Inferring Hidden Financial Volatility with an Exponential OU Model

Summary

This study treats volatility as a hidden, time-varying quantity inferred from observed prices. It models price returns and volatility jointly as a two-dimensional diffusion process, then derives a maximum-likelihood procedure for a broad class of such processes. An exponential Ornstein-Uhlenbeck stochastic-volatility model is selected for estimating the hidden volatility state and the method is applied to the Dow Jones index.

The estimated volatility distribution is reported as lognormal, consistent with the selected model. The study also finds a power-law relationship between volatility and trading volume, with an exponent of 0.55, and reports that future returns are proportional to current volatility, suggesting predictability in the magnitude of returns. These are empirical findings from one index application; the supplied description gives no sample period, uncertainty estimates, or out-of-sample trading test. The return relationship concerns return size and does not by itself establish a profitable directional signal.

Key ideas

• Volatility is inferred as a latent state because prices are observed while volatility is not.
• A joint diffusion model and maximum-likelihood estimation procedure are used to recover volatility.
• The exponential Ornstein-Uhlenbeck model produces an estimated lognormal volatility distribution in the Dow Jones application.
• Estimated volatility scales with trading volume by a reported power law with exponent 0.55.
• The reported link between current volatility and future return size does not establish directional predictability or trading profitability.

Tags

# Volatility: a hidden Markov process in financial time series


# Volatility: a hidden Markov process in financial time series









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 of the form $σ\propto V^{0.55}$; and (iii) future returns are proportional to the current volatility which suggests some degree of predictability for the size of future returns.

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