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.

## Predictor-corrector.

et be a space-direction finite difference operator, Assume that is independent from . We seek for efficient ways to convert to a finite difference scheme in -direction. We integrate over : and approximate the integral where the is the step . Hence, If the operator is positive definite the the norm of is less then and the scheme is stable.

The predictor-corrector way to improve scheme's efficiency is the following. We find a separation for the implicit part of the scheme. Observe that the Crank-Nicolson may be written in two steps as We transform further The last equation is equivalent to We aim to replace the last equation with the equation To see that we preserve the second order of approximation we compute The resulting scheme is These should be more efficient because we split the inversion into two, presumably, simpler components. The last step is explicit.

To explore stability we put all steps together Hence Make the change of function then and the stability follows from the stabilization scheme considerations.

 Notation. Index. Contents.