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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.
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.

Uniqueness of pricing vector.


e would like to derive conditions for the vector $w$ to be unique.

The defining relationship is (we omit the upper index $v$ on $P$ ) MATH or MATH To have uniqueness we must make sure that MATH where the Null MATH is the notation for null set. In general, MATH hence, it is enough to show that $\QTR{cal}{P}$ has the full range.

We introduce the notion of acceptable completeness.

Definition

The market is acceptably complete iff any payoff $h$ may be hedged with existing assets so that the remaining risk is acceptable: MATH

In other words MATH or MATH Hence, it is enough to make sure that $Ph$ spans the full space. This means that the space of cash flows is at least as rich as the structure of the test measures.

Summary

Assuming that $P$ has the maximal rank, the pricing vector is unique iff the market is acceptably complete.





Notation. Index. Contents.


















Copyright 2007