I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 1 Black-Scholes formula.
 2 Change of variables for Kolmogorov equation.
 3 Mean reverting equation.
 4 Affine SDE.
 A. Ricatti equation.
 B. Evaluation of option price.
 C. Laplace transform.
 D. Example: CDFX model.
 a. Definition of CDFX model.
 b. The martingale normalization (CDFX).
 c. Fourier transform (CDFX).
 d. Calculation of Fourier transform (CDFX).
 e. Calculation of Premium Leg of CDS.
 f. Calculation of the protection leg of the CDS.
 5 Heston equations.
 6 Displaced Heston equations.
 7 Stochastic volatility.
 8 Markovian projection.
 9 Hamilton-Jacobi Equations.
 IV. Data Analysis.
 V. Implementation tools.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Calculation of Fourier transform (CDFX).

e summarize the situation as follows. We have a 3-dimentional process given by the equation where we use the notation

We calculate the increment For to be a martingale, we must have We introduce the notations as follows: Consequently, we have We separate terms with every power and coordinate of :

We now recover the expressions for before we substitute it into the above ODEs for and . By comparing definitions we see Similarly for we have definitions hence Also,

We now recover the ODEs for :

We calculate the expression under assumption that is the only non-zero correlation:

Hence, Equivalently,

The equation for resolves to

We summarize the results up to this point as follows

We would like to compare these results with the results of the paper [CarrWu2006a] . Hence, we perform additional transformations.

We calculated earlier Hence, where the is the notation of the paper. We change variable from to , hence, The last expressions compare exactly with the paper [CarrWu2006a] .

 Notation. Index. Contents.