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.

## Stability of one-dim heat equation schemes.

he schemes ( Implicit scheme ),( Explicit scheme ),( Crank Nicolson ) take the following forms: where the notation introduced earlier refers to the -direction. We recall from the ( Laplace spectrum ) that Hence, according to the spectral mapping theorem ( Spectral mapping )

 (Implicit spectrum)
 (Krank Nicolson spectrum)
In every case it is sufficient to have Hence, the explicit scheme is stable if or The other two schemes are always stable.

 Notation. Index. Contents.