e01cec computes, for a given set of data points, the forward values and other values required for monotone convex interpolation as defined in Hagan and West (2008). This form of interpolation is particularly suited to the construction of yield curves in Financial Mathematics but can be applied to any data where it is desirable to preserve both monotonicity and convexity.
The function may be called by the names: e01cec, nag_interp_dim1_monconv_disc or nag_interp_1d_monconv_disc.
e01cec computes, for a set of data points,
, for , the discrete forward rates, , and the instantaneous forward rates, , which are used in a monotone convex interpolation method that attempts to preserve both the monotonicity and the convexity of the original data. The monotone convex interpolation method is due to Hagan and West and is described in Hagan and West (2006), Hagan and West (2008) and West (2011).
The discrete forward rates are defined simply, for ordered data, by
The discrete forward rates, if pre-computed, may be supplied instead of , in which case the original values are computed using the inverse of (1).
The data points need not be ordered on input (though must correspond to ); a set of ordered and scaled values are calculated from and stored.
In its simplest form, the instantaneous forward rates, , at the data points are computed as linear interpolations of the :
If it is required, as a constraint, that these values should never be negative then a limiting filter is applied to as described in Section 3.6 of West (2011).
An ameliorated (smoothed) form of this linear interpolation for the forward rates is implemented using the amelioration (smoothing) parameter . For , equation (2) is used (with possible post-process filtering); for , the ameliorated method described fully in West (2011) is used.
The values computed by e01cec are used by e01cfc to compute, for a given value , the monotone convex interpolated (or extrapolated) value and the corresponding instantaneous forward rate ; the curve gradient at can be derived as for .
Hagan P S and West G (2006) Interpolation methods for curve construction Applied Mathematical Finance13(2) 89–129
Hagan P S and West G (2008) Methods for constructing a yield curve WILLMOTT MagazineMay 70–81
West G (2011) The monotone convex method of interpolation Financial Modelling Agency
1: – IntegerInput
On entry: , the number of data points.
2: – doubleInput
On entry: , the amelioration (smoothing) parameter. Forward rates are first computed using (2) and then, if , a limiting filter is applied which depends on neighbouring discrete forward values. This filter has a smoothing effect on the curve that increases with .
3: – Nag_BooleanInput
On entry: determines whether or not to allow negative forward rates.
Negative forward rates are permitted.
Forward rates calculated must be non-negative.
4: – Nag_BooleanInput
On entry: determines whether the array y contains values, , or discrete forward rates .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, . Constraint: .
The computational errors in the values stored in the array comm should be negligible in most practical situations.