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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.
A. Delta hedging in situation of predictable jump I.
B. Delta hedging in situation of predictable jump II.
C. Backward Kolmogorov's equation for jump diffusion.
D. Risk neutral valuation in predictable jump size situation.
E. Examples of credit derivative pricing.
F. Credit correlation.
a. Generic Copula.
b. Gaussian copula.
c. Example: two dimensional Gaussian copula.
d. Simplistic Gaussian copula.
G. Valuation of CDO tranches.
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.

Example: two dimensional Gaussian copula.


he following is the program of our actions: MATH We start from the uniform on $[0,1]$ random variables MATH and apply the Box-Muller procedure, described in the claim ( Box-Muller procedure ). The step $Q$ is a unitary linear transformation of the iid standard normal variables MATH into a jointly normal variables MATH . The transformation MATH produces two correlated uniform variables according to the ( Sklar_theorem_2 ). The transformation MATH is the application of the ( Sklar_theorem_1 ). We spell out every step below.

We choose the transformation $Q$ , MATH to construct the $\eta$ with the following properties MATH We observe that MATH Hence, it suffices to chose $Q$ according to MATH

The step MATH is performed according to the idea behind the result ( Sklar_theorem_2 ). We note that the cumulative standard normal distribution $\Phi$ produces a uniform on [0,1] random variable $u$ from a standard normal variable $\xi$ according to the rule MATH Indeed, for such $u$ we calculate

MATH

We also introduce the function $\Phi_{\rho}$ and uniform on [0,1] random variables MATH : MATH Hence, MATH

For the final step MATH we are given the cumulative distributions MATH . We simulate the variables $X_{1}$ and $X_{2}$ according to the rules MATH Such variables have marginal cumulative distributions $F_{1}$ and $F_{2}$ respectively. Their correlation is controlled by the parameter $\rho$ . The joined distribution is given by the following calculation: MATH If $\rho$ is 0 then the $\Phi_{\rho}$ splits into product and the $X_{i}$ are uncorrelated. Preservation of sign of correlation takes place.





Notation. Index. Contents.


















Copyright 2007