** Solving partial differential equations related to option pricing
with numerical methods **

Coen Leentvaar

Site of the project:

TU Delft

start of the project: January 2003

The Master project has been finished in December 2003
( Masters Thesis).
For working address etc. we refer to our
alumnipage.

**Summary of the master project:**

Options are widely used on markets and exchanges. The famous
Black-Scholes
model is a convenient way to calculate the price of an option. In this
thesis
a highly accurate numerical method for solving this equation is
presented.
Although the exact solution of the Black-Scholes equation is known, a
numerical method will be proposed. A reason is to create a general
numerical
model for many different types of options. In particular, American
options
are not solvable in an analytic sense. If the numerical method works
for
European style option, then this is the basis to get the solution for
an
American option. Another issue is ``implied volatility''. Volatility
is
a quantitative expression for the randomness in the market. From
newspapers or stock exchanges, the volatility of asset prices in the
future
is not known, so it has to be estimated. If we have a value for the
option
price, it is possible
to calculate the volatility, which is the only unknown parameter in
the
Black-Scholes equation.
The main questions dealt with in this thesis are:
- Can we use fewer than 50 grid points to calculate the value of an
option with a reliable accuracy by setting up a high order
discretization in
space and time and by employing grid stretching around interesting
regions?
- Will the highly accurate numerical scheme also work for
exotic options with discontinuous final conditions?
- Can we calculate the implied volatility by some iterative methods
in
only a few iterations?

## Contact information:
Kees
Vuik

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