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.
 C. Two dimensional heat equation.
 a. Peaceman-Rachford (alternating directions) scheme.
 b. Stability of Peaceman-Rachford.
 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.

## Peaceman-Rachford (alternating directions) scheme.

n this section we present a general way to construct a finite difference approximation for a solution of the heat equation via a procedure with linear dependency of the amount of computation on the size of the lattice.

We are considering the following boundary problem for the heat equation:

We introduce the uniform lattices the lattice function and notation . The introduced above notations and refer to the -variable. Consider the scheme

 (Alternating directions1)
 (Alternating directions2)
in all internal points of the lattice . The refers to the operator acting in the index. The initial conditions are and the boundary conditions are
 (Alternating boundary1)
 (Alternating boundary2)

The key observation about the scheme ( Alternating directions1 )-( Alternating boundary1 ) is that the ( Alternating directions1 ) is a one dimensional implicit scheme in the -direction while the ( Alternating directions2 ) is the implicit scheme in the -direction. Hence, starting from we use ( Alternating directions1 ) to find the through the factorization procedure of the previous section. Afterwards, we similarly use ( Alternating directions2 ) to find .

To understand the boundary condition ( Alternating boundary1 ) subtract ( Alternating directions1 ) from ( Alternating directions2 ) and obtain

 (Alternating boundary)

 Notation. Index. Contents.