- Priv.-Doz. Dr. Matthias Ehrhardt (TU Berlin, WIAS Berlin)

In this thesis,
we seek for an effective numerical solution for the American option pricing problem.
The Black-Scholes equation is transformed to the diffusion equation on an unbounded domain.
An *artificial boundary* is introduced to limit this domain.
The diffusion operator is discretized by using the different schemes
for the interior grid with different boundary conditions.
The considered high order compact schemes Method 1 proposed by MacCartin,
Method 2 proposed by Tangman and R3-Methods belong to weighted scheme of the approximation for the pure heat equation. Moreover, they coincide and this method is called the optimal weighted scheme.
This scheme is unconditionally stable and it has the fourth order approximation in space and the second in time.
Also, we compare Crank-Nicolson and five-point two-level non-compact stencils with compact method.
They show worth numerical result in comparison with high order compact method.
The Crank-Nicolson scheme with different boundary conditions is inferior to the non-compact Heun's method. But the Heun's scheme is conditionally stable,
which is a big disadvantage for this scheme.
On the right boundary of the truncated computational domain we use the Dirichlet boundary condition.
The boundary conditions on the left boundary have influence for the numerical solution of the heat equation.
The numerical results show that the combination of high order compact schemes with the
*Han and Wu boundary condition* is more accurate to the exact solution which obtained by using the Binomial method with large number of the steps.

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ehrhardt@math.uni-wuppertal.de