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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.
4. Market model.
5. Currency Exchange.
6. Credit risk.
7. Incomplete markets.
A. Single time period discrete price incomplete market.
B. Coherent measure.
C. Incomplete market with multiple participants.
D. Example: uncertain local volatility.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Coherent measure.

et MATH be a probability space. It is natural, after results ( Incomplete market ask ) and ( Incomplete market bid ), to consider a mapping MATH where the $\QTR{cal}{D}$ is some set of absolutely continuous with respect to $P$ measures.


Let $L^{0}$ be a set of random variables on $\Omega$ . A coherent measure is a mapping: MATH :

MATH (Coherent measure)


The coherent measure has the following properties:


1. (Subadditivity) MATH .

2. (Monotonicity) MATH .

3. (Positive homogeneity) MATH for MATH .

4. (Translation invariance). MATH .

5. (Fatou property). MATH in MATH .


Given a coherent measure $\rho$ , the largest set $\QTR{cal}{D}$ that enables the relationship ( Coherent measure ) is called the determining set MATH .

Note that if $\Omega$ is finite then the representation ( Coherent measure ) is exactly the formula ( Support function ). Hence, for the finite $\Omega$ the representation ( Coherent measure ) completely describes all functions that have the property 1 and 3. Furthermore, if we introduce a notion of maximal set $\QTR{cal}{D}$ (in inclusion sense) for a measure $\rho$ then, given $\rho$ and $\QTR{cal}{D}$ as in ( Coherent measure ), taking double convex dual of $\rho$ reveals that the MATH is a closure of the convex hull of the original $\QTR{cal}{D}$ . Another description for MATH is MATH where the $\QTR{cal}{P}$ is the set of all absolutely continuous measures with respect to $P$ .

In the case of general $\Omega$ there is a following representation theorem.


A measure MATH satisfies the conditions 1-5 above iff there exists a non empty MATH such that the ( Coherent measure ) holds.

Note that the theorem holds for $L^{\infty}$ . There reason may be easily understood by looking at derivations around the formula ( Support function ). To extend those derivations to the general case we need the duality to the space $L^{\infty}$ in order to make sense of all the scalar products MATH . Such duality needs to have reflexivity. The $L^{\infty}$ has such dual space $L^{1}$ while $L^{0}$ , endowed with the topology of convergence in probability, does not have such structure.

Notation. Index. Contents.

Copyright 2007