This work 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 implementable 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 GARCH 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 Martingale Measure(EMM). 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.

Since J.P.Morgan introduced RiskMetrics in 1994, the normal quantile based VaR has been considered as industry standard for risk management. However VaR itself has inherent inconsistencies which are exacerbated under the assumption of normality. The second part of this thesis considers two frequently referred approaches to non-normality in risk management : extreme value(EV) approach and Levy approach. The idea is to reveal the relative performance of various risk measures under full density based Levy approach and solely tail observation based EV approach. We provide empirical evidence which confirms that though purely tail based risk measures value-at-risk(VaR) and its coherent version expected shortfall(ES) are well comparable under both approaches, entire spectrum based spectral risk measure(SRM) is misleading for EV approach. Backtesting risk measure VaR is considered under both approaches. We plan to improve the computational efficiency of
estimation of Levy coherent risk measures through application of characteristic function based FRFT. Our ultimate goal is to see whether the conditional moment generating functions developed for GARCH-Levy models in the first part of this thesis can be adapted to the characteristic function based FRFT technique in order to estimate the risk measures in analytic fashion.