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Summary I propose a framework to separate the probability of default (PD) and loss given default (LGD) of credit default swap (CDS). While the no arbitrage CDS spread is determined given PD and LGD, the separation of PD and LGD is impossible solely with CDS spread. Thus I develop a joint estimation procedure of PD with stock optiCJ'riS;a:s the probability that the underlying stock price becomes zero. As the option pricing model under credit risk is one of the jump diffusion option pricing model, the calibration of the option pricing model to estimate the PD implied in stock options require the identification of jump diffusion intensity and the diffusion coefficient of Brownian motions. For the stable and precise identification in the calibration, I separate the calibration procedure into two steps, non credit risk factors and credit risk factors. Since the behavior of the term structure model is similar to that of the jump diffusion intensity in the option pricing model, the choice of the term structure model influences the calibration performance. I demonstrate that the adoption of Hull and White (1990) model improve the fitness of the option pricing model and the stability and precision of the PD estimator. While I propose an alternative approach of the PD estimation, I also develop a procedure of LGD derivation of CDS given PD when CDS contract is subject to counterparty risk since CDS pricing model with counterparty risk implies counterparty risk can bias the LGD estimation. Finally I analyze the CDS spread of major UK and US financial institutions in 2008-2009. The results indicate that the market wide systemic risk factors drive the implied LGD although the CDS spread of the large banks reflected strong impact of the government intervention.

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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. 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.

We develop a new method for pricing options on discretely sampled arithmetic average in exponential Lévy models. The main idea is the reduction to a backward induction procedure for the difference Wn between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, Wn is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either flat iFT and (refined) iFFT or the parabolic iFT. Numerical results demonstrate the advantages of the method.