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.

## Definition of CDFX model.

e are modelling the quantities and . The is dollar price of a foreign currency (exchange rate), see the section ( Currency exchange ). The is intensity of sovereign default pertaining to the foreign currency. We use the following equations under the risk neutral measure. where the are stochastic processes, are standard Brownian motions under the risk neutral measure, are constant parameters, is the random jump magnitude and is the martingale normalization constant.

The and are stochastic riskless rates of accrual on dollar and foreign MMAs. We will also use the notations , . These processes are determined by the term structure of bond prices

The is modelled as a normal variable .

The is a Poisson process, see the section ( Poisson process ).

The are subsequently assumed to be 0.

 Notation. Index. Contents.