I. Basic math.
 II. Pricing and Hedging.
 1 Basics of derivative pricing I.
 A. Single step binary tree argument. Risk neutral probability. Delta hedging.
 B. Why Ito process?
 C. Existence of risk neutral measure via Girsanov's theorem.
 D. Self-financing strategy.
 E. Existence of risk neutral measure via backward Kolmogorov's equation. Delta hedging.
 F. Optimal utility function based interpretation of delta hedging.
 2 Change of numeraire.
 3 Basics of derivative pricing II.
 4 Market model.
 5 Currency Exchange.
 6 Credit risk.
 7 Incomplete markets.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Existence of risk neutral measure via Girsanov's theorem.

he is a reference filtration and is a column of -adapted independent standard Brownian motions. Consider a column of prices of several traded assets : where and are column and matrix-valued processes adapted to the filtration . For simplicity we restrict our attention to assets without dividends. We assume a possibility to construct a portfolio such that the value of the portfolio is deterministic during the next infinitesimally small time interval: The is another column-valued process adapted to . It represents trading strategy. By no-arbitrage assumption we conclude that such riskless portfolio must earn the riskless rate : Equivalently, where the is stochastic -adapted riskless rate. We summarize the above argument with the following statement We think of columns , of the matrix as elements of a linear space of some finite dimension. We introduce the linear span of the set . Let be an orthogonal projection to the linear span . The statement (*) reads as We would like to conclude that . Indeed, suppose such conclusion is false and there is a nonzero element We have by the form of the . We arrive to a contradiction by taking in the statement (**). Therefore . Equivalently, there are -adapted processes such that or

 (Market price of risk)
This result is remarkable because the is dependent only on one index. The formula ( Market_price_of_risk ) states that the excess return of the asset over the riskless rate is proportional to the volatilities associated with the driving Brownian motions and the coefficient of proportionality does not depend on the type of the asset but is only dependent on the source of risk. Therefore, the coefficients are called "market prices of risk". We have
 (Risk neutral Brownian motion)
According to the proposition ( Girsanov theorem ), there exists a change of the probability measure that makes the process given by a standard Brownian motion. Such probability measure is called "the risk neutral measure". Note that hence the discounted price of any traded asset is a martingale with respect to the risk neutral measure :
 (Risk neutral pricing)
The above relation is the foundation of classic derivative pricing.

Consider the following special situation: where the and are some deterministic functions of time. The ( Market price of risk ) reads as or

Hence, in the setting (***), if is a deterministic function then the market prices of risk are deterministic functions as well.

A particular vector may be represented as a linear combination of in many different ways if the number of vectors is bigger then the dimensionality of their linear span. If there are more sources of uncertainty than traded assets then the risk neutral measure is not unique. This means that there may be a variety of opinions about prices of contingent claims but still there is no arbitrage.

 Notation. Index. Contents.