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.

## Useful calculation. he previous section hints that we are going to work with ratios of stochastic processes. We will repeatedly perform the ( Ito_formula )-based evaluation of and   Hence, (Useful formula)
Suppose the stochastic processes and are given by the SDEs  where the and are columns. In terms of the SDE's coefficients we obtain Note, if is a Brownian motion in a -numeraire measure then the above drift is zero and It is informative to compare the last relationship with the ( Market prices of risk ).

 Notation. Index. Contents.