Content of present website is being moved to www.lukoe.com/finance . Registration of www.opentradingsystem.com will be discontinued on 2020-08-14.
 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.

## Remark on stability of financial problems. or most of financial problems the payoff function does not vanish for both of the infinities. Hence, if one uses at the lattice boundaries then such boundaries have to be far away to make such positions of the state variable improbable. Such problem is even more apparent if American feature is involved.

One alternative is to use (Second derivative localization)
on the boundary because it is true for most of the practical payoffs. However, such condition, when substituted into the scheme, destroys the normality of the problem leading to the difficulties described in the section ( Stability of general evolution equation ). By using the formula ( Second derivative localization ) we introduce a Jordanian block of at least second order that creates unstable behavior near the boundary. When propagating to the center the error competes with the slight stability of the scheme presumably obtained excluding the difficulty on the boundary.

For some contracts one may use for a known constant and then transform the problem to the homogenous condition by changing the function . This way normality is preserved: However, such change will introduce a RHS component into the equation. If the original problem has a blow up at spacial infinity then the difficulty is likely to migrate into the RHS. The same problem occurs when using for a known linear function and the lattice boundary .

 Notation. Index. Contents.