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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
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I. Basic math.
II. Pricing and Hedging.
1. Basics of derivative pricing I.
2. Change of numeraire.
3. Basics of derivative pricing II.
4. Market model.
5. Currency Exchange.
A. Change of numeraire in currency markets.
B. Invariant form of SDE transformation formula.
C. Delta hedging in currency markets.
D. Example: forward contract to purchase foreign stock for domestic currency.
E. Example: forward currency exchange contract.
F. Example: quanto forward contract.
G. Example: quanto caplet.
H. Example: quanto fixed-for-floating swap.
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.

Invariant form of SDE transformation formula.

reviously we obtained ( Change of drift recipe 1 ) that the drift of an SDE changes according to MATH We now combine the program MATH into a similar expression. Note, that the general formula does not work for the second transformation between the risk neutral measures. We use the explicit result for that case. Let $S_{t}$ be a $\$$ -price of a traded asset. Then MATH We put the recipes together:

MATH Therefore,

MATH (Currency change of numeraire recipe)
The above expression may be understood by considering units of measure. The $\mu_{S.t}$ is an absolute quantity but $B_{t}$ and $A_{t}$ are measured in units of currency: pound for $B_{t}$ and dollar for $A_{t}$ . Hence, there must be a multiplication by $X_{t}$ .

Note that the transition MATH does not involve multiplication of $S_{t}$ by exchange rate. The path of $S_{t}$ remains the same. This is different from the change MATH In such situation SDE transformation is slightly different: MATH thus MATH We use the formula ( X to Y connection ) to convert everything to $\$$ quantities: MATH and conclude MATH

Notation. Index. Contents.

Copyright 2007