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 I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 2 Change of numeraire.
 A. Definition of change of numeraire.
 B. Useful calculation.
 C. Transformation of SDE based on change of measure results.
 D. Transformation of SDE in two asset situation.
 E. Transformation of SDE based on term matching.
 F. Invariant representation for drift modification.
 G. Transformation of SDE based on delta hedging.
 H. Example. Change of numeraire in Black-Scholes economy.
 I. Other ways to look at change of numeraire.
 3 Basics of derivative pricing II.
 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.

## Invariant representation for drift modification.

e already established in ( Change of measure recipe section ) and ( Change of numeraire recipe section ) that the change of numeraire is all about the modification of SDE drift term. However, the lengthy calculation of the ( Two asset change of numeraire section ) creates uneasy feeling. We would like to have a formula for direct transformation of the drift that is independent of the choice of the driving Brownian motion.

We are changing from numeraire to numeraire and look at price of some traded asset : The notation is chosen according to the rule "drift of Z in X-based probability". We perform the change ( Change of Brownian motion ) then the SDE for changes as follows Hence, We represent the result in symmetrical form:

 (Change of drift recipe)
where is the drift of under the -based probability measure, is the drift of under -based probability measure. We obtain another useful representation as follows Therefore,
 (Change of drift recipe 1)
for any traded asset and numeraires and .

 Notation. Index. Contents.