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( )
yi
log ∼ t(ν, 0, exp(−2s i )) (6.4)
y i−1
Thus the full stochastic volatility model is specified by four parameters, each with an associated
prior:
σ ∼ Exponential(50) (6.5)
ν ∼ Exponential(0.1) (6.6)
s i ∼ N (s i−1 , σ −2 ) (6.7)
( )
yi
log ∼ t(ν, 0, exp(−2s i )) (6.8)
y i−1
PyMC3 will now be used to fit this model to a set of historical financial asset pricing data.
6.3 PyMC3 Implementation
The first task is to import the necessary libraries used in the stochastic volatility model. This
consists of NumPy, SciPy, Pandas, Matplotlib and Seaborn. These libraries are used for data
import, manipulation and plotting.
As in previous chapters the PyMC3 library is used to carry out the MCMC procedure. The
GaussianRandomWalk model is imported and is used to model the returns of the daily equity
prices:
# pymc3_bayes_stochastic_vol.py
import datetime
import pprint
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pandas_datareader as pdr
import pymc3 as pm
from pymc3.distributions.timeseries import GaussianRandomWalk
import seaborn as sns
The next step is to download a suitable equities price series. In this example the historical
prices of Amazon will be downloaded from Yahoo Finance.
6.3.1 Obtaining the Price History
The returns of Amazon, Inc. (AMZN) are downloaded from Yahoo Finance using the pandas_datareader
module. However, it is straightforward to utilise S&P500, FTSE100 or any other asset pricing
data.