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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.

## Single time period discrete price incomplete market.

his section follows the article [CarrMadan2000] .

We consider economy that has two time moments and a finite space of future realizations of the world (finite number of "states"). The present is the time moment 0. The future is the time moment 1. The states are indexed by . There is no cost of borrowing or there is a riskless asset and we measure prices in units of such asset. We introduce several points of view on development of the market, indexed by . We use the following notation:

 (Finite space variable incomplete market)
The position with the property is "0-cost portfolio" or "opportunity". An "acceptable opportunity" is a vector such that for some vector of "floors" .

We divide our models into two categories. Valuation category consists of regular models (probability measures) representing reasonable expectations about development of the market. In order for an opportunity to be acceptable we require that for such models. The measures are called "valuation measures". The other category is stress models . Stress models represent extreme point of view about market behavior. These are introduced to exclude infinite risk taking along opportunities accepted by valuation measures. It is also a representation of common risk management practices. We require that

 (Stress tests)
There is no point in introducing a stress measure for a situation that is not considered possible under any reasonable assumptions. Conversely, if we believe that certain situation requires a stress measure then it better have non zero probability under some reasonable model. Hence, we require that the union of supports of all stress measures would be included in the union of supports of all valuation measures: Valuation measures may be obtained via empirical research, hedging argument or via utility function argument. For example, where, as usual, stand for a utility function (with opinion index ) and is a position vector of the market participant, responsible for the opinion.

A strictly acceptable opportunity is an acceptable opportunity s.t. Such opportunity should be taken. Hence, one may assume that market would reach a state when there are no strictly acceptable opportunities. The assumption of "no acceptable opportunity" is stronger then the assumption of "no arbitrage". Hence, the existence of some (not necessarily unique) risk neutral measure is given under the "no strictly acceptable opportunity" assumption. There is however a stronger consequence: We call such a "pricing vector".

The uniqueness statement of the theory is connected with the notion of acceptable completeness of the market. The market is acceptably complete if any time-1 payoff could be hedged so that the residue, after applying of the hedge, is an acceptable opportunity. Under condition that there are more assets (index ) than models (index ), the market is acceptably complete if and only if the vector is unique (see the following sections for justification).

Bid-ask spread is defined as a minimal price of a hedge, transforming an exposure into an acceptable opportunity. Note that under such setting the bid-ask spread is dependent on the current position of the market participant. In particular, a market maker is justified in modifying a bid-ask spread after each trade.

 a. Existence of pricing vector.
 b. Uniqueness of pricing vector.