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Unformatted text preview: Electronic copy available at: Volatility Models: from GARCH to Multi-Horizon Cascades Alexander Subbotin * Thierry Chauveau Kateryna Shapovalova June 15, 2009 Abstract We overview different methods of modeling volatility of stock prices and exchange rates, focusing on their ability to reproduce the empirical properties in the corresponding time series. The properties of price fluctu- ations vary across the time scales of observation. The adequacy of different models for describing price dynamics at several time horizons simultane- ously is the central topic of this study. We propose a detailed survey of recent volatility models, accounting for multiple horizons. These mod- els are based on different and sometimes competing theoretical concepts. They belong either to GARCH or stochastic volatility model families and often borrow methodological tools from statistical physics. We compare their properties and comment on their practical usefulness and perspec- tives. Keywords: Volatility modeling, GARCH, stochastic volatility, volatil- ity cascade, multiple horizons in volatility. J.E.L. Classification: G10, C13. * University of Paris-1 (Panth` eon-Sorbonne) and Altran Technologies. 58, bd Gouvion Saint Cyr, 75017 Paris - France. E-mail: University of Paris-1 (Panth` eon-Sorbonne) E-mail: University of Paris-1 (Panth` eon-Sorbonne) and Paris School of Economics. E-mail: 1 Electronic copy available at: 1 Introduction Modeling stock prices is essential in many areas of financial economics, such as derivatives pricing, portfolio management and financial risk follow-up. One of the most criticized drawbacks of the so-called modern portfolio theory (MPT), including the diversification principle of Markowitz (1952) and the cap- ital asset pricing model by Sharpe (1964) and Lintner (1965), is the non-realistic assumption about stock price variability. Clearly, stock returns are not iid dis- tributed Gaussian random variables, but alternatives to this assumption are numerous, sometimes complicated and application-dependent. In this paper we review empirical properties of stock price dynamics and various models, pro- posed to represent it, focusing on the most recent developments, concerning mainly multi-horizon and multifractal stochastic volatility processes. The subject of this study is the variability of stock prices, referred to as volatility. Usually introduction of scientific terminology aims at making a gen- eral concept more precise, but this is rather an example of the contrary. De- pending on the context and the point of view of the author, the term volatility in finance can stand for the variability of prices (in this sense we used it above), an estimate of standard deviation, financial risk in general, a parameter of a derivative pricing model or a stochastic process of particular form. We willderivative pricing model or a stochastic process of particular form....
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