Monotonicity of the collateralized debt obligations term structure model

TL;DR

This paper investigates conditions ensuring arbitrage-free and monotone CDO term structure models, focusing on the Heath-Jarrow-Morton-Musiela equation, and characterizes models based on volatility and Lévy process properties.

Contribution

It formulates conditions for positivity and monotonicity of CDO models using the Milian result, considering different state spaces and linking model characteristics to arbitrage-free monotonicity.

Findings

Conditions for positivity and monotonicity are established.

Regularity results for pointwise monotonicity are proven.

Models are characterized by volatility and Lévy process features.

Abstract

The problem of existence of arbitrage free and monotone CDO term structure models is studied. Conditions for positivity and monotonicity of the corresponding Heath-Jarrow-Morton-Musiela equation for the $x$-forward rates with the use of the Milian type result are formulated. Two state spaces are taken into account - of square integrable functions and a Sobolev space. For the first the regularity results concerning pointwise monotonicity are proven. Arbitrage free and monotone models are characterized in terms of the volatility of the model and characteristics of the driving L\'evy process.