Markov Switching Model Analysis of Implied Volatility for Market Indexes with Applications to S&P 500 and DAX

We adopt a regime switching approach to study concrete financial time series with particular emphasis on their volatility characteristics considered in a space-time setting. In particular the volatility parameter is treated as an unobserved state variable whose value in time is given as the outcome of an unobserved, discrete-time and discrete-state, stochastic process represented by a suitable Markov chain. We will take into account two different approaches for inference on Markov switching models, namely, the classical approach based on the maximum likelihood techniques and the Bayesian inference method realized through a Gibbs sampling procedure. Then the classical approach shall be tested on data taken from the Standard & Poor’s 500 and the Deutsche Aktien Index series of returns in different time periods. Computations are given for a four-state switching model and obtained numerical results are put beside by explanatory graphs which report the outcomes obtained exploiting both smoothing and filtering algorithms used in the estimation/calibration procedures we proposed to infer on the switching model parameters.

[1]  David B. Dunson,et al.  Bayesian Data Analysis , 2010 .

[2]  J. Durbin,et al.  Testing for serial correlation in least squares regression. I. , 1950, Biometrika.

[3]  James D. Hamilton.  A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle , 1989 .

[4]  N. Shephard,et al.  The ACR Model: A Multivariate Dynamic Mixture Autoregression , 2008 .

[5]  M. F. Fuller,et al.  Practical Nonparametric Statistics; Nonparametric Statistical Inference , 1973 .

[6]  Howell Tong,et al.  Non-Linear Time Series , 1990 .

[7]  J. Durbin,et al.  Testing for serial correlation in least squares regression. II. , 1950, Biometrika.

[8]  G. Casella,et al.  Explaining the Gibbs Sampler , 1992 .

[9]  Anil K. Bera,et al.  A test for normality of observations and regression residuals , 1987 .

[10]  Stochastic Relaxation , 2014, Computer Vision, A Reference Guide.

[11]  Min Chen,et al.  Functional coefficient autoregressive conditional root model , 2012, Journal of Systems Science and Complexity.

[12]  R. Kohn,et al.  On Gibbs sampling for state space models , 1994 .

[13]  Alok Bhargava,et al.  Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk , 1983 .

[14]  Chang‐Jin Kim,et al.  State Space Models with Regime Switching , 1999 .

[15]  Viktor Witkovský,et al.  Computing the Distribution of a Linear Combination of Inverted Gamma Variables , 2001, Kybernetika.

[16]  W. M. Bolstad.  Introduction to Bayesian Statistics , 2004 .

[17]  Susan Holmes,et al.  Three Examples of Monte-Carlo Markov Chains: At the Interface Between Statistical Computing, Computer Science, and Statistical Mechanics , 1995 .

[18]  S. Goldfeld,et al.  A Markov model for switching regressions , 1973 .

[19]  Donald Geman,et al.  Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images , 1984 .

[20]  S. Chib,et al.  Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts , 1993 .