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 be a probability space. It is natural, after results ( Incomplete market ask ) and ( Incomplete market bid ), to consider a mapping where the is some set of absolutely continuous with respect to measures.

Definition

Let be a set of random variables on . A coherent measure is a mapping: :

 (Coherent measure)

Proposition

The coherent measure has the following properties:

Proposition
Proposition

2. (Monotonicity) .

3. (Positive homogeneity) for .

4. (Translation invariance). .

5. (Fatou property). in .

Definition

Given a coherent measure , the largest set that enables the relationship ( Coherent measure ) is called the determining set .

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

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

Proposition

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

Note that the theorem holds for . 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 in order to make sense of all the scalar products . Such duality needs to have reflexivity. The has such dual space while , endowed with the topology of convergence in probability, does not have such structure.

 Notation. Index. Contents.