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.
 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.

## Laplace transform.

e start our consideration from the Fourier transform.

 (Direct Fourier transform)
 (Inverse Fourier transform)
We aim to expand these relationships to a transformation where the is a complex number , , for some . The idea is to split and apply ( Direct Fourier transform ),( Inverse Fourier transform ) to .

Suppose the integral converges absolutely for the given interval of values . We proceed according to the stated above idea: We would like to recover the from . We use ( Inverse Fourier transform ) with the substitution , , : We transform the last relationship: Hence, we invert with

 Notation. Index. Contents.