While the volatility level changes frequently, correlations between various maturity forward rates tend to be more stable. This lends itself to extracting the correlation term structure from historical data and matching the exact level of volatility to market implied volatilities. The volatility associated with factor i , ƒ i ( t , t + j ), may be expressed as:
where ƒ ( t , t + j ) is the total volatility of forward rate with maturity t + j and ˆ i ( t , t + j ) is the weight associated with factor i . These functions ˆ i ( t , ·), are elements of the i th principal component of the correlation matrix; hence [7] :
are the diagonal and off diagonal elements of the correlation matrix respectively. The forward rate volatilities ƒ ( t , t + j ), are the market implied Black volatilities for caplets expiring at time t + j .
[7] As in (15.10), we decompose the correlation matrix C such that C = A A ² and hence define the matrix of principle components as = A ½ where ² = C