stochastic volatility models: bayesian framework
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/1.jpg)
Stochastic Volatility Models: Bayesian Framework
Haolan Cai
![Page 2: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/2.jpg)
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)
![Page 3: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/3.jpg)
Basic Model
2(0, )t tr N
exp( )t tx
(1 )tx AR
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.
![Page 4: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/4.jpg)
Transformation of Model
Previous model is non-linear which creates complications. When we apply the following transformation:
2log( ) / 2t ty rWe get nice linear form:
t t ty x
Where is the error term with the following form:t
log( ) / 2t t 21t
(1 ( , ))tx AR
![Page 5: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/5.jpg)
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.
t
1
( ) ( , )J
t i i ii
p q N b w
In this case the optimal J is equal to 7.
Kim, Shephard and Chib (1998)
![Page 6: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/6.jpg)
Bayesian Framework
Now all the parameters have nice distributions from which they can be sampled using a Gibbs sampling algorithm.
( , )N g G ( , )N c C 1 ( / 2, / 2)ov Ga a av
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.
![Page 7: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/7.jpg)
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.
t
![Page 8: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/8.jpg)
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.
![Page 9: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/9.jpg)
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
![Page 10: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/10.jpg)
Checking Autocorrelation Structure
![Page 11: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/11.jpg)
Results
φ is steady around .956
![Page 12: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/12.jpg)
Results: μ = .0037
![Page 13: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/13.jpg)
Results: ν = .4150
![Page 14: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/14.jpg)
Results:
![Page 15: Stochastic Volatility Models: Bayesian Framework](https://reader035.vdocuments.site/reader035/viewer/2022062301/56815007550346895dbdde40/html5/thumbnails/15.jpg)
Further Analysis
Try to build in autoregressive of high order.
Allow J, the number of normals used to fit the error term, to vary.
What kind of predictive value does this model produce for stock returns?
Does using higher frequency data improve predictive and/or fit value?