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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.
 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.

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

The random variable is the price of riskless asset at time . Suppose that there is an opportunity to sell a contingent claim for price , ask . Because there is not enough money to finance the hedge, such opportunity does not change the According to the result ( Existence of incomplete market pricing ), the last statement is equivalent to Hence, we proved that Therefore, The opposite inequality is derived by similar argument. Indeed, if we are going to offer for ask then we are still not adding the acceptable opportunities and Consequently, Hence,