A general HJM framework for multiple yield curve modelling (Q287657)

In the present paper, the authors provide a general framework for modeling multiple yield curves. Given a stochastic basis \((\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0}, \mathbb{Q})\), they introduce an abstract framework in the spirit of Heath, Jarrow and Morton (HJM) by specifying a family \[ \{ (S(t,T))_{t \in [0,T]} : T \geq 0 \} \] of positive semimartingales. They derive equivalent conditions for the abstract HJM setting to be risk-neutral, which means that \((S(t,T))_{t \in [0,T]}\) is a \(\mathbb{Q}\)-martingale for each \(T \geq 0\). Then they show how a classical HJM model of the form \[ \{ (B(t,T) / B_t)_{t \in [0,T]}, T \geq 0 \} \] with bank account \(B\) given by \[ B_t = e^{\int_0^t r_s ds}, \] where \(r\) denotes the short rate process, can be formulated in terms of this general framework. Moreover, the authors extend the abstract HJM setting by considering families of spreads \[ \{ (S^{\delta}(t,T))_{t \in [0,T]}, T \geq 0, \delta \in \mathcal{D} \}, \] where \(\mathcal{D} = \{ \delta_1,\dots,\delta_m \}\) denotes a family of tenors, with \(0< \delta_1< \delta_2<\dots<\delta_m\), for some \(m \in \mathbb{N}\). They investigate when such an HJM-type multiple yield curve model is risk-neutral. More precisely, they provide equivalent conditions for the property that for every \(T \geq 0\) the spreads \(\{ (S^{\delta}(t,T))_{t \in [0,T]}, \delta \in \mathcal{D} \}\) are \(\mathbb{Q}^T\)-martingales, where \(\mathbb{Q}^T\) denotes the \(T\)-forward measure. Then, the authors show how to construct risk-neutral HJM-type multiple yield curve models with spreads being ordered with respect to the tenor's length. For this purpose, they solve a stochastic partial differential equation for the evolution of the forward spread curves. Furthermore, the authors deal with model implementation and calibration, and they draw relations to other multiple yield curve modeling approaches existing in the literature.