Heston (1993) assumed that the spot price follows the diffusion:
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 and the domestic and foreign interest rates are 5% and 3%, respectively, yielding a drift of
. The volatility in the GBM is constant
, while in Heston's model it is driven by the mean reverting process (7.2) with the initial variance
, the long term variance
, the speed of mean reversion
, and the vol of vol
. The correlation is set to
.
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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).
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Equations (7.1) and (7.2) define a two-dimensional stochastic process for the variables and
. By setting
, we can express it in terms of the centered (log-)return
and
. The process is then characterized by the transition probability
to have (log-)return
and variance
at time
given the initial return
and variance
at time
.
The time evolution of
is governed by the following Fokker-Planck (or forward Kolmogorov) equation:
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