advanced-algorithmic-trading

Chapter 6
Bayesian Stochastic Volatility Model
In this chapter all of the Bayesian statistical theory discussed thus far will be utilised to build
a Bayesian stochastic volatility model. Such a model allows estimation of current and
historical volatility levels of asset returns. Within quantitative trading this is useful from the
perspective of risk management as it provides a "risk filter" mechanism for trade signals.
A Bayesian stochastic volatility model provides a full posterior probability distribution of
volatility at each time point t, as opposed to a single "point estimate" often provided by other
models. This posterior encapsulates the uncertainty in the parameters and can be used to obtain
credible intervals (analogous to confidence intervals) and other statistics about the volatility.
This chapter is heavily influenced by two main sources. The first is a paper written by Hoffman
and Gelman[53], which introduced a highly efficient form of Markov Chain Monte Carlo, known
as the No-U-Turn Sampler (NUTS). The outline of NUTS is beyond the scope of the book. For
a detailed reference, refer to the paper.
The paper describes a Bayesian formulation of a stochastic volatility model, the sampling
of which is carried out using the presented NUTS technique. The presented model is the main
inspiration for the model in this chapter. The second source is the Python PyMC3 library[10]
tutorial article by Salvatier, Fonnesbeck and Wiecki[89] on applying such a model within the
PyMC3 MCMC library.
This chapter will closely follow these two sources but will provide a fully functional end-to-end
script that can be used to estimate volatility for daily equities returns.
6.1 Stochastic Volatility
A stochastic volatility model consists of an underlying generative process for the volatility of
asset returns, which is then used to provide a time-varying variance in the model for the asset
returns distribution itself.
They are used to account for the empirical fact that volatility in financial markets tends to
cluster together. An example of which is the period of excess volatility that occured in the 2008
financial crisis. In time series analysis this "volatility clustering" is known as heteroskedasticity.
Another stochastic volatility model, known as GARCH, will be considered later in the book.
Readers who have also read the C++ For Quantitative Finance ebook by QuantStart will
be aware of the continuous Heston stochastic volatility model. This is specified via a pair of
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