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.

## HJM.

e introduce forward rate connected to defaultless bond price by the relationship where is observation time and is maturity of the bond. We are given a reference filtration and the real world SDE where and are -adapted vector and matrix valued processes.

Our intention is to compute an SDE for by direct differentiation of (*), to produce the risk neutral measure from Girsanov's theorem and the requirement that would drift with riskless rate in the risk neutral world and, finally, to compute the risk neutral world version of the (**).

We introduce the convenience notations and for any function of two variables . We calculate We introduce for some -adapted process and continue By existence of the risk neutral measure, see ( Risk neutral Brownian motion ), there has to be a such that We differentiate the above relationship with respect to the variable : and substitute (***) and (****) into (**):

Summary

If riskless bond and forward rate are given by the real world SDEs (*) and (**) then, in the risk neutral world, the bond and forward rates evolve according to

 (Bond SDE)

 Notation. Index. Contents.