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plt.show()
Figure 6.1: Different realisations of the exponential distribution for various parameters λ.
Note that a much larger parameter value is chosen for σ than ν because there is a lot of
initial uncertainty associated with the scale of the volatility generating process. As more data
is provided to the model the Bayesian updating process will reduce the spread of the posterior
distribution reflecting an increased certainty in the scale factor of volatility.
This stochastic volatility model makes use of a random walk model for the latent volatility
variable. Random walk models are discussed in significant depth within the subsequent time
series chapter on White Noise and Random Walks. However the basic idea is that the latent
volatility at time point i, namely s i is a function only of the previous time point value s i−1 along
with some normally distributed error. In probabilistic terms this is written as:
s i ∼ N (s i−1 , σ −2 ) (6.3)
That is, the value of the latent vol at i, s i , has a normally distributed prior centred at the
previous value (s i−1 ) with variance σ −2 . This variance, as was seen above, is distributed as an
exponential distribution.
It remains only to assign a prior to the logarithmic returns of the asset price series being
modelled. The point of a stochastic volatility model is that these returns are related to the
underlying latent volatility variable. Hence any prior that is assigned to the log returns must
have a variance that involves s.
One approach (utilised in the PyMC3 tutorial) is to assume that the log returns, log(y i /y i−1 )
are distributed as a Student’s t-distribution, with mean zero and variance given as the exponential
of negative of the latent vol variable. Figure 6.2 shows how the PDF of a Student’s t-distribution