e01cef 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 routine may be called by the names e01cef or nagf_interp_dim1_monconv_disc.
e01cef 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 e01cef are used by e01cff 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: – Real (Kind=nag_wp)Input
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: – LogicalInput
On entry: determines whether or not to allow negative forward rates.
Negative forward rates are permitted.
Forward rates calculated must be non-negative.
4: – LogicalInput
On entry: determines whether the array y contains values, , or discrete forward rates .
On entry: , the (possibly unordered) set of data points.
6: – Real (Kind=nag_wp) arrayInput
If , the discrete forward rates corresponding to the data points
, for .
If , the data values corresponding to the data points
, for .
7: – Real (Kind=nag_wp) arrayCommunication Array
On exit: contains information to be passed to e01cff. The information stored includes the discrete forward rates , the instantaneous forward rates , and the ordered data points .
8: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).