On parameter estimation of Heston’s stochastic volatility model: a polynomial filtering method

Abstract

In this paper, we investigate the problem of estimating the volatility from the underlying asset price for discrete-time observations. This topic has attracted much research interest due to the key role of the volatility in finance. In this paper, we consider the Heston stochastic volatility model with jumps and we develop a new polynomial filtering method for the estimation of the volatility. The method relies on a linear filter which uses a polynomial state-space formulation of the discrete version of the continuous-time model. We demonstrate that a higher-order polynomial filtering method can be efficiently applied in the context of stochastic volatility models. Then, we compare our approach with some, well-established, techniques in the literature.

Appendix A

Kalman Filter The state-space form of a dynamical system is defined by the following pair of equations:

$$\begin{aligned} X_{t+1}= & {} A_t X_t+B_t+E_t \end{aligned}$$

(6.1)

$$\begin{aligned} Y_{t}= & {} C_t X_t+D_t+H_t \end{aligned}$$

(6.2)

where $X_t$ and $Y_t$ are vectors in $\mathbb {R}^p$ representing a discrete random variables on the same probability space $(\Omega ,\mathcal{F},\mathbb {P})$, endowed with a discrete filtration $\mathcal{F}_0,\;\mathcal{F}_1,\;,\ldots ,\mathcal{F}_N$. In general, the elements of $X_t$ are not observable, whereas the elements of $Y_t$ are observable. $E_t$ and $H_t$ are $\mathcal{F}_{t+1}$-adapted, square integrale, white noise processes. All vectors $B_t,\;D_t,\;E_t,\;H_t\in \mathbb {R}^p$ and matrices $A_t,\;C_t\in \mathbb {R}^{p\times p}$ can vary with t, but apart from $X_t$ and $Y_t$, that they only vary in a deterministic manner.

Furthermore, let denote $Q_{i,j,t}=\mathbb {E}\left[ E_{i,t}\cdot E_{j,t}\right]$ and $R_{i,j,t}=\mathbb {E}\left[ H_{i,t}\cdot H_{j,t}\right]$, for any $i,\,j=1,2,\ldots ,p$, and let $\mathcal{Y}_s:=[Y_0,\,Y_1,\ldots ,Y_{s}]\in \mathbb {R}^{p\times {s+1}}$, for $s=0,\ldots ,N$. By ${\hat{X}}_{t|s}$, we shall mean the best linear minimum variance estimate of $X_t$, given $\mathcal{Y}_s$. That is, it holds

$$\begin{aligned} \mathbb {E}\left[ |{\hat{X}}_{t|s}-X_{t}|^2\right] \le \mathbb {E}\left[ |L\mathcal{Y}_{s}-X_{t}|^2\right] , \end{aligned}$$

(6.3)

for all matrices $L\in \mathbb {R}^{p\times (p\times p)}$.

We also define the error matrix

$$\begin{aligned} P_{t|s}=\mathbb {E}\left[ (X_t-{\hat{X}}_{t|s})\cdot (X_t-{\hat{X}}_{t|s})^{\top }\right] \in \mathbb {R}^{p\times p}. \end{aligned}$$

(6.4)

When $s = t$, the estimate is called a filtered estimate and is denoted as ${\hat{X}}_t$, when $t > s$, the estimate is called a predicted estimate, and for $t<s$, the estimate is called a smoothed estimate. The discrete Kalman filter (1960) ${\hat{X}}_t$ may be generated recursively by the following relations:

$$\begin{aligned} \text{(predicted } \text{ state) }&{\hat{X}}_{t+1|t}=A_t {\hat{X}}_{t|t}+B_t, \nonumber \\ \text{(predicted } \text{ state } \text{ error } \text{ matrix) }&P_{t+1|t}=A_t P_{t|t} A_t^{\top }+Q_t, \nonumber \\ \text{(predicted } \text{ observation) }&{\hat{Y}}_{t+1|t} = C_{t+1} {\hat{X}}_{t+1|t}+D_{t+1}, \nonumber \\ \text{(predicted } \text{ obs } \text{ error) }&r_{t+1} = Y_{t+1}- {\hat{Y}}_{t+1|t}, \nonumber \\ \text{(predicted } \text{ obs } \text{ error } \text{ matrix) }&\Sigma _{t+1} = C_{t+1} P_{t+1|t}C^{\top }_{t+1}+R_{t+1} \nonumber \\ \text{(Kalman } \text{ gain) }&K_{t+1} = P_{t+1|t}C^{\top }_{t+1}\Sigma _{t+1}^{-1}, \nonumber \\ \text{(next } \text{ filtered } \text{ state) }&{\hat{X}}_{t+1|t+1}={\hat{X}}_{t+1|t}+K_{t+1}r_{t+1}, \nonumber \\ \text{(next } \text{ filtered } \text{ state } \text{ error } \text{ matrix) }&P_{t+1|t+1}=\left[ I-K_{t+1}C_{t+1}\right] P_{t+1|t}.\nonumber \\ \end{aligned}$$

