Option pricing and risk management: analytic approaches with GARCH-Lévy dynamics - Mozumder, Md. Sharif Ullah (2011)
This Ph.D. thesis considers making some contributions to the asset pricing and financial risk management literature. First of all it offers some dynamics in the area of asset pricing which are practically implement able for pricing European style options. More precisely it considers blending GARCH type non-Markovian dynamics with Levy type Markovian innovations to offer analytic valuation of European style derivatives (at this initial stage). Revealing the mathematical underpinnings- required to replace conditional Gaussian innovations in G ARCH option pricing models by innovations coming from some Levy processes( with one sided and both sided jumps)-is the main focus. The necessity for this arises from the fact that the non-normal (Levy) innovations are crucial as heteroskedasticity alone doesn't suffice to capture the option smirk and the analytic valuation is highly expected because it makes the model practically implementable. Thus besides incorporating non-normality particular attention is paid to analytic valuation as well; though the Monte Carlo techniques can be readily applied for the proposed dynamics. However an approximation is required to uphold the analytic pricing, especially for innovations coming from Levy processes which are not Subordinator. These dynamics are capable of overcoming many deficiencies of benchmark Black-Scholes model and can be used to price other derivatives such as Credit, Interest rate, Commodity, Weather etc. The approach is built on a discrete time continuous state space and upholds the no-arbitrage principle of derivative pricing through the use of conditional Esscher transform to configure Equivalent :tviartingale Measure(EMl'vI). Similar to the existing literature, established for GARCH with normal innovations, existence of EMM provides de-facto evidence in support of no-arbitrage argument. Besides the main focus this research has made some complementary contributions to the option pricing literature.