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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.
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 MATH , the set of acceptable opportunities MATH is defined by MATH

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

Remark

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

Definition

Given the market structure MATH the assumption of "no acceptable opportunities" is the requirement that the set MATH 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 MATH MATH

Notation

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

Hence, MATH

Condition

(Basic coherence condition) We always assume the following: MATH

Proposition

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

In other words MATH iff there exists a vector $w$ s.t. MATH

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 1


Incomplete Market 2
Incomplete Market 2

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

It is now clear why we need the condition ( Basic coherence condition ): If all the $\QTR{cal}{P}_{k}$ lie in the plane then the replication is never possible.

Proof

(If there exists $w$ then there are no strictly acceptable x). We assume MATH Thus MATH Since $w_{k}>0$ this implies MATH Therefore MATH

Proof

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

We introduce the transformation $y=\QTR{cal}{P}x$ . Let MATH The starting assumption is equivalent to MATH We would like to conclude that MATH If there exists MATH then MATH and MATH s.t. MATH and MATH . Hence, it suffices to show that MATH We assume existence of $x_{0},x_{1}$ with the properties $x_{0}\in C_{0}$ , $x_{1}\in C_{1}$ , MATH and derive the contradiction: MATH We conclude MATH

The $\QTR{cal}{P}C_{0}$ is an affine set and $\QTR{cal}{P}C_{1}$ is a convex set. Hence, there is a hyperplane that strictly separates these two sets: $\exists w$ such that MATH By definition of $C_{1}$ and $\QTR{cal}{P}$ , for every MATH , the components $y_{k}$ are non-negative. Hence, $w>0$ . Also, for every $x\in C_{0}$ MATH and $C_{0}$ has dimension N-1. Hence, MATH for some constant $C$ . This concludes the proof.

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

If we assume existence of a riskless asset then there is no choice but to conclude $C=1$ . Hence, the $w^{T}P$ represents some convex combination of the original measures and the equality MATH means that $w^{T}P$ is a risk neutral measure.

Definition

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

Corollary

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

Here $conv_{k}P_{k}$ refers to the convex hull across various ways to model the market (across the index $k$ ).

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 $x$ with the property MATH we introduce the set MATH Clearly, by assumption of no acceptable opportunities MATH The next task is to prove that MATH We argue by contradiction. Assume MATH and introduce MATH By the assumption MATH we have MATH By the theorem ( Separating hyperplane theorem ) there exists MATH s.t. MATH From the latter inequality we derive that $u\geq0$ . Then from the former inequality we conclude MATH But MATH is another portfolio with the property MATH . Hence, we arrived to contradiction with the assumption of no acceptable opportunities. We conclude that any finite intersection MATH is non empty.

We next want to prove that the intersection MATH is not empty for any countable collection of $x$ . We introduce the linear subspace MATH The $L$ may be empty. Any x s.t. MATH may be represented as a sum MATH where the $l\in L$ and the y has the property MATH Also, MATH Hence, we will restrict further proof to $y$ only (switch to the orthogonal complement of $L$ ). For $y$ the set $C(y)$ always has an interior. Let $\mu$ be the Lebesgue measure acting in the space of $w$ : MATH . By the structure of $C\left( y\right) $ the mapping MATH is continuous with respect to y. We choose some sequence MATH where $B_{1}$ is the unit ball. By taking a subsequence we have MATH for some $y_{0}$ . We have MATH Since $C\left( y\right) $ always has an interior we also have MATH By taking further subsequence and using continuity of MATH we have MATH where we use the notation MATH for the difference of sets. Note, that MATH has properties of distance between sets. Hence, MATH cannot converge to zero (because MATH and MATH are $\mu-$ small and MATH is at least as large as MATH ). Consequently, MATH cannot converge to zero. Hence, we proved that MATH is not empty for any countable collection MATH .

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

It remains to note that the structure of the set $C\left( x\right) $ is such that the result extends from MATH to general $x$ .

Remark

Note that the last proof expands to the case of continuous random variable $x$ if we have weak compactness of the set of probability measures $\left\{ P\right\} $ .





Notation. Index. Contents.


















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