Bond Option Pricing using the Vasicek Short Rate Model

Abstract

An option is a financial instrument that allows the holder to buy or sell an underlying security in the future at an agreed strike or price set today. Many options are priced under the assumption of constant interest rates as seen in the Black-Scholes (1973) model. In interest rate markets however the underlying security is an interest rate, which cannot be assumed constant. Likewise bond markets have a similar requirement.In what follows the assumption of a constant interest rate is relaxed. Bond option pricing using the Vasicek short rate model is examined in such a way that the methodology could be applied to any short rate model such as the classical Hull-White model (1990a).Firstly we discuss the preliminaries, namely numeraires and measures, where it can be seen that a careful choice of numeraire can simplify option calculations. Secondly we summarize the Vasicek short rate process and a change of numeraire to the terminal-forward measure is outlined, which simplifies bond option pricing calculations. Thirdly we review both pure discount and coupon bond pricing. Fourthly bond option pricing formulae are derived and Jamshidian's Trick outlined.Finally in conclusion practical implementation considerations and model extensions are discussed. The aim of this paper is to provide a general overview of option pricing using short rate models, using the Vasicek model as an important case study.