Discrete LIBOR Market Model Analogy

TL;DR

This paper introduces a discrete-time LIBOR market model analogous to continuous models, enabling arbitrage-free discretization and convergence analysis, with applications to Levy LIBOR models and Euler discretizations.

Contribution

It constructs a discrete LIBOR model based on discrete exponential martingales, proving its weak convergence to continuous Levy LIBOR models under certain conditions.

Findings

Discrete LIBOR model constructed from exponential martingales

Proves weak convergence to continuous Levy LIBOR models

Establishes connection to Euler discretization methods

Abstract

This paper provides a discrete time LIBOR analog, which can be used for arbitrage-free discretization of Levy LIBOR models or discrete approximation of continuous time LIBOR market models. Using the work of Eberlein and Oezkan as an inspiration, we build a discrete forward LIBOR market model by starting with a discrete exponential martingale. We take this pure jump process and calculate the appropriate measure change between the forward measures. Next we prove weak convergence of the discrete analog to the continuous time LIBOR model, provided the driving process converges weakly to the continuous time one and the driving processes are PII's. This especially implies the weak convergence of the model to a Levy LIBOR market model if the driving process variables are infinitely divisible distributions. This also relates our model to an Euler discretization.