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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.

Bid and ask.

f we are selling a derivative with the cash flow MATH at time 1 then we would like to hedge it with the traded assets. We are assuming that the derivative $h$ is not among components of MATH and is not a linear combination of such components. We require that for some hedge MATH MATH Hence, our asking price is MATH Similarly, the bid is MATH

The random variable MATH is the price of riskless asset at time $k$ . Suppose that there is an opportunity to sell a contingent claim $h$ for price $x_{0}\in\U{211d} $ , $x_{0}\leq$ ask $\left( h\right) $ . Because there is not enough money to finance the hedge, such opportunity does not change the MATH MATH According to the result ( Existence of incomplete market pricing ), the last statement is equivalent to MATH Hence, we proved that MATH Therefore, MATH The opposite inequality MATH is derived by similar argument. Indeed, if we are going to offer $h$ for $x_{1},x_{1}>$ ask $\left( h\right) $ then we are still not adding the acceptable opportunities and MATH Consequently, MATH Hence,

MATH (Incomplete market ask)
MATH (Incomplete market bid)

Notation. Index. Contents.

Copyright 2007