I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 A. Generation of random samples.
 B. Acceleration of convergence.
 C. Longstaff-Schwartz technique.
 D. Calculation of sensitivities.
 a. Pathwise differentiation.
 b. Calculation of sensitivities for Monte-Carlo with optimal control.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Pathwise differentiation.

e consider the original SDE and a perturbed SDE We introduce the difference The difference is given by the following SDE We assume that the functions and are such that the solution of the original SDE changes slightly when the perturbation of the initial condition is slight. Hence, the is small if is small. Consequently, We substitute these approximations into the SDE for and note that such SDE scales with the magnitude of . We introduce the process : We arrive to the following system of SDEs: The derivative of interest may be expressed in terms of and :

 Notation. Index. Contents.