Joint Estimation

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with 9,111.76, the SV-MJ model with 9,079.55, and then the SV model with 9,052.64.

This ranking result is consistent with AIC.

However, when we rank the order of model for BIC, we find that the SV-DEJ-VCJ model still have the highest BIC value 18,261.49, followed by the SV-MJ-VCJ model with 18,228.48, and then the SV-NIG model with 18,217.31. From this result, we find the SV-DEJ-VCJ model has the best fitting performance. However, when we consider the penalty with both the number of parameters and the sample size, the SV-NIG model have better performance than the SV-DEJ-VIJ model which is more complex than the SV-NIG model.

Overall, the SV-DEJ-VCJ model provides the best goodness-of-fit with S&P500 in-dex returns during this sample period. Interesting, we find that the finite jumps models with correlated volatility jumps (SV-MJ-VCJ and SV-DEJ-VCJ) have better fitting per-formance than infinite jumps models (SV-VG and SV-NIG), and then the finite jumps models (SV-MJ and SV-DEJ). This result is consisted with KS test statistics, QQ-plots, and kernel densities of those models. That is to say,the models with the finite retun jumps and the correlated volatility jumps perform well in capturing the left and right heavier tails.

7.2.3 Joint Estimation

Table 13 reports the results of dynamic joint estimation discussed in subsection 6.6 for each model. The sample period is rolling from January 1, 2005 to August 31, 2017 for the daily Standard & Poor’s 500 index (S&P500 index) and January 1, 2007 to August 31, 2017 for the weekly cross sectional S&P500 call options. The average parameters are reported first, followed by its standard errors in parentheses. Panel A of Table 13 reports the average parameters of stochastic volatility dynamic process of each model, the parameters are estimated on daily S&P500 index returns over the past two years of each trading date. Panel B and Panel C of Table 13 report the average parameters of return and volatility jump dynamic process of each model, the parameters are also estimated on daily S&P500 index returns over the past two years of each trading date. Panel D of Table 13 reports parameters of volatility and jump risk premiums of each model, the

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parameters are calibrated jointly through the S&P500 call option prices.

Moreover, the last five rows of Table 13 summarizes the log-likelihood values of fitting S&P500 index dynamic process (L_P), the log-likelihood values of fitting call option prices (L_Q), the joint log-likelihood values (L(Joint)) discussed in subsection 6.6, the Akaike information criterion values (AIC) and the Bayesian information criterion values (BIC) for each model. The bold font represents the top three log-likelihood function values of nine models, the last three AIC values of nine models, and the last three BIC values of nine models.

We summarize the empirical findings from Table 13 as follows. First, we look at the log-likelihood values of fitting S&P500 index dynamic process L_P, the SV-DEJ-VCJ model has the best performance with 1436.23, following by the SV-DEJ-VIJ model with 1426.92, and then the SV-MJ-VCJ model with 1425.02. That is, the models with the volatility jumps provide better fitting than those models without volatility jumps. This result is consisted with ranking of the log-likelihood values estimated under only the physical measure in Table 5. Second, under the risk-neutral measure, the log-likelihood values of fitting call option prices L_Q, the SV-DEJ-VIJ model has the best performance with 114.22, following by the SV-DEJ-VCJ model with 114.21, and then the SV-NIG model with 114.19. This result is slightly different to ranking of the mean of weekly RIVRMSE values in Panel A of Table 8. Third, we look at the joint log-likelihood values L(Joint) of fitting S&P500 index returns and call options simultaneously, the SV-DEJ-VCJ model has the best performance with 1554.61, following by the SV-DEJ-VIJ model with 1549.74, and then the SV-MJ-VCJ model with 1546.98. This result is also consisted with ranking of the log-likelihood values estimated under only the physical measure in Table 5.

