|
Abstract:
|
In this thesis I show that finite difference methods are a very good alternative
to the much used Monte Carlo simulations for financial problems considering
three dimensional PDEs. The ADI scheme produces stable results within
one standard deviation of the Monte Carlo price faster than the Monte Carlo
simulation. I show two applications of the ADI scheme, one on Asian options
where the price depend on the arithmetic average of the underlying, and the
Heston model, an option pricing model with stochastic volatility. For the
Heston model I also show how to implement a mixed derivative term when
the correlation between the two underlying processes differs from zero, and
how such terms can impose problems to the stability.
In chapter three I spend time on explaining the Black – Scholes PDE as
it is the basis of both the Asian PDE and the Heston PDE, and show finite
difference on this two dimensional equation. In chapter four I show the
general setup for finite differenced in three dimensions, the ADI scheme. In
chapter five I apply the ADI setup on the Asian PDE and than the Heston
PDE. I go through stability conditions for both applications and show
numerical results on how it converges. I also compare the numerical results
against Monte Carlo simulations. Last, in chapter seven the matlab code for
both applications are printed.
All numerical results are computed on a quad-core 3.4GHz computer with
8GB RAM, and all code are run in matlab. |
