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.

Change of numeraire in currency markets.

e have description of market under some numeraire $A_{t}$ in $\$$ -denomination and we would like to change to some $\U{a3}$ -denominated numeraire $B_{t}$ (The $A_{t}$ is measured in $\$$ and the $B_{t}$ is measured in $\U{a3})$ . We introduce pound price of a dollar $Y_{t}=$ MATH . A $\$$ -amount should be multiplied by MATH to obtain a $\U{a3}$ -amount. We also introduce the reciprocal quantity MATH .

We proceed to calculate the drift of $X_{t}$ . Suppose we have one pound at time $t$ . We may invest into pound bonds MATH and convert to dollars at maturity. We may also convert to dollars right away and invest into dollar bonds MATH . We get a dollar outcome in both situations and the dollar risk neutral expectation of both strategies should be the same. We express such conclusion below: MATH We move the time $t$ -known quantities out of the expectation sign and obtain MATH Let $T=t+dt$ . We get MATH and consequently MATH Note that the expectation MATH is the drift that we are calculating and the bonds have expansions MATH The above is to be compared with the formula ( Bond SDE ) under the condition MATH . Hence, MATH or MATH where the $W_{t}^{\ast,\$}$ is standard Brownian motion with respect to risk neutral probability measure on dollar market. The result agrees with the intuition that when the dollar MMA rate is higher than the pound MMA rate then the exchange rate should drift against dollar (otherwise there would be arbitrage).

By similar argument MATH We also have MATH Hence,

MATH (X to Y connection)

We now collect results for general case. We want to change numeraire from MATH to MATH , where $A_{t}$ is a price of a traded asset denominated in $\$$ . We execute the program MATH Hence, MATH or MATH

Notation. Index. Contents.

Copyright 2007