In this talk, we show how the generalised forward market model (FMM) introduced by Lyashenko and Mercurio (2019) can be extended to make it a complete term-structure model describing the evolution of all points on a yield curve, as well as that of the bank account.
The ability to model the bank account, in addition to the forward curve, is going to be of crucial importance once Libor rates are replaced with setting-in-arrears rates in derivative and cash contracts, where fixings are defined in terms of the realised bank account values.
To achieve our goal, we “embed” the FMM into a Markovian HJM model with separable volatility structure by aligning the HJM and FMM dynamics of the forward term rates modelled by the FMM. This FMM-aligned HJM model is effectively a hybrid between an instantaneous forward-rate model and a LMM, and shares the advantages of both approaches, with the caveat that the number of variables to simulate could be too high.
A more efficient approach is then derived by expressing the zero-coupon bonds and the bank account as functions of the modelled forward term rates and their volatilities. In this FMM-HJM construct, FMM acts as a “coarse” model capturing a “macro” structure of the market such as the covariance structure of the set of modelled rates, while the FMM-aligned HJM serves as a finer modelling environment used to fill the gaps left by the coarser FMM.
The problem of recovering the whole yield curve evolution from the modelled set of Libor rates has been extensively discussed in the LMM literature, and is often referred to as Libor-rate interpolation, or front- and back-stub interpolations. Contrary to existing methods, the approach we propose is not only arbitrage-free by construction, but it also allows for the generation of bank account values.