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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.

Definition of change of numeraire.

e build on results of the section ( Change_of_measure_definition_section ).

Suppose that $M_{t}\,$ is a numeraire with respect to the probability measure $P^{M}$ and filtration $\QTR{cal}{F}_{t}$ . By definition, this means that MATH and for any traded asset $S_{t}$ the ratio MATH is a martingale: MATH with respect to some probability $P^{M}$ . We will refer to the positivity condition $M_{t}>0$ as the "acceptable numeraire" condition. Suppose that $N_{t}$ is a price of another acceptable ( MATH ) instrument. We would like to produce a probability measure $P^{N}$ such that for any traded asset $S_{t}$ MATH We write MATH MATH Such transformation has the form ( Main property of change of measure ) with MATH . To determine the constant we recall that $a_{0}=1$ . Hence,

MATH (Change of numeraire kernel)

Notation. Index. Contents.

Copyright 2007