I. Basic math.
 II. Pricing and Hedging.
 III. Explicit techniques.
 IV. Data Analysis.
 V. Implementation tools.
 1 Finite differences.
 A. Finite difference basics.
 B. One dimensional heat equation.
 a. Finite difference schemes for heat equation.
 b. Stability of one-dim heat equation schemes.
 c. Remark on stability of financial problems.
 d. Lagrangian coordinate technique.
 e. Factorization procedure for heat equation.
 C. Two dimensional heat equation.
 D. General techniques for reduction of dimensionality.
 E. Time dependent case.
 2 Gauss-Hermite Integration.
 3 Asymptotic expansions.
 4 Monte-Carlo.
 5 Convex Analysis.
 VI. Basic Math II.
 VII. Implementation tools II.
 VIII. Bibliography
 Notation. Index. Contents.

## Finite difference schemes for heat equation. onsider the following boundary problem:    where the is the unknown function, the functions are given and regular, and the variable and lie in the domain We set up the lattice and approximate with the . We arrive to the following ODE problem    for , where , , .

Let us consider the boundary conditions. The matrix of the Laplacian has the form Note what happens to the finite difference approximation of the second derivative on the edges of the matrix. Obviously, we do not approximate it if  are some non zero values. We perform the following trick. We set Then the equation will describe exactly the same if we choose according to (Boundary trick)  Set up the lattice covering the and integrate the -th equation over the interval . We have where   The integral may be approximated by one of the quadrature formulas   The resulting schemes are called implicit, explicit and Crank-Nicolson schemes respectively. The expressions for the schemes are (Implicit scheme) (Explicit scheme) (Krank Nicolson)
with the boundary conditions in every case.

 Notation. Index. Contents.