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.

## Transformation of SDE in two asset situation.

n the formula ( Change of Brownian motion ) the and are columns and the are scalar products. We intend to apply the result ( Change of Brownian motion ) to a situation of two assets given by one dimensional correlated diffusion terms. where the expression stands for 's Brownian motion with respect to taken as numeraire. We assume that for some number . The increments are jointly normal. We are seeking a matrix such that where is a column of independent standard Brownian motions.

Observe that the Brownian motions given by are standard and satisfy the condition Hence, it suffices to set We write The formula ( Change of Brownian motion ) takes the form Therefore, we write SDE for , in the -measure as follows where the Brownian motions are standard under the measure of numeraire and satisfy the relationship

 Notation. Index. Contents.