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