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 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.
 a. Peaceman-Rachford (alternating directions) scheme.
 b. Stability of Peaceman-Rachford.
 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 Peaceman-Rachford.

e aim to convert the scheme ( Alternating directions1 ),( Alternating directions2 ) to the canonical form by eliminating the . Add the equations ( Alternating directions1 ),( Alternating directions2 ): and substitute from ( Alternating boundary ): Note that and are commutative. We transform the last expression as follows: Finally, we write the evolution equation where For stability it is sufficient to have Such results follow from the spectral considerations of the Crank-Nicolson ( Crank Nicolson spectrum ) and implicit ( Implicit spectrum ) schemes and the minimax theorem ( Minmax theorem ).

One may ask "Wait a moment, how about the boundary conditions?". The answer is "Check the trick around the ( Boundary trick )".

 Notation. Index. Contents.