Pricing in-arrears caps and ratchet caps under LIBOR market model with multiplicative basis

Abstract

In Zhong (2018), LIBOR market model with multiplicative basis, International Journal of Financial Engineering, 5(2), we proposed a LIBOR market model with multiplicative basis, namely, the LMM-MB model, to model the joint evolution of the LIBOR rates and the OIS forward rates. This model leads to tractable pricing formulas for the standard interest rate derivatives such as the (vanilla) caplet, swaption and futures. In this paper, we study the pricing of some non-standard interest rate derivatives under the LMM-MB model, specifically the in-arrears (IA) cap and the ratchet cap. Similar to the vanilla caplet, we show that the pricing of the IA caplet can be readily computed by a proper integral of real-valued functions. We then derive an analytical approximation for the ratchet cap. In the case of non-zero spread, the ratchet cap can be approximated by using a two-dimensional fast Fourier transform method. In the case of zero spread, the ratchet cap can be computed from a proper integral of a single variable function. Numerical results reveal a good match of our close-form formulas with the Monte Carlo simulation method.