 
Summary: Validation of Volatility Models
MALIK MAGDONISMAIL* AND YASER S. ABUMOSTAFA
Caltech, Pasadena, CA, USA
ABSTRACT
In forecasting a Žnancial time series, the mean prediction can be validated
by direct comparison with the value of the series. However, the volatility or
variance can only be validated by indirect means such as the likelihood
function. Systematic errors in volatility prediction have an `economic value'
since volatility is a tradable quantity (e.g. in options and other derivatives)
in addition to being a risk measure. We analyse the Ždelity of the likelihood
function as a means of training (in sample) and validating (out of sample) a
volatility model. We report several cases where the likelihood function
leads to an erroneous model. We correct for this error by scaling the
volatility prediction using a predetermined factor that depends on the
number of data points. # 1998 John Wiley & Sons, Ltd.
KEY WORDS validation; volatility prediction; maximum likelihood
INTRODUCTION
Consider the time series depicted in Figure 1. Each point x(t) is drawn from a distribution whose
mean is m(t) and variance s2
(t). While we can only observe x(t), we wish to learn about the mean
