Financial Market Volatility Definition(s)

What is Volatility?

In words volatility refers to the degree to which financial prices fluctuate. Large volatility means that returns (that is: the relative price changes) fluctuate over a wide range of outcomes. For more precise definitions, see below.

Volatility Clustering

 The graph below illustrates what is often referred to as time varying volatility.

<<Fig: S&P 500 daily close-to-close returns>>
The graph displays the daily percentage changes in the value of the S&P 500 index over the period 1988-2006. There are clearly periods of small fluctuations (for example 1992-1996 and 2003-2006) and periods of large fluctuations (for example 1997-2003). This phenomenon is known as volatility clustering, and is the object of a great deal of research in financial economics, as well as econometrics and mathematics. Volatile periods are hectic periods with large price fluctuations. Intuitively, such periods reflect investor uncertainty and one may suspect that such uncertainty is caused by uncertainty about the fundamentals in the economy. This has proven true to a certain extent. Uncertainty about fundamentals, however, is known to explain only a moderate portion of the observed financial market volatility. If one is interested in measuring and forecasting volatility, then simple time series models, or implied volatilities, still do a better job than looking at the fundamentals.

Volatility is Unobservable

Common to all models of volatility, is that the volatility itself is not observed. The situation may be compared to rolling a dice: you observe the realisations of the dice {1,2,3,4,5,6}, but you won't observe the tendency of the (unfair) dice to yield extreme outcomes, say "1" or "6". This tendency may estimated though, and after sufficiently many observations there remains only a small uncertainty about the probability of obtaining a "6". The situation of estimating volatility is comparable: here too, one observes the realisations (the returns), and not the tendency to yield extreme returns. However, the situation is less encouraging in an important respect: in contrast to the setting of rolling a dice, the estimate of volatility does not necessarily improve as one collects more data, since the volatility itself changes. That is, data one month from now may have little to do with the current volatility. So there will always be uncertainty about the current and past values of volatility.

Volatility Definition(s)

Volatility is a theoretical construct. Models for volatility often use an unobservable variable that controls the degree of fluctuations of the financial return process. This variable is usually called the volatility. Generally, two different volatility models, will lead to different concepts of volatility. For example, in GARCH models the volatility is thought of as conditional variance (or standard deviation) of the return, whereas in diffusion models (stochastic differential equations) the volatility refers to either the instantaneous diffusion coefficient or the quadratic variation over a given time period (often called the integrated volatility). It is useful to keep the following questions in mind in the context of any volatility definition:
 Each bullet below treats a different volatility definition.
Given the information up until now F_n, what is the variance of the financial return r_n+1 over the next period:

conditional variance       (1).

The square root of this quantity is the conditional standard deviation. Note that this variance depends on the information set. To obtain an explicit number, one has to make model assumptions for the returns. If one assumes that the financial returns are iid Normal, then the volatility is constant and the usual variance estimator is appropriate:

variance estimator

Here average return denotes the average of the returns. Time series models are designed to deal with the situation of time varying volatility.
Discrete time models for time varying volatility often have a product structure,

       discrete time product structure

The financial return  r_n over period n is the product of the volatility sigma_n and the mean zero, variance one innovation epsilon_n. The innovations are for example iid Gaussian. Models for sigma_n  include ARCH/GARCH, Stochastic Volatility, Long Memory, Markov switching.
The instanteneous volatility. One needs a model describing the continuous time price movements. Consider a stochastic differential equation like

SDE,         (2)

where B denotes standard Brownian motion and p(t) the log price process. The variable sigma(t) is called the spot volatility. Models from this class are often used in option pricing.
Consider a continuous time stochastic process over a given time period. Divide the time period into small, adjacent intervals. Determine the sum of squared returns over these intervals:

rqv          (3)

The quadratic variation is defined as the limit of these sums as the length of the sampling intervals goes the zero. This limit is well defined in case the log price process p(t) is a semimartingale. The quadratic variation may be seen as a model free quantity. In the general semimartingale case, assuming some (mild) restrictions on the types of leverage, the quadratic variation is an unbiased estimator of conditional variance as in (1). Although the quadratic variation uses the full price path over a given time period, it does not necessarily lead to the most efficient estimator of conditional variance.

In the special case of the SDE in (2) the quadratic variation equals the so-called integrated volatility

integrated volatility,

which is of importance in option pricing. If one assumes that the volatility process sigma(t) is independent of the Brownian motion B(t), then the log return r_n is normally distributed with variance quadratic_variation_n, the quadratic variation over the n-th period:

gaussian_QV_n

The assumption that the volatility process sigma(t) is independent of the Brownian motion is a restrictive assumption, which rules out leverage effects and leads to symmetric return distributions. It seems to be more appropriate for foreign exchange markets, then it is for equity markets.
The finite sample quantities in equation (3) are often called 'realized volatility' or 'realized variance'. Popular sampling frequencies are every 5 minutes and every 30 minutes. It is possible to construct alternative proxies for volatility, for example using the high low ranges over intraday intervals.
Given a specific asset price model, with option pricing formulas. Determine the volatility that matches the theoretical option prices from the model to the real life option prices in the market. This volatility is called the implied volatility. The Black Scholes model is often used for determining implied volatilities. There also exist 'model free implied volatilities', see the references below.

References

  Volatility Proxies and GARCH models:
    Thesis cover -- Volatility Proxies and GARCH Models
   Dissertation University of Amsterdam, 2009. Develops an approach for incorporating intraday high-frequency data into discrete time GARCH models. You may want to access the Summary (2 pages), the Introduction (24 pages), the Full Text,  or  the Marcel P. Visser UvA Homepage.

Other literature

For ARCH/GARCH models:
-Bollerslev, T., Engle, R.F., and Nelson, D.B. (1994). ARCH Models. Chapter in handbook of econometrics, volume IV, pages 2959-3038.

For discrete time stochastic volatility models:
-Shephard, N. (1996). Statistical aspects of ARCH and stochastic volatility. Chapter in Time Series Models in Econometrics, Finance and Other Fields. Pages 1-67.

For an overview of forecast comparisons:
-Poon, S.H. and Granger, C.W.J. (2003). Forecasting volatility in financial markets: A review. Journal of Economic Literature 41. Pages 478-539.

For realized volatility and quadratic variation:
-Barndorff-Nielsen, O.E. and Shephard, N. (2002). Estimating Quadratic Variation using realized variance. Journal of Applied Econometrics 17. Pages 457-477.
-Andersen, T.G. and Bollerslev, T. and Diebold, F.X. and Labys, P. (2003).  Modeling and forecasting realized volatility. Econometrica 71. Pages 579-625.

For model free implied volatility:
-Britten-Jones, M. and Neuberger, A. (2000). Option Prices, Implied Price Processes, and Stochastic Volatility. Journal of Finance 55. Pages 839-866.
-Jiang,  G.J. and Tiang, Y.S. (2005). The Model-Free Implied Volatility and its Information Content. Review of Financial Studies 18(4). Pages 1305-1342.

For asset returns stylized facts:
-Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1. Pages 223-236

For textbook  treatments of  most of the above:
-Taylor, S.J. (2005). Asset Price Dynamics, Volatility, and Prediction. Princeton University Press.
-Andersen, T.G. and Davis, R.A. and Kreiss, J.-P. and Mikosch, T. (2009). Handbook of Financial Time Series. Springer.