7.2 Estimated Parameters
7.2.1 Model Parameters and Latent Volatility/Jump Variables
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options. First, we estimate the parameters of stochastic volatility (SV) model with L´evy jumps type using the particle filtering (PF), smoothing filtering (SF) and expectation-maximization (EM) algorithm under the physical measures discussed in section 6.5. Table 5 reports the daily estimated parameters and the implied spot standard deviation under the physical probability measure P with respect to each model. Second, we use the joint estimation and the rolling time-window method to estimate the parameters with PF, SF and EM algorithm under the risk-neutral measures Q discussed in section 6.6.
7.2.1 Model Parameters and Latent Volatility/Jump Variables
Take SV-DEJ-VCJ as example in the Table 5, for the parameters of the dynamic process of log return, the instantaneous expected rates of daily return µ = 0.0342 percent is also known as trend factor of log returns which presents that the log returns increase year-by-year and close to the sample volatility mean of 0.0200 percent in Panel B of Table 3. The average upside exponential jump size is η+ = 0.0569 in the log asset returns, downside exponential jump size is η− = 0.0588, upside jump probability p = 0.456 and the jump frequency is λy = 0.0149. This means that about 3.7548 (≈ 0.0149 × 252) days jumps once year, that implies about 67 = (≈ 1/0.0149) days occur once jump and the left-skewed distribution of the return.
For the parameters of the dynamic process of the stochastic volatility, the long-term mean level is θ = 0.00726 percentage mean-reverting speed κ = 0.0343 tells us that the stochastic volatility will return to the long-term mean level needing about 9 (≈ 0.0343 × 252) days once year if the current volatility level diverge from the long-term mean level. The volatility of the variance is σv = 0.00269. The correlation coefficient of Wiener processes of return and volatility is ρ = −0.457 which has a significant lever-age effect. The averlever-age implied-standard-deviation is 18.4957 percent and is close to the sample volatility variance of 18.6583 = √
0.0159 × 252 percent in Panel B of Table 3.
The average of the jump amplitude of the stochastic volatility is µv = 8.23E − 05 and the correlation coefficient of the return jump size and volatility jump size is ρJ = −0.2420.
Overall, we observe parameters of the return variance (volatility) equation in Table 5.
Figure 3 and Figure 4 show that the latent stochastic volatility variables of all the nine
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models. These models are estimated by the particle filtering (PF), smoothing filtering and expectation-maximization (EM) algorithm with 300 particles for each time step from the corresponding model using parameters estimated in Table 5 and daily returns of the S&P500 index between January 1, 2007 and August 31, 2017, respectively. We find that the estimates of ρ for the nine models range from -0.421 to -0.523. This mean all nine models exhibit strong negative correlations between volatility and returns. The models share similar estimates of the volatility of return variance σv of the volatility processes range from 2.22E-03 to 2.69E-03. The models also share similar estimates of the long-run mean θ of the volatility processes range from 7.26E-05 to 1.71E-04.
The nine models also differ from each other. For example, the volatility process of the SV-DEJ-VCJ model has the strongest mean-reversion κ with 0.0343, followed by the SV-MJ-VCJ model with 0.0242, the SV-DEJ-VIJ model with 0.0218, the SV model with 0.0214, the SV-MJ-VIJ model with 0.0206, the SV-DEJ model with 0.0156, the SV-NIG model with 0.0137, the SV-MJ model with 0.0135, and finally the SV-VG model with 0.0127. Clearly, we find that the model obtain stronger mean-reversion κ than others seen from Figure 3 and Figure 4 when we add the correlated volatility jumps into the model.
Moreover, if we add the return jumps into the model without volatility jumps, the model have smaller κ than the SV model seen from Figure 3 and Figure 4. In other words, the return jumps can replace some part of return variance. Note that there are significant volatility clustering effects in the financial crisis period (from August 2007 to June 2009) and the European sovereign debt crisis period (from January 2010 to December 2012 ) from Figure 3 and Figure 4.
Next, we discuss the parameters of the compound Poisson returns (volatility) jump and the L´evy jump model. The estimated return (correlated volatility) jump intensi-ties λy(λ) for SV-MJ, SV-MJ-VIJ, SV-MJ-VCJ, SV-DEJ, VIJ, and SV-DEJ-VCJ models suggest that on average there are about 1.5674 (0.00622×252) to 3.5784 (0.0142×252) jumps per year. From Figure 8 and Figure 9 show that there are a few large return (correlated volatility) jump numbers in finite-activity SV-MJ, SV-MJ-VIJ, SV-MJ-VCJ, SV-DEJ, SV-DEJ-VIJ, and SV-DEJ-VCJ models. The estimated volatility jump intensities λv for SV-MJ-VIJ and SV-DEJ-VIJ suggest that on average there are
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about 0.97776 (0.00388×252) to 2.7468 (0.0109×252) jumps per year.
