Joining the SABR and Libor models together

Fabio Mercurio and Massimo Morini propose a Libor market model consistent with SABR dynamics and develop approximations that allow for the use of the SABR formula with modified inputs. They verify that the approximations are acceptably precise, imply good fitting of market data and produce regular Libor rate parameters. They finally show that the correct assessment of the no-arbitrage volatility drift leads to a more sensible pricing of derivatives not included in the calibration set

The SABR model is a stochastic volatility dynamics for a single asset under its natural probability measure. However, when pricing general term structure payouts, we need to model the joint evolution of relevant rates, as in the Libor market model (LMM) of Brace, Gatarek & Musiela (1997). Moreover, if the stochastic volatility factors are correlated with the term structure of rates (as in Hagan et al, 2002, the underlying is correlated to its stochastic volatility), no-arbitrage constraints indicate that the stochastic differential equation governing the volatility evolution will be different under different swap and forward measures. This implies that when one assumes a factor structure for the stochastic volatility, as is usually done in the market for computational reasons, even the pricing of simple derivatives such as caps and swaptions is not trivial, because different volatility dynamics must be considered for different underlying rates.

Joining the SABR and Libor models together (PDF, 321KB)