Pricing by Change of Measure and Numeraire

Abstract

There are several ways to derive the no-arbitrage price of a contingent claim, such as following a replicating portfolio strategy or solving a partial differential equation. Another prominent approach is martingale pricing, which is the method we deal with in this chapter. We briefly review well-known facts on equivalent measures, the Radon-Nikodym derivative, martingale measures, and the change of numeraires following Geman, El Karoui, and Rochet [27]. The only measures we consider within this thesis are the ones equivalent to the physical measure. The ultimate goal in deriving the pricing formula for a claim is to write it in terms of possibly different artificial probabilites. It is known that the choice of different numeraires allows for a convenient computation of the claim’s fair price. This can be seen when looking at the BS formula: The easiest method for the valuation of a standard call is to choose appropriate normalizing assets and corresponding martingale measures. This will be reviewed to motivate the concept of different numeraires.