Quantitative Analysis.
Trading Platform.
Python for Excel.
Author.

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.The 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 the 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.Generation of random samples.
5.Monte-Carlo.
6.Convex Analysis.
VI.Basic Math II.
VII.Implementation tools II.
Bibliography.
Forum 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 the problems when such difficulty is particularly dire. Consider the following problem (asian option).

We wish to evaluate the expectationMATH where the processes $X_{t}$ and $A_{t}$ are given by the SDEsMATH and the function $\phi$ is a given final payoff. The $A_{t}$ is the integralMATH 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 aplified on every step. Consequently, one cannot expect that the straightforward discretisation of this equation would be stable.

The equation MATH is well behaved. We would like to find representationMATH at least locally 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 propertiesMATH 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 toMATH

3. We set

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

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

Normally, one would have to make sense of this procedure in the limit $dt\rightarrow0$. But we are not goingt o do it. The process $A_{t}$ was introduced through an SDEMATH because the such SDE allows to write a two-dimenstional 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 onMATH

Summary

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

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

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

2. Set MATH

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

We haveMATH





Forum Notation Index Contents


















Copyright 2007.