Bonds and Options in Exponentially Affine Bond Models

Abstract

In this article we apply the Flesaker–Hughston approach to invert the yield curve and to price various options by letting the randomness in the economy be driven by a process closely related to the short rate, called the abstract short rate. This process is a pure deterministic translation of the short rate itself, and we use the deterministic shift to calibrate the models to the initial yield curve. We show that we can solve for the shift needed in closed form by transforming the problem to a new probability measure. Furthermore, when the abstract short rate follows a Cox–Ingersoll–Ross (CIR) process we compute bond option and swaption prices in closed form. We also propose a short-rate specification under the risk-neutral measure that allows the yield curve to be inverted and is consistent with the CIR dynamics for the abstract short rate, thus giving rise to closed form bond option and swaption prices.

Key Words:

• Interest rates
• yield curve
• term structure
• bonds
• bond options
• swaptions
• square-root process
• exponentially affine measure

Acknowledgements

I thank Eric Fournier for introducing me to the Flesaker–Hughston approach and Magnus Hyll for the valuable discussions regarding the CIR model.

The ideas expressed herein are mine and do not necessarily reflect those of JPMorgan Chase.