However, when we use the Akaike information criterion (AIC) for model selection to solve this problem by introducing a penalty term for the number of parameters in the model, we find that the SV-DEJ-VCJ model still has the best performance with -3081.95, following by the SVDEJVIJ model with 3073.48, and then the SVNIG model with -3072.68. On the other hand, when we use the Bayesian information criterion (BIC) for

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model selection to solve this problem by introducing a penalty term for the number of parameters with sample size in the model, we find that the SV-NIG model become the best performance with -3029.44, following by the SV-DEJ-VCJ model with -3027.01, and then the SV-DEJ model with -3022.09. Interestingly, this sorting result of BIC is different to the Panel D of Table 5 estimated only under physical measure. That is, the SV-DEJ model become the best model, when we jointly estimate the parameters with S&P500 index returns and call options.

Figure 17, 18 and 19 report the time series of log-likelihood values in joint estimation for each model we discussed, respectively. Those figures show that the adaptability to daily S&P500 index returns and weekly cross sectional call options. The dotted vertical lines indicate the financial crisis (from August 2007 to June 2009) and the European sovereign debt crisis (from January 2010 to December 2012 ). Clearly, the joint log-likelihood value start to become lower than before after the September 25, 2008 during the financial crisis and the August 3, 2011 during the European sovereign debt crisis.

After the European sovereign debt crisis, the joint log-likelihood values recover to higher level than before. Those result show that the model fitting is better when the volatility of the S&P500 index returns is lower.

The volatility risk premium coefficient h₂ estimates in Panel D of Table 13 are all statistically significant for each model. We use the term h2× vt to represent the price of volatility risk premiums in Heston (1993), where v_t is stochastic volatility for each day t. Panel A of Table 14 report the annualized volatility risk premiums in each time inter-vals and the volatility risk premiums range from 0.0224 to 0.0511 percent. Note that we divide the total time period from January 1, 2007 to August 31, 2017 into five intervals.

The first time interval (January 2007-July 2007) is before the financial crisis. The sec-ond time interval (August 2007-June 2009) is during the financial crisis. The third time interval (July 2009-December 2009) is after the financial crisis and before the European sovereign debt crisis. The fourth time interval (January 2010-December 2012) is during the European sovereign debt crisis. The fifth time interval (January 2013-August 2017) is after the European sovereign debt crisis. It seems that the volatility risk premiums have increased significantly after the financial crisis (August 2007June 2009).

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Take the SV model as example, we find that the average annualized volatility risk pre-mium is 0.0255% during the second time interval (August 2007-June 2009), ,i.e., during the financial crisis. However, in the third time interval (July 2009-December 2009), the average annualized volatility risk premium become significantly higher than the second time interval to 0.1160%. Panel B of Figure 20 show that the volatility risk premiums of the SV model increase after the large volatility of S&P500 index returns during the financial crisis and decrease before the large volatility of S&P500 index returns during the European sovereign debt crisis. Interestingly, this result show that the investor require the higher volatility risk premium after the financial crisis rather than the financial crisis occurred presently.

On the other hand, the jump risk premium coefficient h3 estimates in Panel D of Table 13 are not all statistically significant for each model. We use the term M_J^P− M_J^Q to repre-sent the estimated jump risk premium in index returns for each model in Li et al. (2011), where M_J^P = E^P e^Jy
, M_J^Q= E^Q e^Jy
and J_y is return jump. Panel B of Table 14 report the annualized jump risk premiums in each time intervals. It seems that the jump risk premiums have increased significantly after the financial crisis (August 2007June 2009).

Take the SV-DEJ-VCJ model as example, we find that the average annualized jump risk premium is 0.0170% during the second time interval (August 2007-June 2009), i.e. during the financial crisis.

However, in the third time interval (July 2009-December 2009), the average annual-ized volatility risk premium become significantly higher than the second time interval to 0.110%. Panel D of Figure 24 show that the jump risk premiums of the SV-DEJ-VCJ model increase after the large volatility of S&P500 index returns during the financial crisis and decrease before the large volatility of S&P500 index returns during the Euro-pean sovereign debt crisis. Interestingly, this result show that the investor require the higher jump risk premium after the financial crisis rather than the financial crisis oc-curred presently. However, the jump risk premium of the SV-VG and SV-NIG models are smaller than other models which vary between 0.00171 to 0.0906 percent. That is, the infinite-activity jump model cannot capture the jump risk exactly, because most of the

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small jumps can be explained by the stochastic volatility which can capture the volatility risk well.