7.2 Heston's Model

Heston (1993) assumed that the spot price follows the diffusion:

$\displaystyle dS_t = S_t\left(\mu\,dt+\sqrt{v_t}dW_t^{(1)}\right),$ (7.1)

i.e. a process resembling geometric Brownian motion (GBM) with a non-constant instantaneous variance $ v_t$. Furthermore, he proposed that the variance be driven by a mean reverting stochastic process of the form:

$\displaystyle dv_t=\kappa(\theta-v_t)\,dt+\sigma\sqrt{v_t}dW_t^{(2)},$ (7.2)

and allowed the two Wiener processes to be correlated with each other:

$\displaystyle dW_t^{(1)}dW_t^{(2)}=\rho\,dt.
$

The variance process (7.2) was originally used by Cox, Ingersoll, and Ross (1985) for modeling the short term interest rate. It is defined by three parameters: $ \theta$, $ \kappa$, and $ \sigma$. In the context of stochastic volatility models they can be interpreted as the long term variance, the rate of mean reversion to the long term variance, and the volatility of variance (often called the vol of vol), respectively.

Surprisingly, the introduction of stochastic volatility does not change the properties of the spot price process in a way that could be noticed just by a visual inspection of its realizations. In Figure 7.1 we plot sample paths of a geometric Brownian motion and the spot process (7.1) in Heston's model. To make the comparison more objective both trajectories were obtained with the same set of random numbers. Clearly, they are indistinguishable by mere eye. In both cases the initial spot rate $ S_0=0.84$ and the domestic and foreign interest rates are 5% and 3%, respectively, yielding a drift of $ \mu=2\%$. The volatility in the GBM is constant $ \sqrt{v_t}=\sqrt{4\%}=20\%$, while in Heston's model it is driven by the mean reverting process (7.2) with the initial variance $ v_0=4\%$, the long term variance $ \theta=4\%$, the speed of mean reversion $ \kappa=2$, and the vol of vol $ \sigma=30\%$. The correlation is set to $ \rho=-0.05$.

Figure 7.1: Sample paths of a geometric Brownian motion (dotted red line) and the spot process (7.1) in Heston's model (solid blue line) obtained with the same set of random numbers (left panel). Despite the fact that the volatility in the GBM is constant, while in Heston's model it is driven by a mean reverting process (right panel) the sample paths are indistinguishable by mere eye.
\includegraphics[width=0.7\defpicwidth]{STFhes01a.ps} \includegraphics[width=0.7\defpicwidth]{STFhes01b.ps}

A closer inspection of Heston's model does, however, reveal some important differences with respect to GBM. For example, the probability density functions of (log-)returns have heavier tails - exponential compared to Gaussian, see Figure 7.2. In this respect they are similar to hyperbolic distributions (Weron; 2004), i.e. in the log-linear scale they resemble hyperbolas (rather than parabolas).

Figure 7.2: The marginal probability density function in Heston's model (solid blue line) and the Gaussian PDF (dotted red line) for the same set of parameters as in Figure 7.1 (left panel). The tails of Heston's marginals are exponential which is clearly visible in the right panel where the corresponding log-densities are plotted.
\includegraphics[width=0.7\defpicwidth]{STFhes02a.ps} \includegraphics[width=0.7\defpicwidth]{STFhes02b.ps}

Equations (7.1) and (7.2) define a two-dimensional stochastic process for the variables $ S_t$ and $ v_t$. By setting $ x_t = \log(S_t/S_0) - \mu t$, we can express it in terms of the centered (log-)return $ x_t$ and $ v_t$. The process is then characterized by the transition probability $ P_t(x,v\,\vert\,v_0)$ to have (log-)return $ x$ and variance $ v$ at time $ t$ given the initial return $ x=0$ and variance $ v_0$ at time $ t=0$. The time evolution of $ P_t(x,v\,\vert\,v_0)$ is governed by the following Fokker-Planck (or forward Kolmogorov) equation:

$\displaystyle \frac{\partial}{\partial t}P$ $\displaystyle =$ $\displaystyle \kappa\frac{\partial}{\partial v}\left\{(v-\theta)P\right\}
+ \frac12\frac{\partial}{\partial x}(vP) +$  
    $\displaystyle + \, \rho\sigma\frac{\partial^2}{\partial x\,\partial v}(vP)
+ \f...
...l^2}{\partial x^2}(vP)
+ \frac{\sigma^2}{2}\frac{\partial^2}{\partial v^2}(vP).$ (7.3)

Solving this equation yields the following analytical formula for the density of centered returns $ x$, given a time lag $ t$ of the price changes (Dragulescu and Yakovenko; 2002):

$\displaystyle P_t(x) = \frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{i\xi x + F_t(\xi)} d\xi,$ (7.4)

with
    $\displaystyle F_t(\xi)
= \frac{\kappa\theta}{\sigma^2}\, \gamma t - \frac{2\kap...
...ega^2 -\gamma^2 + 2\kappa\gamma}{2\kappa\Omega}
\sinh\frac{\Omega t}{2}\right),$  
    $\displaystyle \gamma = \kappa + i\rho\sigma \xi, \quad \textrm{and} \quad
\Omega = \sqrt{\gamma^2 + \sigma^2(\xi^2-i\xi)}.$  

A sample marginal probability density function in Heston's model is illustrated in Figure 7.2. The parameters are the same as in Figure 7.1, i.e. $ \theta=4\%$, $ \kappa=2$, $ \sigma=30\%$, and $ \rho=-0.05$. The time lag is set to $ t=1$.