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: We start from the uniform on random variables and apply the Box-Muller procedure, described in the claim ( Box-Muller procedure ). The step is a unitary linear transformation of the iid standard normal variables into a jointly normal variables . The transformation produces two correlated uniform variables according to the ( Sklar_theorem_2 ). The transformation is the application of the ( Sklar_theorem_1 ). We spell out every step below.

We choose the transformation , to construct the with the following properties We observe that Hence, it suffices to chose according to

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

We also introduce the function and uniform on [0,1] random variables : Hence,

For the final step we are given the cumulative distributions . We simulate the variables and according to the rules Such variables have marginal cumulative distributions and respectively. Their correlation is controlled by the parameter . The joined distribution is given by the following calculation: If is 0 then the splits into product and the are uncorrelated. Preservation of sign of correlation takes place.

 Notation. Index. Contents.