After the crisis that has affected financial markets in 2007-2009, the Financial Stability Board (FSB) initiated a process of reform for the principal interest rate benchmarks. This operation aimed to address concerns about the robustness and integrity of the existing benchmarks, the Interbank Offered Rates (IBORs), such as the London Interbank Offered Rate (LIBOR), which had been found to be vulnerable to manipulation. As part of this process, the FSB introduced new Risk-Free term Rates (RFRs) as alternative benchmarks to LIBOR. These new RFRs, such as the Overnight Indexed Swap rate (OIS), are based on transactions between market participants, rather than on banks’ submissions, making them less susceptible to manipulation. The main difference between the old and the new rates is that the former are term rates while the latter have a Overnight (O/N) maturity. However, the possibility of building a term structure also for the RFRs made it possible to develop an extension of the traditional LIBOR Market Model (LMM), which is strongly related to the old interest rates, to a new framework designated as Forward Market Model (FMM). Such model is able to simultaneously describe the evolution of both the forward-looking (LIBOR-like) and the new backward-looking (setting-in-arrears) term rates using the same stochastic process for both. It is due to note that, when switching from forward-looking to backward-looking term rates, the properties of the standard interest-rate modeling framework are not only maintained but also enriched. Setting-in-arrears rates, in fact, own all the relevant analytical features held by the IBORs, such as the martingale property under the related forward measure, plus other nice qualities, such as a simple analytic formula for the drift under the risk-neutral measure. Hence, they show to be a very valid replacement for the old term rates from an analytical point of view. Our research paper presents a novel Heath-Jarrow-Morton (HJM) model that exploits the potentialities of these new risk-free term rates. Clearly, the HJM framework is a powerful tool for pricing derivatives, but it has its limitations. The new RFRs present a unique challenge, as they require a model that can take into account the specific characteristics of these new benchmarks. Our HJM model uses a parsimonious (finite-dimensional) Markovian framework, with a separable volatility form, that generates the dynamics of the extended forward rates, that are equivalent to the FMM ones. Such model exploits a reduced number of free parameters compared to the complete framework of Lyashenko and Mercurio, thus demanding a much lower computational effort. Moreover, it constitutes a single-curve framework where all the structures are generated starting from a single rate, the RFR, thus differing from the previous multi-curve model of Moreni and Pallavicini based on the old IBOR rates. We would like to remark here that this model, which combines precision and flexibility, is specifically designed to take advantage of the new RFRs, making it the go-to choice for pricing vanilla derivatives on the new O/N interest rate benchmarks. Indeed, due to its ability to handle the complexities of the new rates, while maintaining a high level of accuracy, it is well suited to meet the needs of today’s market participants, which are seeking a more robust, efficient, and reliable way to achieve this goal. Specifically, in this paper, after a brief introduction on the fundamental definitions of the FMM, we draw final expressions for the risk-free forward rate dynamics and the (approximated) risk-free swap rate dynamics as well as valuation Black-like formulas for derivatives on these rates (caps and European swaptions), by adopting a specific model realization with a deterministic volatility. It is worth noticing that, thanks to the concept of extended zero-coupon bond, on which our FMM formulation is based, the forward and swap rate dynamics are defined for all times, even those beyond their natural expiries. As an additional thing, by restricting ourselves to a HJM two-factor model, we derive explicit pricing formulas for European (payer and receiver) swaptions
