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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 $S_{t}$ given by the SDE MATH under the original probability measure. There is possibility of short time borrowing at some deterministic riskless rate $r\left( t\right) $ . We want to determine the present $t=0$ price of a derivative paying MATH at maturity $T$ . Let $V_{t}$ be the time $t$ price of the derivative. To apply the Ito formula we need to know the functional dependence of $V_{t}$ . If we make the simplifying assumption MATH MATH MATH then MATH because the filtration $\QTR{cal}{F}_{t}$ of the model is generated by $S_{t}$ : MATH and the $S_{t}$ is Markovian. Suppose that at the time moment $t$ we are short one unit of derivative and long MATH units of the underlying asset $S_{t}.$ The value of the position is MATH At the infinitesimally close moment $t+dt$ the value of the portfolio becomes MATH Hence, the infinitesimal change in portfolio's value is MATH by ( Ito_formula ), MATH The choice MATH makes the value $\Sigma_{t}$ instantaneously deterministic. Hence, it has to perform as the money market account (MMA) during the small interval MATH : MATH where MATH and MATH Therefore, MATH We compare the last result with the proposition ( Backward Kolmogorov for discounted payoff ) and conclude MATH MATH

Notation. Index. Contents.

Copyright 2007