Simulated likelihood inference for stochastic volatility models using continuous particle filtering

Abstract

Discrete-time stochastic volatility (SV) models have generated a considerable literature in financial econometrics. However, carrying out inference for these models is a difficult task and often relies on carefully customized Markov chain Monte Carlo techniques. Our contribution here is twofold. First, we propose a new SV model, namely SV–GARCH, which bridges the gap between SV and GARCH models: it has the attractive feature of inheriting unconditional properties similar to the standard GARCH model but being conditionally heavier tailed. Second, we propose a likelihood-based inference technique for a large class of SV models relying on the recently introduced continuous particle filter. The approach is robust and simple to implement. The technique is applied to daily returns data for S&P 500 and Dow Jones stock price indices for various spans.

Appendices

Appendix A

Appendix A deals with the specific implementation of the particle filter for the case of leverage and jumps. This relates to Sect. 3.3. Specifically, we are concerned with Step (1a) of Algorithm: PF. We describe how to sample continuously (via inversion of the cumulative distribution function) from the mixture,

$$\begin{aligned} f(\epsilon _{t}|h_{t},y_{t})=\sum \limits _{j=0}^{1}f(\epsilon _{t} |J_{t}=j;h_{t},y_{t})\,\hbox {Pr}(J_{t}=j|h_{t},y_{t}) \end{aligned}$$

where the conditional probability of a jump is given by

$$\begin{aligned} \hbox {Pr}(J_{t}=1|h_{t},y_{t})&= \frac{\hbox {Pr}(y_{t}|h_{t},J=1)\hbox {Pr}(J=1)}{\hbox {Pr}(y_{t}|h_{t},J=1)\hbox {Pr}(J=1)+\hbox {Pr}(y_{t}|h_{t},J=0)\hbox {Pr}(J=0)},\\&= \frac{\text {N}(y_{t}|0;\exp (h_{t})+\sigma _{J}^{2})p}{\text {N} (y_{t}|0;\exp (h_{t})+\sigma _{J}^{2})p+\text {N}(y_{t}|0;\exp (h_{t}))(1-p)}. \end{aligned}$$

and

$$\begin{aligned} f(\epsilon _{t}|J=1;h_{t},y_{t})\propto f(y_{t}|J=1,h_{t},\epsilon _{t})f(\epsilon _{t}). \end{aligned}$$

As we have $f(\epsilon _{t}|J=1;h_{t},y_{t})\propto$ N$(y_{t}|\epsilon _{t} \exp (h_{t}/2);\sigma _{J}^{2})\times$N$(\epsilon _{t}|0;1)$, it follows that

$$\begin{aligned}&f(\epsilon _{t}|J_{t}=1;h_{t},y_{t})=\text {N}\left( \upsilon _{\epsilon _{1} },\sigma _{\epsilon _{1}}^{2}\right) \quad \text {where }\upsilon _{\epsilon _{1}} =\frac{y_{t}\exp (h_{t}/2)}{\exp (h_{t})+\sigma _{J}^{2}} \,\, \nonumber \\&\quad \text { and } \sigma _{\epsilon _{1}}^{2}=\frac{\sigma _{J}^{2}}{\exp (h_{t})+\sigma _{J}^{2}}. \end{aligned}$$

If the process does not jump, there is a Dirac-delta mass at the point $\epsilon _{t}=y_{t}\exp (-h_{t}/2)$. We therefore have the expression (14). If we denote $p_{t}^{*}\equiv \hbox {Pr}(J_{t}=j|h_{t},y_{t}),$ then this mixture is

$$\begin{aligned} f(\epsilon _{t}|h_{t},y_{t})=(1-p_{t}^{*})\,\,\delta _{y_{t}\exp (-h_{t}/2)}\left( \epsilon _{t}\right) +p_{t}^{*}\text { N}\left( \epsilon _{t}|\upsilon _{\epsilon _{1}},\sigma _{\epsilon _{1}}^{2}\right) . \end{aligned}$$

(15)

We may invert the corresponding distribution function $F(\epsilon _{t} |h_{t},y_{t})$ straightforwardly allowing for draws which are continuous as a function of our parameters.

Assume we have generated a uniform random variate $U\sim \text {UID}(0,1)$. We show how to generate a single sample $\epsilon _{t}=F^{-1}(U|h_{t},y_{t})$ accordingly, where $\epsilon _{t}^{*}=y_{t}\exp (-h_{t}/2)$,

$$\begin{aligned} K=\Phi \left( \frac{\epsilon _{t}^{*}-\upsilon _{\epsilon _{1}}^{1}}{\sigma _{\epsilon _{1}}^{1}}\right) p_{t}^{*}, \end{aligned}$$

$p_{t}^{*}\equiv \hbox {Pr}(J_{t}=j|h_{t},y_{t})$ again, and $\Phi (.)$ denotes the standard normal distribution function. The following scheme is applied:

• If $\ U\le K$, set $\epsilon _{t}=\upsilon _{\epsilon _{1}} +\sigma _{\epsilon _{1}}\Phi ^{-1}\left( \frac{u}{p_{t}^{*}}\right)$.

• If $K<U\le K+(1-p_{t}^{*}),\$ set $\epsilon _{t}=y_{t}\exp (-h_{t}/2)$.

• If $U>K+(1-p_{t}^{*}),\$ set $\epsilon _{t}=\upsilon _{\epsilon _{1} }+\sigma _{\epsilon _{1}}\Phi ^{-1}\left( \frac{U-(1-p_{t}^{*})}{p_{t}^{*}}\right)$.

The above probability integral transform procedure is repeated for each of the uniform $u_{1},\ldots ,u_{M}$ to obtain the required sample $\epsilon _{t}^{i}\,\sim f(\epsilon _{t}^{i}|h_{t}^{i},y_{t}),\,i=1,\ldots ,M$.

Appendix B

Particle filter estimation of SV–GARCH model

We start at $t=0$ with samples from the stationary distribution of GARCH, $v_{0}^{i}\sim f(v_{0}),\,i=1,\ldots ,M$.

Algorithm : PF for $t=0,\ldots ,T-1$:

We have samples $v_{t}^{i}\sim f(v_{t}|Y_{t})$ for $i=1,\ldots ,M$.

• 1. For $i=1:M$, sample $\widetilde{v}_{t+1}^{i}\sim f(v_{t+1}|v_{t}^{i}).$

• 2. For $i=1:M$ calculate normalized weights,

  $$\begin{aligned} \lambda _{t+1}^{i}&= \frac{\omega _{t+1}^{i}}{\sum _{k=1}^{M}\omega _{t+1}^{k} },\quad \text { where }\omega _{t+1}^{i}=f\left( y_{t+1}|\widetilde{v}_{t+1}^{i}\right) \nonumber \\&= \left\{ 2\pi \widetilde{v}_{t+1}^{i}\right\} ^{-\frac{1}{2}}\exp \left( -\frac{1}{2}\frac{y_{t+1}^{2}}{\sqrt{\widetilde{v}_{t+1}^{i}}}\right) . \end{aligned}$$

• 3. For $i=1:M$, sample $v_{t+1}^{i}\sim \sum _{k=1} ^{M}\lambda _{t+1}^{k}\delta _{\widetilde{v}_{t+1}^{k}}(v_{t+1})$. As in the case of SV with leverage and jumps, we replace Step 3 with the continuous resampling scheme described in Malik and Pitt (2011). Parameters of the SV–GARCH $\theta =(\mu ,\alpha ,\beta ,\varphi )$ can be estimated by maximizing the simulated log-likelihood function.