Volatility: A Hidden Markov Process in Financial Time Series

17 Pages Posted: 27 Jul 2007

See all articles by Zoltan Eisler

Zoltan Eisler

Capital Fund Management

Josep Perelló

University of Barcelona - Department of Physics

Jaume Masoliver

University of Barcelona - Department of Physics

Date Written: July 27, 2007

Abstract

The volatility characterizes the amplitude of price return fluctuations. It is a central magnitude in finance closely related to the risk of holding a certain asset. Despite its popularity on trading floors, the volatility is unobservable and only the price is known. Diffusion theory has many common points with the research on volatility, the key of the analogy being that volatility is the time-dependent diffusion coefficient of the random walk for the price return. We present a formal procedure to extract volatility from price data, by assuming that it is described by a hidden Markov process which together with the price form a two-dimensional diffusion process. We derive a maximum likelihood estimate valid for a wide class of two-dimensional diffusion processes. The choice of the exponential Ornstein-Uhlenbeck (expOU) stochastic volatility model performs remarkably well in inferring the hidden state of volatility. The formalism is applied to the Dow Jones index. The main results are: (i) the distribution of estimated volatility is lognormal, which is consistent with the expOU model; (ii) the estimated volatility is related to trading volume by a power law; and (iii) future returns are proportional to the current volatility which suggests some degree of predictability for the size of future returns.

Keywords: random diffusion, hidden markov process, stochastic volatility

Suggested Citation

Eisler, Zoltan and Perello, Josep and Masoliver, Jaume, Volatility: A Hidden Markov Process in Financial Time Series (July 27, 2007). Available at SSRN: https://ssrn.com/abstract=951173 or http://dx.doi.org/10.2139/ssrn.951173

Zoltan Eisler

Capital Fund Management ( email )

23 rue de l'Université
Paris, 75007
France

Josep Perello (Contact Author)

University of Barcelona - Department of Physics ( email )

Diagonal, 647
Barcelona, E-08028
Spain
+34 9 34021150 (Phone)
+34 34021149 (Fax)

Jaume Masoliver

University of Barcelona - Department of Physics ( email )

Barcelona, E-08028
Spain
00 34 3 402 11 59 (Phone)
00 34 3 402 11 49 (Fax)

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