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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Printable PDF file
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 MATH where the processes $X_{t}$ and $A_{t}$ are given by the SDEs MATH and the function $\phi$ is a given final payoff. The $A_{t}$ is the integral MATH Note that the straightforward Backward Kolmogorov's equation for this problem has the form

MATH (Asian PDE)
The term MATH is large on the lattice boundary because the $x$ 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 MATH is well behaved. We would like to find representation MATH 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 MATH .

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

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

1. We set the final condition for $v$ at $t+dt$ MATH

2. We calculate the $v$ at $t$ according to MATH

3. We set

MATH We seek MATH that provides the property MATH for all $x,a$ .

According to our calculation procedure MATH Note that the function MATH satisfies the condition MATH for all $x,a$ because its defining equation MATH is the backward Kolmogorov's equation for the problem MATH Hence, it suffices to choose $f$ according to MATH Under such choice the expressions MATH and MATH have the same third argument. Consequently, MATH and MATH We recall that by definition of $A_{t}$ MATH Hence, MATH

Normally, one would have to make sense of this procedure in the limit $dt\rightarrow0$ . We are not going to do it. The process $A_{t}$ was introduced through an SDE MATH because the such SDE allows to write a two-dimensional PDE ( Asian PDE ) for the function $u$ with subsequent transformation into finite difference scheme. We just discovered that we do not need the ( Asian PDE ). In financial applications the $A_{t}$ is a finite sum. The contract determines some set of times MATH and set of weights MATH . The contract's payoff depends on MATH


Let $X_{t}$ be the process defined by MATH The set MATH and function MATH are given, $t_{k}\equiv T$ . We calculate the function MATH via the following procedure:

A. At the final time $T\equiv t_{K}$ set MATH B. For every MATH do the following:

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

2. Set MATH

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

We have MATH

Notation. Index. Contents.

Copyright 2007