The Hitchhiker's Guide to the Risk-Neutral Galaxy

Abstract

The initial reason for writing this essay was to provide the background and a derivation of the IR quanto adjustment, as a mean of expressing the Hull-White dynamics of foreign currency interest rates under a domestic risk-neutral measure. However, one thing led to another, and the essay ended up including three different applications of the change of measure (or, Girsanov Theorem); namely, quanto adjustment, stock price dynamics under the risk-neutral measure, and pricing IR derivatives under the T-forward measure. The presented approach is neither meant to be original nor mathematically rigorous: it just aims at building an intuition of the employed methodology, hence facilitating a quick understanding of change of measure technique and some of its important applications. It is structured as a “user manual” and targets an audience with little background on the matter.

Keywords: probability spaces, filtrations, Brownian motion, Ito's formula, Doleans-Dade exponential, Girsanov theorem, change of measure, quanto adjustment, Arrow-Debrew securities, risk-neutral probability, Cameron-Martin-Girsanov theorem, change of numeraire, T-forward measure

JEL Classification: C01, C02, C18, C65, G12, Z23