I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 A. Option pricing formula for an economy with stochastic riskless rate.
 B. T-forward measure.
 C. HJM.
 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.

## Option pricing formula for an economy with stochastic riskless rate.

e use the notation

 (MMA numeraire)
The is the worth of one unit of the reference currency invested in the MMA at time . It is a traded asset suitable as a numeraire ( Suitable numeraire ).

According to the martingale result ( Risk neutral pricing ), the price of a call stroke at written on the traded asset is given by Let , . The last formula for transforms to The expressions and may be regarded as parts of a kernel for the changes of numeraires ( Change of numeraire definition ). Indeed, the may be regarded as where and the Prob is the probability changed to the numeraire . Similarly, where the is the price of riskless bond with maturity as observed at time and Prob is the probability with respect to the numeraire . We compute the probabilities as follows. where the is a martingale under Prob . where the is a martingale under Prob . We are unable to proceed further without some assumptions about the distribution. Under the log-normal assumption, we compute both of the probabilities by setting where the variables are standard normal with respect to the corresponding measures and the volatility is the same in both of the expressions as shown in the ( Change of measure recipe section ).

 Notation. Index. Contents.