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Quantitative Analysis
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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.

Example: uncertain local volatility.


his section follows [Avellaneda1995] .

Let $X_{t}$ be a traded asset and the only state variable $X_{t}$ is given by the SDE MATH under the risk neutral measure. We are considering a situation when the analytical form of the function MATH is not known. However, we do assume that MATH for all values of arguments. Following the conclusion ( Incomplete market ask ) we are seeking MATH for some final payoff function MATH .

We introduce the notation MATH for the value of the derivative $\phi$ dependent on the particular assumption $\sigma$ about the volatility. For any $\sigma$ and all $t,x$ we have MATH The supremum is approached by some sequence MATH . Under sufficient regularity restrictions on the class of $\sigma,\mu$ we pass the above PDE to the limit: MATH Clearly, the supremum is achieved if MATH MATH Since the $X_{t}$ is the only state variable, the delta hedging is performed in the regular way using the MATH .





Notation. Index. Contents.


















Copyright 2007