Stochastic Volatility model

In [1]:
import numpy as np
import pymc3 as pm
from pymc3.distributions.timeseries import GaussianRandomWalk

from scipy import optimize

%pylab inline
Populating the interactive namespace from numpy and matplotlib

Asset prices have time-varying volatility (variance of day over day returns). In some periods, returns are highly variable, while in others very stable. Stochastic volatility models model this with a latent volatility variable, modeled as a stochastic process. The following model is similar to the one described in the No-U-Turn Sampler paper, Hoffman (2011) p21.

\[\sigma \sim Exponential(50)\]
\[\nu \sim Exponential(.1)\]
\[s_i \sim Normal(s_{i-1}, \sigma^{-2})\]
\[log(\frac{y_i}{y_{i-1}}) \sim t(\nu, 0, exp(-2 s_i))\]

Here, \(y\) is the daily return series and \(s\) is the latent log volatility process.

Build Model

First we load some daily returns of the S&P 500.

In [2]:
n = 400
returns = np.genfromtxt(pm.get_data("SP500.csv"))[-n:]
array([-0.00637 , -0.004045, -0.02547 ,  0.005102, -0.047733])
In [3]:
[<matplotlib.lines.Line2D at 0x7fae8c869a20>]

Specifying the model in pymc3 mirrors its statistical specification.

In [4]:
model = pm.Model()
with model:
    sigma = pm.Exponential('sigma', 1./.02, testval=.1)

    nu = pm.Exponential('nu', 1./10)
    s = GaussianRandomWalk('s', sigma**-2, shape=n)

    r = pm.StudentT('r', nu, lam=pm.math.exp(-2*s), observed=returns)

Fit Model

For this model, the full maximum a posteriori (MAP) point is degenerate and has infinite density. To get good convergence with NUTS we use ADVI (autodiff variational inference) for initialization.

In [9]:
with model:
    trace = pm.sample(2000, tune=1000)[1000:]
Auto-assigning NUTS sampler...
Initializing NUTS using ADVI...
Average Loss = 7.9136: 100%|██████████| 200000/200000 [01:14<00:00, 2691.66it/s]
Finished [100%]: Average Loss = 7.896
100%|██████████| 2000/2000 [04:47<00:00,  8.48it/s]/home/jovyan/pymc3/pymc3/step_methods/hmc/ UserWarning: Chain 0 reached the maximum tree depth. Increase max_treedepth, increase target_accept or reparameterize.
  'reparameterize.' % chain)
/home/jovyan/pymc3/pymc3/step_methods/hmc/ UserWarning: The acceptance probability in chain 0 does not match the target. It is 0.688505126364, but should be close to 0.8. Try to increase the number of tuning steps.
  % (chain, mean_accept, target_accept))

In [10]:
pm.traceplot(trace, model.vars[:-1]);
In [11]:
plot(trace[s][::10].T, 'b', alpha=.03);
ylabel('log volatility')
<matplotlib.text.Text at 0x7fae6019f8d0>

Looking at the returns over time and overlaying the estimated standard deviation we can see how the model tracks the volatility over time.

In [12]:
plot(np.exp(trace[s][::10].T), 'r', alpha=.03);
sd = np.exp(trace[s].T)
ylabel('absolute returns')
<matplotlib.text.Text at 0x7fae64197c50>