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

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

 Notation. Index. Contents.