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.KeywordsContingent ClaimMartingale MeasureIncomplete MarketUnderlying AssetPrice EquationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.