ABSTRACT
If there are no market instruments implying future correlation, backward looking analysis of historical data is the only reasonable source of information. Hence the topic of this chapter is estimating historical correlation within a shifted BGM framework. That said, increasingly in some markets like EUR, there is implied correlation information available in the form of options on differences in short-term and long-term swap rates (CMS spread options), and in the future that will probably become the dominant source of correlation information. Once determined, correlation can either be input directly into the model
becoming part of the parameter set, or it can be used as a desirable target for optimization routines that bestfit cap and swaption prices. Our historical data will be assumed to consist of day-by-day quarterly (cov-
erage a uniform δ = 14 year) readings at maturities jδ (j = 0, 1, ..,N − 1) off either yield-to-maturity, forward or swaprate curves. To get quarterly readings, one might use any of the standard curves readily available in banks, designed to fit relevant data and produce discount functions for all maturities. A practical difficulty is that the (relative) lack of smoothness in these
standard curves tends to create phantom principal components. Thus special curves designed to be super-smooth and bestfit data are required to give the best results for correlation; for example, use a forward curve specified at quarterly intervals and a bipartite objective function with one part designed to bestfit market data and the other part to minimize the second derivative measured as quarterly second differences. Another practical difficulty is that the natural objects for analyzing cor-
relation in shifted BGM are forward curves, which must be created from combinations of cash, futures, bond and swap curves by interpolation, differencing and bestfitting. Apart from smoothness problems, when a day-by-day filmshow of forward curves is run (always a good test of routines producing periodic curves or surfaces), one often observes flapping of the long end of the forward curve, where data is relatively sparse and errors compound. To analyze day-by-day data like yieldcurves with maturities fixed relative
to calendar time t, we introduce the concept of relative maturity x = T − t and relative forward
K (t, x) = K (t, T )|T=t+x K (t, T ) = K (t, x)|x=T−t
K (t, creating a new range of variables. Our convention will be that absolute maturities use capitals and the variables in which they appear, like K (t, T ), are also absolute, while relative maturities use small letters and the variables in which they appear, like K (t, x), are relative. Thus expressions like
K (t, t+ x) or K (t, T − t)
will be avoided as being confusing or meaningless. The cth maturity on a yieldcurve changing with calendar time t, will be at the relative maturity xc with reference to the moving root of that instantaneous yieldcurve, and we will be interested in the set of relative maturities xc = cδ (c = 0, .., N − 1). To statistically analyze historical correlation, the underlying models de-
scribing the day-to-day movement of yield curves must necessarily be stationary. So we assume the shift is a constant a and the r-factor shifted BGM historical volatility is homogeneous and a function of relative maturity x = (T − t) only, that is
a (T ) = a, ξ (t, T ) = ξ (T − t) = ξ (x) = ³ ξ(1) (x) , ξ(2) (x) , .., ξ(r) (x)
´∗ .
