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 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 the protection leg of the CDS.

e would like to calculate the expression We proceed similarly to the previous section. We calculate the drift Hence, We conclude that Following logic of the previous section we will be seeking for a function of the form for some deterministic functions having the property We have We introduce notation with

then Hence we calculate the drift part as We want the above expression to be Hence, it suffices to have where we made an assertion that and are the same as in the previous section and and are given by the same expressions Consequently, We separate the coordinates We substitute the expressions for the . The equations for have been calculated before. The equations for resolve to

 Notation. Index. Contents.