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Encyclopedia of Complexity and Systems Science pp 1–42Cite as

Stochastic Volatility

  • Living reference work entry
  • First Online:

Definition of the Subject

Given the importance of return volatility on a number of practical financial management decisions, the efforts to provide good real-time estimates and forecasts of current and future volatility have been extensive. The main framework used in this context involves stochastic volatility models. In a broad sense, this model class includes GARCH, but we focus on a narrower set of specifications in which volatility follows its own random process, as is common in models originating within financial economics. The distinguishing feature of these specifications is that volatility, being inherently unobservable and subject to independent random shocks, is not measurable with respect to observable information. In what follows, we refer to these models as genuine stochastic volatility models.

Much modern asset pricing theory is built on continuous-time models. The natural concept of volatility within this setting is that of genuine stochastic volatility. For example,...

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Notes

  1. 1.

    (Discrete-time SV models go farther back in time, at least to the paper by Rosenberg (1972) recently reprinted in Shephard (2004))

Abbreviations

Implied volatility:

The value of asset return volatility which equates a model-implied derivative price to the observed market price. Most notably, the term is used to identify the volatility implied by the Black and Scholes (1973) option pricing formula.

Quadratic return variation:

The ex post sample-path return variation over a fixed time interval.

Realized volatility:

The sum of finely sampled squared asset return realizations over a fixed time interval. It is an estimate of the quadratic return variation over such time interval.

Stochastic volatility:

A process in which the return variation dynamics include an unobservable shock which cannot be predicted using current available information.

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Acknowledgments

We are grateful to Olena Chyruk, Bruce Mizrach (the Section Editor), and Neil Shephard for helpful comments and suggestions. Of course, all errors remain our sole responsibility. The views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Chicago or the Federal Reserve System. The work of Andersen is supported by a grant from the NSF to the NBER and support from CREATES funded by the Danish National Research Foundation.

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  1. Kellogg School of Management, Northwestern University, Evanston, USA

    Torben G. Andersen

  2. Federal Reserve Bank of Chicago, Chicago, USA

    Luca Benzoni

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Correspondence to Torben G. Andersen .

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    Robert A. Meyers

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Andersen, T.G., Benzoni, L. (2015). Stochastic Volatility. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_527-3

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