On the simultaneous calibration of multifactor lognormal interest rate models to Black volatilities and to the correlation matrix

Abstract

ABSTRACT It is shown in this paper that it is not only possible, but indeed expedient and advisable, to perform a simultaneous calibration of a lognormal Brace-Gabrek-Musiela interest-rate model to the percentage volatilities of the individual rates and to the correlation surface. One of the contributions of the paper it to show that the task can be accomplished in two separate and independent steps: the first part of the calibration (i.e. to cap volatilities) can always be accomplished exactly thanks to straightforward geometrical relationships; the fitting to the correlation surface, thanks to a simple theorem, can then be carried out in a numerically efficient way so that the calibration to the volatilities is not spoiled by the second part of the procedure. The ability to carry out the two tasks separately greatly simplifies the overall task. Actual calculations are shown for a three- and four-factor implementation of the approach, and the quality of the overall agreement between the target and model correlation surfaces is commented upon. Finally, the dangers of overparametrization, i.e. of forcing (near) exact fitting to certain portions of the correlation matrix, are analyzed by looking at the cases of a trigger swap, a Bermudan swaption, and a one-way floater (resettable cap).

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