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Semi-Markov Models in High Frequency Finance: A Review

In this paper we describe three stochastic models based on a semi-Markov chains approach and its generalizations to study the high frequency price dynamics of traded stocks. The three models are: a simple semi-Markov chain model, an indexed semi-Markov chain model and a weighted indexed semi-Markov chain model. We show, through Monte Carlo simulations, that the models are able to reproduce important stylized facts of financial time series as the persistence of volatility. In particular, we analyzed high frequency data from the Italian stock market from the first of January 2007 until end of December 2010 and we apply to it the semi-Markov chain model and the indexed semi-Markov chain model. The last model, instead, is applied to data from Italian and German stock markets from January 1, 2007 until the end of December 2010.