Stochastic Volatility Models: Bayesian Framework

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Stochastic Volatility Models: Bayesian Framework. Haolan Cai. Introduction. Idea: model returns using the volatility Important: must capture the persistence of the volatilities (i.e. volatility clusters) along with other characteristics

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Stochastic Volatility Models: Bayesian Framework

Haolan Cai

Introduction

Idea: model returns using the volatility

Important: must capture the persistence of the volatilities (i.e. volatility clusters) along with other characteristics

Use: a class of Hidden Markov Models (HMM) known as Stochastic Volatility Models (SV models)

Basic Model

Where θ = (φ, v) is the parameter space for the AutoRegressive process of order 1 (i.e. linear). φ is the persistence of the model.

Transformation of Model

Previous model is non-linear which creates complications. When we apply the following transformation:

We get nice linear form:

Where is the error term with the following form:

The Problem Child

does not have a close form from which it is easy to sample. However it can be accurately approximated with a discrete mixture of normals.

In this case the optimal J is equal to 7.

Kim, Shephard and Chib (1998)

Bayesian Framework

Now all the parameters have nice distributions from which they can be sampled using a Gibbs sampling algorithm.

Use semi-informative priors (above) with parameters loosely developed from data. Imposes some but little structure to the sampling.

The algorithm was ran for 500 iterations with a burn in period of 50.

The Problem Child (again)

In order to sample we sample from the mixture of normals. This is done by a Forward Filtering, Backwards Sampling (FFBS) algorithm. A Kalman filter is applied from t = 0 to t = n. Then the states (xn, xn-1 … x0) are simulated in the backwards order.

The reasoning for this more complicated sampling measure is the high AR dependence of this type of data. φ is close to 1.

Initial Conditions

For the mixture of normals, 7 normals are chosen to fix the log chi-squared distribution.

For the other parameters, initial values were chosen to sufficiently cover the parameter space as to be semi-informative but not restrictive.

For example, parameters for μ are g and G; where g is the mean and G the standard deviation. Here there are chosen to be 0 and 9 respectively.

Data

1-minute prices from General Electric and Intel Corporation

• GE: April 9, 2007 9:35 am to Jan 24, 2008 3:59 pm
• Used Daily Returns for SV model
Results