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 in -denomination and we would like to change to some -denominated numeraire (The is measured in and the is measured in . We introduce pound price of a dollar  . A -amount should be multiplied by to obtain a -amount. We also introduce the reciprocal quantity .

We proceed to calculate the drift of . Suppose we have one pound at time . We may invest into pound bonds and convert to dollars at maturity. We may also convert to dollars right away and invest into dollar bonds . 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: We move the time -known quantities out of the expectation sign and obtain Let . We get and consequently Note that the expectation is the drift that we are calculating and the bonds have expansions The above is to be compared with the formula ( Bond SDE ) under the condition . Hence, or where the 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 We also have Hence, (X to Y connection)

We now collect results for general case. We want to change numeraire from to , where is a price of a traded asset denominated in . We execute the program Hence, or Notation. Index. Contents.