Lognormal approximations to Libor market models

ABSTRACT We study several lognormal approximations for Libor market models, where special attention is paid to their simulation by direct methods and lognormal random fields. In contrast to conventional numerical solution of SDEs, this approach simulates the solution directly at a desired point in time and therefore may be more efficient. As such, the proposed approximations provide valuable alternatives to the Euler method, in particular for long-dated instruments. We carry out a pathwise comparison of the different lognormal approximations with the “exact” SDE solution obtained by the Euler scheme using sufficiently small time steps. Also we test approximations obtained via numerical solution of the SDE by the Euler method, using larger time steps. It turns out that, for typical volatilities observed in practice, improved versions of the lognormal approximation proposed by Brace, Gatarek and Musiela (1997) appear to have excellent pathwise accuracy. We found that this accuracy can also be achieved by Euler-stepping the SDE using larger time steps; however, from a comparative cost analysis it follows that, particularly for long-maturity options, the latter method is more time-consuming than the lognormal approximation. We conclude with applications to some example Libor derivatives.