Path Calculations and Option Pricing

Abstract

The thesis is worked in the areas of the intersection of probability, combinatorics and analytical combinatoric. The research is motivated from the need of producing new methodologies and financial models in global market resulted from the lesson of 2007-2009 global financial market and a quantum tool called Feynman path integral method which has been applied to model path-dependent option pricing model by Hao and Utev. Path calculation method deal with models by analysing each possible individual asset price paths which broaden the methodology of modelling financial market and can solve some unusual or complex models which is difficult to model by using non path-dependent calculation method. My research has focused on developing combinatorial structure and path calculation methods and then apply them to model individual share price path and calculate option prices. The share price can be modelled as a path with a given share price changes and the expiry date. We have applied Flajolet symbolic method, generating functions and path calculation method to model a set of typical finite restricted share price paths with restriction not allowing k consecutive down steps and derived a calculation of option prices in the model. Besides, applying the Flajolet symbolic method, we constructed a relationship between individual share price and generating function, analysed the transformed share price paths via different operations on generating functions. In addition, we applied path calculation method to solve winning probability in the classical gambler ruin problem which contributes the same result as the solution solved by establishing the recurrence equation method. Furthermore, we have solved a different gambler's ruin problem using the path calculation method which cannot be solved by the recurrence equation method. Counting paths with combinatoric can be studied from two ways, one way is to label and the other is computation. Labelling is a part of representation of objects. We have developed a graphical theoretical construction of individual share price path via general binary trees and matroid. In addition, We have developed a method to solve some kinds of pattern avoiding path counting combinatorial problem by modifying certain probability methods. Two working papers including modelling of paths via matroids and counting via Markov-type technique is now being produced. </div