. (7), then the conditional cha-racteristic function , ; , log satisfies the following analytic expression:
log , ; , log log , ; log , ; (25)
where
11 Further details of the Kolmogorov (forward-) backward equation and their applications are presented in Ma and Yong (1999).
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, , , , , ;
, ; log
2 2
Proof: See Appendix.
Proposition 5: If the asset percentage return is introduced as Eqn. (8), then the conditional cha-racteristic function , ; , log , satisfies the following analytic expression:
log , ; , log , log
2 Furthermore,
log , ; , log log log
, ; (26) where
, , , , , ,
, ; 2
Proof: Refer to Cont and Tankov (2004, p. 115) and the Appendix of the Proposition 4.
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The results of Propositions 4 and 5 easily lead to the following expression regarding the con-ditional characteristic function of the general time-changed asset price dynamics with asymme-tric volatility.
Proposition 6: If the asset price dynamics is given as Eqn. (21), then the conditional characte-ristic function , ; , log satisfies the following analytic expression:
log , ; , log log log , ; , ;
log , ; (27) where
, , , , , , , , , ,
, ;
, ; log
, ; 2
With the closed-form conditional characteristic functions as described above, spectral GMM estimation can be applied in the proposed general stochastic asymmetric volatility asset return
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Table 1: The spectral GMM Estimates of the General Continuous-time Time-changed Asymme-tric volatility Model
Model parame-ters
Daily Data Weekly Data
Estimates SE p-value Estimates SE p-value
0.811855** 0.036300 0.000000 0.942102** 0.014586 0.000000 82.615909** 30.721756 0.007179 66.147332** 15.646848 0.000025
0.767493** 0.367373 0.036729 0.754793 0.461611 0.102236
-0.348467** 0.004590 0.000000 -0.434598** 0.133070 0.001116
0.829387** 0.371334 0.025543 0.772264* 0.412224 0.061210
63.253463** 0.538977 0.000000 48.152096** 20.409658 0.018440 74.069214** 10.143394 0.000000 101.865132** 27.397853 0.000208
0.539658** 0.058467 0.000000 0.880970** 0.152962 0.000000 0.446344** 0.075333 0.000000 0.421849** 0.082872 0.000000
0.013523** 0.003640 0.000204 0.057222* 0.033777 0.090460
0.078087** 0.022087 0.000410 0.073922** 0.025903 0.004381
*denote an estimate that is significant at 95% level.
** denote an estimate that is significant at 90% level.
Table 1 lists the model parameter estimates, together with their standard errors (SE) and p-values. This table clearly indicates that stochastic asymmetric volatility and infinite-jump structure need to be included to describe the asset price dynamics. The coefficients for the pure-continuous and infinite-jump time-changed components are statistically significant in daily
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frequency and are almost statistically significant in weekly frequency. This work evidently re-jects the assumption that index dynamics lack a diffusion term, an assumption that may be present in the dynamics of individual assets. The estimation results demonstrate that the diffu-sion term is an essential determinant of financial modeling in situations where involve infi-nite-activity jump structure affecting asset returns. In the finance literature, Huang and Wu (2004) show that both infinite-activity jump structure and stochastic volatility should be in-cluded in option pricing models. The empirical findings of this study resemble those of Huang and Wu (2004) in that the diffusion term is necessary for the time-changed Lévy process model when generating dependence on the diffusive state variable. Meanwhile, my findings promote the belief that the diffusion term in the asset return model is still required if one state variable follows a pure-jump process and the correlation between asset return and changes in volatility is established.
The parameter is both less than one at two different frequencies, satisfying the required condition 0,1 for the subordinating process. Both the stochastic time changes, and , obey the dampened power law (see Wu (2006)). Notice that the dam-pening parameter in the weekly data is less than the daily data. The influence of information arrival describing changes in the financial market is strengthened relative to daily frequency, thus implying the frequency of large jumps of index process in weekly is high. The findings are intuitively consistent with what are expected.
Additionally, the correlation coefficients, and , capture the leverage effect for the rela-tionship between asset returns and changes in asset volatility and skewness in the distribution of
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returns. Table 1 clarifies this relationship. The estimate for is negative while that for is positive. One explanation for this is that the stochastic time change in the diffusion component mainly produces the positive leverage effect, the well-known stylized fact documented in the empirical literature. This inference is accord with the findings of present study regarding the implications of asymmetric volatility for optimal portfolio choices.
The estimates for , the rate of mean reversion of the diffusion component, are much lower than those for , the rate of the mean reversion of jump component, for both daily and weekly return data. This shows that the volatility feedback effect resulting from the diffusion component probably has the profound effect upon asymmetric volatility. Furthermore, this study enhances the previous studies’ findings, such as Chacko and Viceira (2003) and Chernov et al. (2003), based on the fact that correlation coefficient related to volatility term decisively contributes to explain negative skewness in asset price dynamics.
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