A time-homogeneous, SABR-consistent extension of the LMM

Riccardo Rebonato proposes an extension of the Libor market model (LMM) that recovers the stochastic, alpha, beta, rho (SABR) caplet prices almost exactly for all strikes and maturities. The dynamics of the volatility are chosen so as to be consistent across expiries, to be financially motivated and to make the evolution of the implied volatilities as time-homogeneous as possible. Given the SABR parameters, the associated LMM parameters are found with minimal numerical work

The stochastic, alpha, beta, rho (SABR) model and the Libor market model (LMM) have become industry standards for pricing plain-vanilla and complex interest rate products, respectively. (For a description of the SABR model, see, for example, Hagan et al, 2002. The LMM is described, for example, in Brace, Gatarek & Musiela (BGM), 1996 and Jamshidian, 1997.) While similar, the two models do not directly 'talk to each other'. Ultimately, the SABR approach is not a consistent dynamical model for a