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.
 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.

## Time dependent case.

uppose the operator of the finite-difference problem is time dependent According to the convergence theorem ( Lax convergence theorem ) we need to prove approximation on the solution and stability. The stability is proved with the same techniques as in the time independent case. We assume that the spacial operator has a correct approximation in spacial variables. In this section we concentrate on the problem of -directional approximation. We assume that the solution possesses smoothness in -variable and invoke the Taylor decomposition where the higher index refers to the point on the uniform time mesh, the low index is the derivative in and the is the time step. We repeatedly use the fact that is the solution of : With the above results we would like to construct a Crank-Nicolson scheme: for some operator that remains to be determined. We expand the above scheme in power of and collect the terms: Hence we need to have and the -term expression gives Hence, the following requirement should be sufficient for approximation in direction Some of the possibilities are

 Notation. Index. Contents.