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stochastic differential equations (SDE). Using the same notation from that book the model is
given below:
dS t = µS t dt + √ ν t S t dW S t (6.1)
dν t = κ(θ − ν t )dt + ξ √ ν t dW ν t (6.2)
This model says that the change in the volatility ν is a mean-reverting process, with its own
variance given by ξ. Thus the volatility has a mean of κθ but can take on more extreme values
occasionally.
The asset paths themselves, given by S t , follow a Geometric Random Walk model with the
variance given by the square root of the volatility process.
In this section a simpler stochastic volatility model will be provided that does not assume
mean reversion of volatility. Instead the volatility ν will follow a Gaussian Random Walk with
the returns distributed as a Student’s t-distribution, with variance derived from ν.
6.2 Bayesian Stochastic Volatility
In order to implement the Bayesian approach to stochastic volatility it is necessary to select priors
for the model parameters. These parameters include σ, which represents the scale of the volatility
and ν, which represents the degrees of freedom of the Student’s t-distribution. Priors must also
be selected for the latent volatility process and subsequently the asset returns distribution.
At this stage the uncertainty on the parameter values is large and so the selected priors
must reflect that. In addition σ and ν must be real-valued positive numbers, so we need to
use a probability distribution that has positive support. As an initial choice the exponential
distribution will be chosen for σ and ν. The Python code to visualise a set of PDFs for the
exponential distribution is given below and plotted in Figure 6.1:
# exponential_plot.py
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
if __name__ == "__main__":
sns.set_palette("deep", desat=.6)
sns.set_context(rc={"figure.figsize": (8, 4)})
x = np.linspace(0.0, 5.0, 100)
lambdas = [0.5, 1.0, 2.0]
for lam in lambdas:
y = lam*np.exp(-lam*x)
ax = plt.plot(x, y, label="$\\lambda=%s$" % lam)
plt.xlabel("x")
plt.ylabel("P(x)")
plt.legend(title="Parameters")