An almost Markovian LIBOR market model calibrated to caps and swaptions

Abstract

A new variant of the LIBOR market model is implemented and calibrated simultaneously to both at-the-money and out-of-the-money caps and swaptions. This model is a two-factor version of a new class of the almost Markovian LIBOR market models with properties long sought after: (i) the almost Markovian parameterization of the LIBOR market model volatility functions is unique and asymptotically exact in the limit of a short time horizon up to a few years, (ii) only minimum plausible assumptions are required to derive the implemented volatility parameterization, (iii) the calibration yields very good results, (iv) the calibration is almost immediate, (v) the implemented LIBOR market model has a related short-rate model. Numerical results for the two-factor case show that the volatility functions for the LIBOR market model can be imported into its short-rate model cousin without adjustment.

Keywords:

• Term structure
• Interest rate derivatives
• Volatility modelling
• Correlation structures
• Local volatility theory
• Correlation modelling
• Multi-factor models

JEL Classification:

• E4
• E43
• G1
• G12
• G13

Notes

† For a quick reference, see section 21.1 of the textbook of Hull (2002).

‡ See section 21.3 of the textbook of Hull (2002).

† Scaling and adiabatic continuation are two basic notions based on which many theories and understandings of natural phenomena have been successfully developed. A good reference is Anderson (1997). Although it is very plausible that the LIBOR market model should have a consistent continuous limit as , the author is not aware of such a proof, but it is assumed so in this paper.