Pricing discretely monitored barrier options under exponential-Levy processes - Camroodien, Ayesha (African Institute of Financial Markets and Risk Management, 2020)
One of the main factors in pricing barrier options is deciding whether to monitor the underlying asset price in continuous time or for a fixed set of time points. Most actively traded barrier options are monitored in discrete time due to reasons such as regulation and practical implementation. This dissertation presents transform methods for pricing discretely monitored barrier options under exponential-Levy ´ processes. Single-barrier knock-out options are evaluated under the Black-Scholes framework, the normal inverse Gaussian model and the Variance Gamma model. These models are widely implemented when dealing with pricing options sensitive to jumps. A diffusion component is included in the Variance Gamma model for comparison purposes. We focus on the COS method using Fourier-cosine series expansions and the Hilbert transform method to obtain prices fast and accurately. These option pricing approaches are suitable for Levy processes where the ´ analytical form of their characteristic function is available. Furthermore, standard Monte Carlo pricing is used as a reference and an outline of the pricing algorithms is presented. Both methods are easy to implement across the different asset price dynamics. In particular, the COS method produces results faster than the Hilbert transform method, however, the truncation assumptions under the COS method derived in (Fang and Oosterlee, 2009) prove to be unreliable. We observe the truncation range requires adjustment under the different asset price dynamics, as well as the different types of knock-out barrier options.