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.
 D. General techniques for reduction of dimensionality.
 a. Stabilization.
 b. Predictor-corrector.
 c. Separation of variables for Crank-Nicolson scheme.
 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.

## Stabilization.

ote that for any pair of operators and we can write Hence, the scheme is equivalent to the Crank-Nicolson after grouping some -order terms. Therefore, if Crank-Nicolson approximates with second order then so does the scheme (*).

Claim

The scheme preserves stability.

Proof

Make the change of function then or where by the Kelly theorem because

To transform the scheme (*) into a calculation recipe observe that We conclude Hence, we replaced inversion of with consecutive inversions and .

 Notation. Index. Contents.