Derivative pricing for a multi-curve extension of the Gaussian, exponentially quadratic short rate model

TL;DR

This paper introduces a multi-curve extension of Gaussian quadratic short rate models, enabling more accurate derivative pricing post-financial crisis by incorporating spreads and leveraging the advantages of Gaussian factors.

Contribution

It develops an exponentially quadratic multi-curve short rate model with Gaussian factors, offering a novel approach to derivative pricing in the post-crisis financial environment.

Findings

Derived an adjustment factor for transitioning from single to multi-curve derivative prices.

Demonstrated the model's tractability and advantages for pricing linear and optional derivatives.

Provided a framework that simplifies derivative valuation in complex multi-curve settings.

Abstract

The recent financial crisis has led to so-called multi-curve models for the term structure. Here we study a multi-curve extension of short rate models where, in addition to the short rate itself, we introduce short rate spreads. In particular, we consider a Gaussian factor model where the short rate and the spreads are second order polynomials of Gaussian factor processes. This leads to an exponentially quadratic model class that is less well known than the exponentially affine class. In the latter class the factors enter linearly and for positivity one considers square root factor processes. While the square root factors in the affine class have more involved distributions, in the quadratic class the factors remain Gaussian and this leads to various advantages, in particular for derivative pricing. After some preliminaries on martingale modeling in the multi-curve setup, we concentrate on pricing of linear and optional derivatives. For linear derivatives, we exhibit an adjustment factor that allows one to pass from pre-crisis single curve values to the corresponding post-crisis multi-curve values.