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.

## Factorization procedure for heat equation.

here is a way to solve a three-diagonal finite difference scheme with an linear number of algebraic operations. Suppose some finite difference scheme takes the form

 (Factorization1)
with the properties
 (Factorization3)
 (Factorization4)
We look for a recursive relationship
 (Factorization2)
with some coefficients that we determine through the substitution into the scheme ( Factorization1 ): We express all the through the using the relationship ( Factorization2 ): Hence, to satisfy the ( Factorization1 ) we must have and Such conditions lead to the recursive relationships
 (Factorization5)
 (Factorization6)
Under conditions ( Factorization3 ),( Factorization4 ) we have for all . Therefore, we start with and produce according to the ( Factorization5 ),( Factorization6 ). At the final step of the iteration we use the boundary condition Afterward, with already known we compute with recursion in opposite direction according to the ( Factorization2 ).

 Notation. Index. Contents.