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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 in two asset situation.

n the formula ( Change of Brownian motion ) the $\sigma$ and $dW$ are columns and the $\sigma^{T}dW~\ $ 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. MATH MATH where the expression $dB_{Y,t}^{X}$ stands for $Y$ 's Brownian motion with respect to $X$ taken as numeraire. We assume that MATH for some number MATH . The increments $dB$ are jointly normal. We are seeking a matrix MATH such that MATH where MATH is a column of independent standard Brownian motions.

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

Notation. Index. Contents.

Copyright 2007