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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.

## Transformation of SDE based on delta hedging. he goal of this section is to understand the change of numeraire from trading point of view. We perform the delta-hedging argument and connect the change of numeraire to the change of variables in the backward Kolmogorov's equation.

The state of the world is given by the processes and suitable as numeraires, see ( Suitable numeraire ). Thus and There is a traded derivative priced at . Assuming that the derivative is defined by the final payoff , the function has two descriptions. The notations are explained below.

First description:   Second description:   We assume above that , are described by the SDEs  with respect to any numeraire .

The notation stands for (XY bracket) According to the formula ( Change of Brownian motion ), the terms are connected by the relationships   Similarly,  We would like to see connection of these results to the delta hedging argument.

We consider a derivative given by the state variable . We form a portfolio and perform hedging calculation (see the argument of the chapter ( Delta hedging )):   We change the unknown function calculate the derivatives    and substitute these into the PDE : We conclude  By symmetry of and variables, we obtain   We compare with the proposition ( Multidimensional backward Kolmogorov equation ) and derive    These results agree with the previously stated goals and Hence, from PDE point of view, change of numeraire is the particular multiplicative change of the unknown function. From -hedging point of view, it is a change of units of measure. For example, an interest rate contract may be priced in units of currency or in units of treasury bills.

 Notation. Index. Contents.