From Figure 11 show that there are a few volatility jump numbers in SV-MJ-VIJ and SV-DEJ-VIJ models. The return jump mean sizes of Merton jump are negative between -0.00726 to -0.0259 in Table 5. The return jump mean sizes of double exponential jump are negative between -0.0056 to -0.0069 in Table 5. From Figure 5 and Figure 6 show that there are a few large return jump sizes in returns in finite-activity SV-MJ, SV-MJ-VIJ, SV-MJ-VCJ, SV-DEJ, SV-DEJ-VIJ, and SV-DEJ-VCJ. The volatility jump mean sizes are between 8.23E-05 to 7.74E-04 in Table 5. From Figure 10 show that there are a few volatility jump sizes in SV-MJ-VIJ, SV-MJ-VCJ, SV-DEJ-VIJ, and SV-DEJ-VCJ models.
On the other hand, estimated return jump sizes for SV-VG and SV-NIG suggest that there are many frequent small jump and a few large jump in returns. The return jump mean sizes of L´evy jump model SV-VG and SV-NIG are 5.50E-05 and 5.22E-05 in Table 5. From Figure 7 show that there are many small return jump sizes in infinite-activity models SV-VG and SV-NIG. Note that there are significant volatility clustering effects with a few large jumps in the financial crisis period (from August 2007 to June 2009) and the European sovereign debt crisis period (from January 2010 to December 2012 ) from Figure 5, Figure 6 and Figure 7.
Panel A of Figure 5 show that the SV-MJ model has the largest jump magnitude about -9.502 percent in September 29, 2008 during the 2008 financial crisis and -4.846 percent in August 9, 2011 during European sovereign debt crisis. However, we can find that the SV-MJ model has difficulty capturing large positive and negative returns simul-taneously in Panel A of Figure 5. This finding is likely due to the jump structure of Merton jump. Because the jumps in returns tend to have a negative mean seen in Table 5 (see Eraker et al. 2003, Li et al. 2008), a normal distribution which is used to model the jump sizes has to shift its location position and this result may lead to nonmonotone density for negative jumps and small density for positive jumps. The nonmonotonicity might be problematic in modeling small jumps and small density for positive jumps might lead difficultly to capture large negaitive jump and positive jump simultaneously.
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On the other hand, Panel B of Figure 6 show that the SV-DEJ model can simultane-ously capture large negative return jump magnitude about -6.058 percent in September 29, 2008 and positive return jump magnitude about 4.069 percent in October 13, 2008 during the 2008 financial crisis. This result may come from that the heavy-tail feature of the double-exponential distribution lead to a better performance than the model with Merton jumps. However, Figure 6 show that the SV-VG and the SV-NIG models have not significantly capturing the large and small jumps simultaneously. This result might actually stem from the fact that the most of the small return variations are explaimed by the stochastic volatility. Thus, there is a issue for return variations how much per-centage of return variations is caused by return jumps. First, we set return rt to follow the discrete-time version of the dynamic process
rt = where ∆ is the time interval, µ is the drift term. MJPy is the compensator of the return jump and the average jump amplitude, i.e., MJPy = EP (Jy,t). vtis the stochastic volatility.
εPy,t is the Wiener process which follow a standard normal distribution. Jy,tP is the jump sizes which is a random variable. Thus, rt follow a normal distribution given by
rt∼ N With the return mean is
E (rt) =
And the return variation is
V ariance (rt) = vt−1∆ +
Jy,tP − MJPy∆
2
(7.4) Thus, we can divide return variation into two parts vt−1∆ and
Jy,tP − MJP
y∆2
. Then, we can calculate how much the percentage of return jump account for the return variation P J Vt and the mean of P J Vt, M P J V . That is,
Panel D of Table 5 show that all the M P J V of models are smaller than 0.6. Moreover, Panel A of Figure 12 show that the P J Vt of the SV-MJ model up to 62.12 percent in
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September 15, 2008 and 91.10 percent in September 29, 2008 during the early stages of a financial crisis. After these dates, the P J Vt of the SV-MJ model declined by 100 percent into zero percent. That is to say, most of the return variations are caused by the stochastic volatility and return jumps only affect at the beginning period of the 2008 financial crisis.
However, Panel B and Panel C of Figure 14 show that the P J Vt of the SV-VG and the SV-NIG models approximate only 0.0844 percent and 21.26 percent in September 15 and 17, 2008, respectively. In other words, the return variation is explained mostly by the stochastic volatility when we use the SV-VG and the SV-NIG model. Although the infinite-activity model can fit small jumps in the equity index returns well, in our empirical results report that most of the small jumps of returns are captured by the stochastic volatility. As a result, the infinite-activity models does not significantly fit better than finite-activity models. Thus, it is hard to use the L´evy-type jumps models to fit large and small jumps simultaneously when we consider the stochastic volatility to the dynamic process of return.