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 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.
 a. An economy with one risky asset.
 b. An economy with two risky assets.
 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.

An economy with one risky asset.

uppose the economy has one risky asset given by the SDE under the original probability measure. There is possibility of short time borrowing at some deterministic riskless rate . We want to determine the present price of a derivative paying at maturity . Let be the time price of the derivative. To apply the Ito formula we need to know the functional dependence of . If we make the simplifying assumption then because the filtration of the model is generated by : and the is Markovian. Suppose that at the time moment we are short one unit of derivative and long units of the underlying asset The value of the position is At the infinitesimally close moment the value of the portfolio becomes Hence, the infinitesimal change in portfolio's value is by ( Ito_formula ), The choice makes the value instantaneously deterministic. Hence, it has to perform as the money market account (MMA) during the small interval : where and Therefore, We compare the last result with the proposition ( Backward Kolmogorov for discounted payoff ) and conclude

 Notation. Index. Contents.