(6.5)

If the initial state $X_0$ and the innovations $E_t$, $H_t$ are multivariate Gaussian, then the forecast ${\hat{X}}_{t|s}$ is the minimum variance estimator.

To run the Kalman filter, we begin with the pair ${\hat{X}}_{0|0}$, $P_{0|0}$. A difficulty with the Kalman filter is the determination of these initial conditions. In many real applications, the distribution for $X_0$ is unknown. Several approaches are possible. For example, one may treat $X_0$ as a fixed vector, $P_{0|0} = 0$, and estimate its components by treating them as extra parameters in the model.¹ The details are more involved. A general rule is that for long time series, the initial state conditions will have little impact.

If the initial state $X_0$ and the innovations $E_t$, $H_t$ are multivariate Gaussian, then the distribution of $Y_t$ conditional on the set $\mathcal{Y}_{t-1}$ is also Gaussian $N({\hat{Y}}_{t|t-1},\Sigma _t)$. Now let us suppose that the state-space vectors and matrices depend on certain unknown parameters, defined by a vector $\Theta$. We may form the log-likelihood function by taking the joint probability density function:

$$\begin{aligned} \log \ell (Y;\Theta )=-\frac{Np}{2}\log (2\pi )-\frac{1}{2}\sum _{t=1}^N \left[ \log (\det \Sigma _t)+r_t^{\top }\Sigma ^{-1} r_t^{\top }\right] . \end{aligned}$$

(6.6)

By maximizing $\log (\ell (Y; \Theta ))$ with respect to $\Theta$ for a particular realization of $Y=\{Y_0,Y_1,\ldots ,Y_N\}$, we obtain the maximum likelihood estimates for the parameters. Moreover, $\log (\ell (Y; \Theta ))$ may be computed via the Kalman filter, since the algorithm naturally computes both $r_t$ and $\Sigma _t$.

Proof of Theorem 3.1

Let $\mathcal{F}_T^R=\sigma \left( R_s:\;s\le T\right)$. Filtrations $\mathcal{F}_T^N=\sigma \Big (R_{t\Delta }: t=0,,1,\ldots N\Big )$, $\Delta =T/N$, are increasing with N, and they are certainly bounded by $\mathcal{F}_T^R$:

$$\begin{aligned} \mathcal{F}_T^N\subset \mathcal{F}_T^{N+1}\subset \cdots \subset \mathcal{F}_T^R. \end{aligned}$$

(6.7)

Let $\mathcal{F}_T^\star$ be the minimal $\sigma$-algebra generated by $\mathcal{F}_T^N$, for all $N\ge 1$. Since $\mathcal{F}_T^\star =\mathcal{F}_T^R$, it suffices to apply Corollary 2.4 (page 59) in Revuz and Yor (1991), to deduce that

$$\begin{aligned} \lim _{N\rightarrow \infty }\mathbb {E}\left[ g(V_T)\Big | \,\mathcal{F}_T^{N} \right] = \mathbb {E}\left[ g(V_T)\Big |\,\mathcal{F}_T^{\star }\right] =\mathbb {E}\left[ g(V_T)\Big |\,\mathcal{F}_T^{R}\right] , \end{aligned}$$

(6.8)

$\mathbb {P}$-a.s., for any bounded continuous function g. In order to prove that the conditional expectation in the right-hand side of (6.8) coincides with $g(V_T)$, we show that V is $\mathcal{F}^{R}$-adapted. In fact, the quadratic variation of R is given by

$$\begin{aligned}{}[R]_t=\int _0^t V_s ds+\sum _{\xi \le t}\left( \Delta R_\xi \right) ^2, \end{aligned}$$

(6.9)

and it holds $[R]_t=R_t^2-2\int _0^t R_{\xi ^-} dR_\xi$; hence, [R] is $\mathcal{F}^R$-adapted. From (6.9), we argue that the process $\int _0^{(\cdot )} V_s ds$ is $\mathcal{F}^{R}$-adapted, implying that V is $\mathcal{F}^{R}$-adapted. $\square$