I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 A. Option pricing formula for an economy with stochastic riskless rate.
 B. T-forward measure.
 C. HJM.
 4 Market model.
 5 Currency Exchange.
 6 Credit risk.
 7 Incomplete markets.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## T-forward measure.

he measure associated with the numeraire is called the " forward measure". It is particularly useful when evaluating a price of a derivative. Indeed, the regular risk neutral measure corresponds to the defined above, see ( MMA numeraire ), numeraire , and Hence, the transformation to the T-forward measure moves the discounting outside of the expectation term.

Suppose an asset and the riskless bond are given by the equations with respect to the risk neutral measure. Here and are correlated increments of the standard Brownian motions, We perform transformation of the SDEs to the T-forward measure. The old numeraire is : and the new numeraire is . Hence, according to the formula ( Change of drift recipe ), the drift of in the -measure is Therefore where the is increment of the standard Brownian motion with respect to the -forward probability.

 Notation. Index. Contents.