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.

- What is the asset price model?

- What is the time horizon for this volatility?
- Is the volatility forward looking, backward looking, or instantaneous?
- Is the volatility a model variable, or the estimator of a model variable?

- In case of an estimator of the model volatility: is the estimated volatility extracted from returns data (price fluctuations), or from option prices (implied volatilities)?

- Conditional variance / conditional standard deviation

Given the information up until now , what is the variance of the financial return over the next period:

(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:

Here denotes the average of the returns. Time series models are designed to deal with the situation of time varying volatility.

(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:

Here denotes the average of the returns. Time series models are designed to deal with the situation of time varying volatility.

- Time series volatility

Discrete time models for time varying
volatility often have a product structure,

The financial return over period n is the product of the volatility and the mean zero, variance one innovation . The innovations are for example iid Gaussian. Models for include ARCH/GARCH, Stochastic Volatility, Long Memory, Markov switching.

- Spot volatility

The instanteneous volatility. One needs
a model describing the continuous time price movements. Consider a
stochastic differential equation like

, (2)

where B denotes standard Brownian motion and p(t) the log price process. The variable is called the spot volatility. Models from this class are often used in option pricing.

, (2)

where B denotes standard Brownian motion and p(t) the log price process. The variable is called the spot volatility. Models from this class are often used in option pricing.

- Quadratic Variation

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:

(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

,

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

The assumption that the volatility process 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.

(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

,

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

The assumption that the volatility process 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.

- Realized Volatility Measures

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.

- Implied volatility

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.

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.