Dynamic Refinement of the Term Structure - Time Homogenous Term Structure Modeling
28 Pages Posted: 29 Dec 2016 Last revised: 6 Mar 2017
Date Written: December 12, 2016
In this note we consider a classical term structure model framework, that is, a HJM framework on a time-discrete tenor, like the LIBOR market model, using a sequence of tenor discretization, where the tenors are valid for a specific simulation time interval.
The setup then allows to model dynamic refinements of the tenor structure and, as a special case, a quasi time-homogenous tenor structure.
A time-homogenous tenor structure has some relevance in exposure simulations, where two requirements come together: it is desirable to model a finer tenor structure on the short end of the rate curve compared to the long end of the rate curve and, this property should persist in the simulation at a future time.
The property is easily fulfilled by models, which uses an equally fine discretization at all times, e.g., a short rate model, where simulation time discretization and tenor time discretization coincide.
As we will demonstrate, a refinement of the tenor structure via a simple interpolation of forward rates would either introduce a strong restriction on the model's volatility structure (as it is the case for a classical fixed tenor LIBOR market model) or introduce an arbitrage violation. The challenge in the refinement is to simulate the right stochastic drifts. Under a (milder) condition on the model's volatility structure, the drifts can be reconstructed using a single additional state variable. The additional state variable is only needed on the coarse discretization tenors, limiting the computational resources needed to implement the model.
In a limit case, the approach can be used to glue together a short rate model for the short end of the rate curve and a term-structure model for the long end of the rate curve in a time-homogenous way.
Keywords: LIBOR Market Model, Heath-Jarrow-Morton Framework, Tenor Discretization, Term-Structure Models
JEL Classification: C15, G13
Suggested Citation: Suggested Citation