Using LMM-SABR Model to Manage Smile Risk of Fixed Income Portfolio

LMM (LIBOR Market Model) (Brace, Gatarek, & Musiela, 1997) was unquestionably a major breakthrough in financial engineering, for the fact that it manages to unite two liquid yet segregated fixed income markets (caps and swaptions) under one single rigorous term structure setup. Yet it is far from being impeccable. One key drawback of LMM is that it has no smiles embedded in the model, meaning consequently market smiles and skews can only be properly managed by other models such as SABR (stochastic αβρ) model (Hagan, Kumar, Lesniewski, & Woodward, 2002). However, unlike LMM, SABR model can only treat individual forward rate and swaption instead of whole term structure. Recent studies found combined LMM-SABR models (Rebonato & White, 2007) would bring out best part of each of the two benchmark models and provide a consistent model for pricing, hedging and risk management of the entire portfolio. Still, the combined model will be overwhelmingly complex as it substantially enlarges covariance structure (forward rates/forward rates, forward rates/volatilities and volatilities/volatilities), meaning the calibration process and general hedging process are non-trivial tasks, even for big institutions. This paper attempts to enhance quality of approximation form and simplify calibration process on the basis of previous studies, (Rebonato & White, 2007) in particular, in the hope that we can get closer to the ultimate goal of discovering realistic “true” dynamics, if there is such one