Martingale Measures & Change of Measure Explained

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. European options are often 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. To relax such an assumption option payoffs and prices can be evaluated as the expectation of a stochastic martingale process.

In this paper we illustrate how to use the change of measure technique to evaluate the dynamics of a stochastic process. Firstly we discuss the preliminaries, namely Martingale measures and numeraires. Secondly we model interest rates as a Vasicek short rate process. Finally we outline how to apply a change of measure technique, where it can be seen that a change of measure to the terminal-forward measure allows us to evaluate model dynamics and simplify the calculation.

Keywords: Martingales, Numeraires, Measures, Change of Measure, Girsanov Theorem

JEL Classification: C02, C65, G12