Two dimensional COS method for pricing early-exercise and discrete barrier options under the Heston Model

We focus on the pricing of Bermudan and barrier options under the dynamics of the Heston stochastic volatility model. The two-dimensional nature of the Heston model makes the pricing of these options problematic, as the risk-neutral expectations need to be calculated at each exercise/observation date along a continuum of the two state spaces. We examine the 2D-COS method, which makes use of Fourier-cosine expansions in each of the two dimensions in order to approximate the integrals. Using the fast Fourier transform, we are able to efficiently calculate the cosine series coefficients at each exercise/observation date. A construction of this method is provided and we conduct numerical experiments to evaluate its speed and accuracy. Un ive rsi ty of Ca pe To wn The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgement of the source. The thesis is to be used for private study or noncommercial research purposes only. Published by the University of Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. Un ive rsi ty f C ap e T ow n Acknowledgements I owe many thanks to my supervisor, Professor Ronald Becker, for stepping in to supervise at a time when I really needed it. I am very grateful for the help which he provided. Plagiarism declaration • I know that plagiarism is wrong. Plagiarism is to use another’s work and to pretend that it is one’s own. • Each significant contribution to, and quotation in, this report from the work of other people has been attributed, and has been cited and referenced. • This dissertation is my own work. • I have not allowed, and will not allow, anyone to copy my work with the intention of passing it off as his or her own work.