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.
|Educations||MSc in Applied Economics and Finance, (Graduate Programme) Final Thesis|
|Number of pages||58|