Indexed Markov Chains for financial data: testing for the number of states of the index process

TL;DR

This paper introduces a method for modeling high-frequency financial returns using Indexed Markov Chains, allowing endogenous consideration of market volatility, long memory, and volatility clustering, with a new approach for determining the index process's state space.

Contribution

It proposes a change-point based method for optimally determining the state space of the index process in Indexed Markov Chains for financial data.

Findings

Effective modeling of intra-day prices of an Italian firm

Explicit formula for the first change of state distribution

Demonstrates the model captures volatility clustering

Abstract

A new branch based on Markov processes is developing in the recent literature of financial time series modeling. In this paper, an Indexed Markov Chain has been used to model high frequency price returns of quoted firms. The peculiarity of this type of model is that through the introduction of an Index process it is possible to consider the market volatility endogenously and two very important stylized facts of financial time series can be taken into account: long memory and volatility clustering. In this paper, first we propose a method for the optimal determination of the state space of the Index process which is based on a change-point approach for Markov chains. Furthermore we provide an explicit formula for the probability distribution function of the first change of state of the index process. Results are illustrated with an application to intra-day prices of a quoted Italian firm from January $1^{st}$, 2007 to December $31^{st}$ 2010.