Numerical Approach on American Options

Specialeforsvar ved David Radevic

Titel: Numerical Approach on American Options
Focus on different fixed-point systems and initial guesses

 

 

Abstract: There exists no closed-form analytical price formula for the American put option, and therefore one must use numerical approximation methods in order to compute the American put options price. Under the assumption of being within the Black-Scholes models’ framework, the American put options price can be expressed as an optimal stopping problem. This can also be converted into a free-boundary problem, which leads to the free-boundary equation also known as the non-linear Volterra integral equation. The free-boundary equation describes the options price through the optimal exercise boundary, and this is solved using different iterative fixed-points methods respectively given as fixed-point system A and B. Since the fixed-point method is an iterative process, there is a need for an initial guess. Therefore, we will also be introducing different initial guesses of more and less complicated character described by either a non-linear systems of equations or not. Implementing a stopping criteria on the optimal exercise boundary under different error margins, the optimal exercise boundary is approximated with different initial guesses and the computational time including constructing the actual guess are calculated to see the effect on the computational time. This shows us that the complexity of the non-linear equation system influences the time, which can be seen in the computational time of approximation method one compared to the others. Besides for testing the different initial guesses influence on the computational time, the two fixed-point systems A and B’s precision to compute the American premium are tested, and this gives an indicator for what is obtained using a more complex fixed-point system

 

Vejleder:  Jesper Lund Pedersen
Censor:    Bjarne Astrup Jensen, CBS