The study on the multiple-curve interest rate models becomes increasingly active since the 2007 credit crunch, for which one curve, typically the OIS curve, is used for discounting purpose, while the LIBOR curves (associated with various market tenors) are used for projecting the future cash flows. In this work, we extend the standard LIBOR market model to accommodate such multiple-curve setting by means of a multiplicative basis. The multiplicative basis is modeled as an exponential function of multi-factor square-root processes. Under the multiplicative basis setup, the OIS forward rates are correlated with the implied (additive) LIBOR-OIS spreads. We then derive closed-form pricing formulas for caplet, swaption, and interest rate futures in the multiplicative basis framework. In particular, we show that the valuation of caplet and swaption can be easily computed by a proper integral of real-valued functions, which facilitates the calibration of our model. Finally, we discuss a slight modification of our model to allow for negative interest rates.
Extension of SABR Libor Market Model to handle negative interest rates