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.

## Lagrangian coordinate technique.

n the previous section ( Remark on stability of financial problems ) we pointed out the boundary difficulty when setting up a finite difference problem in financial applications. There is at least one class of problems when such difficulty is particularly dire. Consider the following problem (Asian option).

We wish to evaluate the expectation where the processes and are given by the SDEs and the function is a given final payoff. The is the integral Note that the straightforward Backward Kolmogorov's equation for this problem has the form

 (Asian PDE)
The term is large on the lattice boundary because the is large. Hence, the loss of normality would be amplified on every step. Consequently, there is no hope that straightforward discretization of this equation would be stable.

The equation is well behaved. We would like to find representation at least locally in time so that we could safely make one step of the finite difference scheme. We now proceed to derive the form of such .

Note, that the function has the following properties The second property is obvious. The first property is the consequence of the ( Chain_rule ). These properties are also sufficient: if has both of these then it solves the original problem. Indeed,

Suppose we already calculated the solution at the time step and would like to derive the solution at the time . This is our calculation scheme:

1. We set the final condition for at

2. We calculate the at according to

3. We set

We seek that provides the property for all .

According to our calculation procedure Note that the function satisfies the condition for all because its defining equation is the backward Kolmogorov's equation for the problem Hence, it suffices to choose according to Under such choice the expressions and have the same third argument. Consequently, and We recall that by definition of Hence,

Normally, one would have to make sense of this procedure in the limit . We are not going to do it. The process was introduced through an SDE because the such SDE allows to write a two-dimensional PDE ( Asian PDE ) for the function with subsequent transformation into finite difference scheme. We just discovered that we do not need the ( Asian PDE ). In financial applications the is a finite sum. The contract determines some set of times and set of weights . The contract's payoff depends on

Summary

Let be the process defined by The set and function are given, . We calculate the function via the following procedure:

A. At the final time set B. For every do the following:

1. Evaluate the function given by the problem or, equivalently,

2. Set

We proceed to verify the validity of the above summary directly.

We have

 Notation. Index. Contents.