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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 two risky assets.

uppose the economy has the two risky assets $S_{1},$ $S_{2}$ given by the SDEs MATH MATH and the MMA $r_{t}$ . The $W_{i,t}$ are standard Brownian motions. Similarly to the previous section we again assume that MATH is deterministic and MATH MATH . Therefore, the price of the derivative $V_{t}$ with the payoff MATH at $T$ has the functional form MATH . We construct the portfolio $\Sigma$ MATH and compute the increment MATH MATH MATH MATH Similarly to the previous section, set MATH then MATH MATH or MATH We introduce the function MATH : MATH then MATH MATH We compare the last result with the proposition ( Multidimensional backward Kolmogorov equation ) and conclude MATH MATH MATH

Notation. Index. Contents.

Copyright 2007