Bayesian estimation of the stochastic volatility model with double exponential jumps

Abstract

This paper generalizes the stochastic volatility model to allow for the double exponential jumps. To derive the jumps and time-varying volatility in returns, we implement an efficient Markov chain Monte Carlo approach based on the band and sparse matrix algorithms used in Chan and Hsiao (SSRN Electron J., 2013, https://doi.org/10.2139/ssrn.2359838) to estimate this model. We illustrate the the methodology using the daily data for the Shanghai Composite Index, Hangseng Index, Nikkei 225 Index and Kospi Index. We find that the stochastic volatility model with double exponential jumps provide better fitness in sample period.

Appendix A

In this section, we focus on sampling from the conditional posterior $p(h|\Theta ,J,Z,Y)$ following Chan and Hsiao (2013). Since

$$\begin{aligned} p(h|\Theta ,J,Z,Y)\propto p(Y|\Theta , h, J, Z)p(h|\Theta ), \end{aligned}$$

(13)

we first derive the conditional densities $p(Y|\Theta , h, J, Z)$ and $p(h|\Theta )$. Equation (2) can be written as

$$\begin{aligned} h_t-\phi _hh_{t-1}=(1-\phi _h)\mu _h+\sigma _h\varepsilon _t, \end{aligned}$$

(14)

and let $H_{\phi _h}$ be a lower triangular matrix:

$$\begin{aligned} H_{\phi _h}= \left( \begin{array}{ccccc} 1 &{} 0 &{} 0 &{} \ldots &{}0\\ -\phi _h &{} 1 &{} 0 &{} \ldots &{}0\\ 0&{} -\phi _h &{} 1 &{} \ldots &{}0\\ \ldots &{} \ldots &{} \ldots &{} \ldots &{} \ldots \\ 0 &{} 0 &{} \ldots &{} -\phi _h &{} 1 \end{array} \right) . \end{aligned}$$

Then (14) can be expressed by matrix:

$$\begin{aligned} H_{\phi _h}h=\tilde{\delta _h}+\xi _t, \xi _t\sim N(0,\Sigma _h), \end{aligned}$$

(15)

where $\tilde{\delta _h}=(\mu _h, (1-\phi _h)\mu _h, \ldots , (1-\phi _h)\mu _h)'$, $\Sigma _h=\mathrm{diag}(\frac{\sigma _h^2}{1-\phi _h^2}, \sigma _h^2, \ldots , \sigma _h^2).$ Hence we have $(h|\Theta )\sim N(\delta _h, (H_{\phi _h}'\Sigma _h^{-1}H_{\phi _h})^{-1})$, where $\delta _h =H_{\phi _h}^{-1}\tilde{\delta _h}$. Then its log-density is

$$\begin{aligned} \log p(h|\Theta )=-\frac{1}{2}(h'H_{\phi _h}'\Sigma _h^{-1}H_{\phi _h}h -2h'H_{\phi _h}'\Sigma _h^{-1}H_{\phi _h}\delta _h)+c_1, \end{aligned}$$

(16)

where $c_1$ is a constant independent of h. On the other hand,

$$\begin{aligned} \log p(Y|\Theta , h, J, Z)=\sum \limits _{t=1}^T \log p(y_t|\Theta , h_t, J_t, Z_t) \end{aligned}$$

(17)

and

$$\begin{aligned} \log p(y_t|\Theta , h_t, J_t, Z_t)=-\frac{1}{2}\log (2\pi )-\frac{1}{2}h_t-\frac{1}{2}e^{-h_t}(y_t-\mu -J_tZ_t)^2. \end{aligned}$$

(18)

We can compute

$$\begin{aligned}&f_t=\frac{\partial \log p(y_t|\Theta , h_t, J_t, Z_t)}{\partial h_t}|_{h_t ={\tilde{h}}_t}=-\frac{1}{2}+\frac{1}{2}e^{-{\tilde{h}}_t}(y_t-\mu -J_tZ_t)^2,\\&G_t=-\frac{\partial ^2\log p(y_t|\Theta , h_t, J_t, Z_t)}{\partial h_t^2}|_{h_t={\tilde{h}}_t}=\frac{1}{2}e^{-{\tilde{h}}_t}(y_t-\mu -J_tZ_t)^2, \end{aligned}$$

where ${\tilde{h}}_t$ is chosen to be the mode of $p(h|\Theta ,J,Z,Y)$. Then the second order Taylor expansion of $\log p(y_t|\Theta , h_t, J_t, Z_t)$ at point ${\tilde{h}}$ is

$$\begin{aligned} p(Y|\Theta , h, J, Z)\approx & {} \log p(Y|\Theta , {\tilde{h}}, J, Z) +(h-{\tilde{h}})'f-\frac{1}{2}(h-{\tilde{h}})'G(h-{\tilde{h}})\nonumber \\= & {} \frac{1}{2}(h'Gh-2h'(f+G{\tilde{h}}))+c_2, \end{aligned}$$

