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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.
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 $f(t,s)$ connected to defaultless bond price MATH by the relationship MATH where $t$ is observation time and $T$ is maturity of the bond. We are given a reference filtration $\QTR{cal}{F}_{t}$ and the real world SDE MATH where MATH and MATH are $\QTR{cal}{F}_{t}$ -adapted vector and matrix valued processes.

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

We introduce the convenience notations MATH and MATH for any function of two variables $h$ . We calculate MATH MATH MATH We introduce MATH for some $\QTR{cal}{F}_{t}$ -adapted process $\theta_{t}$ and continue MATH MATH By existence of the risk neutral measure, see ( Risk neutral Brownian motion ), there has to be a $\theta_{t}$ such that MATH We differentiate the above relationship with respect to the variable $T$ : MATH and substitute (***) and (****) into (**): MATH

Summary

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

MATH (Bond SDE)





Notation. Index. Contents.


















Copyright 2007