Structure preserving stochastic integration schemes in interest rate derivative modeling

In many applications, differential equation models require geometric integration, i.e., the application of structure-preserving integration schemes. In computational finance, for example, the numerical simulation of extended Libor market models used to value structured interest rate derivatives has to preserve positivity or boundedness of the underlying stochastic processes used to model mean-reverting volatility or forward rates. This paper discusses how stochastic integration schemes can be constructed in order to maintain these properties of the analytical solution. Milstein-type methods prove to be the method-of-choice with respect to both efficiency and preservation of structural properties, as they turn out to dominate the increments of Brownian motions. These theoretical results are confirmed by numerical tests.