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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 $X_{t}$ and $Y_{t}$ suitable as numeraires, see ( Suitable numeraire ). Thus MATH and MATH There is a traded derivative priced at MATH . Assuming that the derivative is defined by the final payoff MATH , the function MATH has two descriptions. The notations are explained below.

First description: MATH MATH MATH Second description:


We assume above that $X_{t}$ , $Y_{t}$ are described by the SDEs MATH MATH with respect to any numeraire $Z$ .

The notation MATH stands for

MATH (XY bracket)
MATH According to the formula ( Change of Brownian motion ), the $\mu$ terms are connected by the relationships MATH MATH MATH Similarly, MATH MATH

We would like to see connection of these results to the delta hedging argument.

We consider a derivative $V$ given by the state variable $\left( X,Y\right) $ . We form a portfolio MATH and perform $\Delta-$ hedging calculation (see the argument of the chapter ( Delta hedging )): MATH MATH MATH We change the unknown function MATH calculate the derivatives MATH MATH MATH MATH and substitute these into the PDE MATH : MATH We conclude MATH MATH By symmetry of $x$ and $y$ variables, we obtain MATH MATH MATH We compare with the proposition ( Multidimensional backward Kolmogorov equation ) and derive MATH MATH MATH MATH These results agree with the previously stated goals MATH and MATH Hence, from PDE point of view, change of numeraire is the particular multiplicative change of the unknown function. From $\Delta$ -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.

Copyright 2007