  SABR models have been used to incorporate stochastic volatility to LIBOR
market models (LMM) in order to describe interest rate dynamics and price
interest rate derivatives. From the numerical point of view, the pricing of
derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out
with Monte Carlo simulation. However, this approach could involve excessively
long computational times. For first time in the literature, in the present
paper we propose an alternative pricing based on partial differential equations
(PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs
proposed by Hagan \cite{haganSABRLIBOR}, Mercurio \& Morini
\cite{mercurioMorini} and Rebonato \cite{rebonatoWhite}. Moreover, as the PDEs
associated to these SABR/LMMs are high dimensional in space, traditional full
grid methods (like standard finite differences or finite elements) are not able
to price derivatives over more than three or four underlying interest rates. In
order to overcome this curse of dimensionality, a sparse grid combination
technique is proposed. A comparison between Monte Carlo simulation results and
the ones obtained with the sparse grid technique illustrates the performance of
the method.
