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.
 a. Existence of pricing vector.
 b. Uniqueness of pricing vector.
 c. Bid and ask.
 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.

## Existence of pricing vector.

efinition

Given the market structure , the set of acceptable opportunities is defined by

The is matrix multiplication. The notation is k-th component.

Remark

The assumption that the market does not have acceptable opportunities is equivalent to the assumption that for any such that either or or Observe that if the last inequality is the only one that prevents from being an acceptable opportunity then the may be simply scaled down (remember that the is negative): Such , taken in place of , becomes an acceptable opportunity. Hence, we introduce the following definition.

Definition

Given the market structure the assumption of "no acceptable opportunities" is the requirement that the set is empty. Note the absence of stress measures in the present definition (see the last remark).

We are going to drop the stress measures from further considerations of this chapter.

Notation

We introduce the following notation

Notation

We will use the notation or for a vector to represent the situation

Hence,

Condition

(Basic coherence condition) We always assume the following:

Proposition

(Basic existence of incomplete market pricing) There are no acceptable opportunities iff there exists a pricing vector.

In other words iff there exists a vector s.t.

We explain the geometrical meaning of the proposition ( Basic existence of incomplete market pricing ) using the pictures ( Incomplete Market 1 ),( Incomplete Market 2 ).

Incomplete Market 1

Incomplete Market 2

If the projections of vectors on the plane are pointing in all directions then may be replicated by a positive linear combination of . Otherwise, if the projections of are in the same half-plane with respect to any subdivision to half-planes in then such replication is not possible because then would have a non-zero projection on the plane.

It is now clear why we need the condition ( Basic coherence condition ): If all the lie in the plane then the replication is never possible.

Proof

(If there exists then there are no strictly acceptable x). We assume Thus Since this implies Therefore

Proof

(If there are no strictly acceptable then there exists ). We assume that the conditions are never satisfied together.

We introduce the transformation . Let The starting assumption is equivalent to We would like to conclude that If there exists then and s.t. and . Hence, it suffices to show that We assume existence of with the properties , , and derive the contradiction: We conclude

The is an affine set and is a convex set. Hence, there is a hyperplane that strictly separates these two sets: such that By definition of and , for every , the components are non-negative. Hence, . Also, for every and has dimension N-1. Hence, for some constant . This concludes the proof.

Note that the vector is defined as a normal vector to some separating hyperplane. Set and consider the consequence of :

If we assume existence of a riskless asset then there is no choice but to conclude . Hence, the represents some convex combination of the original measures and the equality means that is a risk neutral measure.

Definition

Given the set of available instruments we introduce the set of risk neutral probability measures:

Corollary

If there are no acceptable opportunities and there is a riskless asset then

Here refers to the convex hull across various ways to model the market (across the index ).

Proof

Even though the corollary above clearly follows from considerations of the present section, such considerations are not easily extensible to more complex situations. Hence, we give a more generic proof.

For any portfolio with the property we introduce the set Clearly, by assumption of no acceptable opportunities The next task is to prove that We argue by contradiction. Assume and introduce By the assumption we have By the theorem ( Separating hyperplane theorem ) there exists s.t. From the latter inequality we derive that . Then from the former inequality we conclude But is another portfolio with the property . Hence, we arrived to contradiction with the assumption of no acceptable opportunities. We conclude that any finite intersection is non empty.

We next want to prove that the intersection is not empty for any countable collection of . We introduce the linear subspace The may be empty. Any x s.t. may be represented as a sum where the and the y has the property Also, Hence, we will restrict further proof to only (switch to the orthogonal complement of ). For the set always has an interior. Let be the Lebesgue measure acting in the space of : . By the structure of the mapping is continuous with respect to y. We choose some sequence where is the unit ball. By taking a subsequence we have for some . We have Since always has an interior we also have By taking further subsequence and using continuity of we have where we use the notation for the difference of sets. Note, that has properties of distance between sets. Hence, cannot converge to zero (because and are small and is at least as large as ). Consequently, cannot converge to zero. Hence, we proved that is not empty for any countable collection .

To see that the same statement holds for any collection we note that there exists a countable everywhere dense set on and is continuous. Hence, any is infinitely close to some point and we can always have for however small . Since we conclude

It remains to note that the structure of the set is such that the result extends from to general .

Remark

Note that the last proof expands to the case of continuous random variable if we have weak compactness of the set of probability measures .

 Notation. Index. Contents.