Stochastic Volatility

Imports:

import numpy as np
import pioupiou as pp; pp.restart()

Model

The evolution of the asset price \(Y_t\) at time \(t\) is given by

\[ Y_t \sim \mathcal{N}(0, (\exp(H_t/2))^2) \]

where initially

\[ H_0 \sim \mathcal{N}\left(\mu, \left(\frac{\sigma}{\sqrt{1 - \phi^2}}\right)^2\right) \]

and then

\[ H_t = \mu + \phi(H_{t-1}- \mu) + \Delta_t \; \mbox{ with } \; \Delta_t \sim \mathcal{N}(0,\sigma^2). \]

mu, phi, sigma = -1.02, 0.95, 0.25
n = 100
ts = np.arange(n)
H, Y = np.zeros(n, dtype=object), np.zeros(n, dtype=object)
for t in ts:
    if t == 0:
        H[t] = pp.Normal(mu, sigma**2 / (1 - phi*phi))
    else:
        H[t] = mu + phi * (H[t-1] - mu) + pp.Normal(0, sigma**2)
    Y[t] = pp.Normal(0, pp.exp(0.5 * H[t])**2)

Simulation

Given that our model is deeply nested, we need to increase the recursion limit:

import sys
sys.setrecursionlimit(10000)

Then, as usual:

>>> omega = pp.Omega(100)
>>> y = np.array([Yt(omega) for Yt in Y])
>>> y
array([[-0.03467247, -0.34340423,  0.25390125, ...,  0.96859772,
         1.98091379,  0.19754246],
       [ 0.81756718,  0.08954963, -0.57286052, ..., -2.04860967,
        -0.18998793, -0.25094848],
       [-0.81096151,  0.60424628,  0.44987883, ..., -0.8988355 ,
         1.18628077, -0.664733  ],
       ...,
       [-0.01718594, -0.5785354 , -0.39955856, ..., -1.4898044 ,
        -0.59212422,  0.22604821],
       [ 0.08299539,  0.98580976, -1.09395039, ...,  1.61315771,
         1.12770729, -0.44484522],
       [ 0.40431131, -0.68520976,  0.4213367 , ...,  0.52895462,
         0.75351984,  0.12524886]])

Visualization

import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns

data = {"trajectory": [], "Time": [], "Price": [], "type": []}
for t in np.arange(len(y)):
    for omega in np.arange(np.shape(y)[1]):
        data["trajectory"].append(omega)
        data["Time"].append(t)
        data["Price"].append(y[t, omega])
        data["type"].append("aggregate")

data = pd.DataFrame.from_dict(data)

_ = sns.lineplot(
    data=data, x="Time", y="Price", 
    markers=True, dashes=False,
    style="type", hue="type")

_ = plt.title("Evolution of the Asset Price")
plt.gcf().subplots_adjust(left=0.20, top=0.90)
plt.savefig("volatility.svg")

References

• Stochastic Volatility Models in Stan User's Guide

---

Last update: 2022-05-25