Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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 reference currency invested in MMA at time . It is a traded asset suitable as a numeraire, see ( Suitable numeraire ).

According to the martingale result ( Risk neutral pricing ), price of a call stroke at written on the traded asset is given by Let , . We rewrite the last formula for : The expressions and may be regarded as parts of kernels for numeraire changes, see ( Change of numeraire definition ). Indeed, where and the Prob is the probability changed to the numeraire . Similarly, where 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 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.