Building Econometric Models

Simple Covariance Models: Historical and Implied

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• In exactly the same fashion as for volatility, the historical covariance or correlation between two series can be calculated from a set of historical data
• Implied covariances can be calculated using options whose payoffs are dependent on more than one underlying asset
• The relatively small number of such options that exist limits the circumstances in which implied covariances can be calculated
• Examples include rainbow options, ‘crack spread’ options for different grades of oil, and currency options.

Implied Covariance Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• To give an illustration for currency options, the implied variance of the cross-currency returns is given by
• where and are the implied variances of the x and y
• returns respectively, and is the implied covariance between
• x and y
• So if the implied covariance between USD/DEM and USD/JPY is of interest, then the implied variances of the returns of USD/DEM and USD/JPY and the returns of the cross-currency DEM/JPY are required.

EWMA Covariance Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• A EWMA specification gives more weight in the covariance to recent observations than an estimate based on the simple average
• The EWMA model estimates for variances and covariances at time t in the bivariate setup with two returns series x and y may be written as
• hij,t = λhij,t−1 + (1 − λ)xt−1yt−1
• where i  j for the covariances and i = j; x = y for the variances
• The fitted values for h also become the forecasts for subsequent periods
• λ (0 < λ < 1) denotes the decay factor determining the relative weights attached to recent versus less recent observations
• This parameter could be estimated but is often set arbitrarily (e.g., Riskmetrics use a decay factor of 0.97 for monthly data but 0.94 for daily).