(19)

where $c_2$ is a constant independent of h, $f=(f_1, \ldots , f_T)'$, and $G=(G_1, \ldots , G_T)'$. Lastly, combining (16) and (19), we have

$$\begin{aligned} \log p(h|\Theta ,J,Z,Y)= & {} \log p(Y|\Theta , h, J, Z)+\log p(h|\Theta )\nonumber \\= & {} -\frac{1}{2}(h'K_hh-2h'k_h)+c_3, \end{aligned}$$

(20)

where $c_3$ is a constant independent of h, $K_h=H_{\phi _h}'\Sigma _h^{-1}H_{\phi _h}+G$, and $k_h=f+G{\tilde{h}}+H_{\phi _h}'\Sigma _h^{-1}H_{\phi _h}\delta _h$. Therefore, $h|_{\Theta ,J,Z, Y}\sim N({\hat{h}}, K_h^{-1})$, where ${\hat{h}}=K_h^{-1}k_h.$ Because $K_h$ is a band matrix, ${\hat{h}}$ can be computed quickly.

The estimations of other parameters $(\mu , \mu _h, \phi _h,\sigma _h^2)$ are the same as those of the standard stochastic volatility model. Their posterior distributions are as follows:

$\bullet$ Posterior of $\mu$. $\mu \sim N({\hat{\mu }}, D_{\mu })$, where

$$\begin{aligned}&{\hat{\mu }}=D_{\mu }\left( \frac{\mu _0}{V_{\mu }}+\sum \limits _{t=1}^T \frac{y_t-J_tZ_t}{e^{h_t}}\right) ,\\&D_{\mu }=\left( \frac{1}{V_{\mu }}+\sum \limits _{t=1}^T e^{-h_t}\right) ^{-1}. \end{aligned}$$

$\bullet$ Posterior of $\mu _h$. $\mu _h\sim N({\hat{\mu }}_h, D_{\mu _h})$, where

$$\begin{aligned}&{\hat{\mu }}_h=D_{\mu _h}\left( \frac{\mu _{h0}}{V_{\mu _h}}+X_{\mu _h}'\Sigma _h^{-1} Z_{\mu _h}\right) ,\\&D_{\mu _h}=\left( \frac{1}{V_{\mu _h}}+X_{\mu _h}'\Sigma _h^{-1}X_{\mu _h}\right) ^{-1},\\&X_{\mu _h}=(1,1-\phi _h, \ldots ,1-\phi _h)',\\&Z_{\mu _h}=(h_1, h_2-\phi _hh_1, \ldots , h_T-\phi _hh_{T-1})'. \end{aligned}$$

$\bullet$ Posterior of $\phi _h$. $\phi _h\sim N(\hat{\phi _h}, D_{\phi _h})1(|\phi _h|<1)$, where

$$\begin{aligned}&\hat{\phi _h}_h=D_{\phi _h}(\frac{\phi _{h_0}}{V_{\phi _h}} +\frac{X_{\phi _h}'Z_{\phi _h}}{\sigma _h^2}),\\&D_{\phi _h}=(\frac{1}{V_{\phi _h}}+\frac{X_{\phi _h}'X_{\mu _h}}{\sigma _h^2})^{-1},\\&X_{\phi _h}=(h_1-\mu _h, \ldots ,h_{T-1}-\mu _h)',\\&Z_{\mu _h}=(h_2-\mu _h, \ldots ,h_{T}-\mu _h)'. \end{aligned}$$

$\bullet$ Posterior of $\sigma _h^2$. $\sigma _h^2\sim IG(\nu +\frac{T}{2}, {\hat{s}})$, where

$$\begin{aligned} {\hat{s}}=s+\frac{1}{2}(h_1-\mu _h)^2(1-\phi _h^2)+\frac{1}{2}\sum \limits _{t=2}^T(h_t-\phi _hh_{t-1}-(1-\phi _h)\mu _h)^2. \end{aligned}$$