Essays On Credit Derivatives Pricing

This doctoral thesis studies several derivative pricing models under uncertainty.We consider three kinds of derivatives: the first is portfolio credit derivatives (multi-name credit derivatives), in which we focus on collateralized debt obligation (CDO)tranches and nth-to-default CDS. The second is defaultable corporate bond, in whichthe underlying is value of corresponding firm. The third is volatility swap, in which wefocus on the optimal dynamic hedging problem.Portfolio credit derivatives experienced a rapid growth in the last decade. Com-pared with other common credit derivatives, such as credit default swap (CDS), theunderlying of portfolio credit derivative is an assets portfolio, and the cash flows ofportfoliocreditderivativedependonhowmuchandwhenassetsfromtheportfoliohavedefaulted. The difficulty in portfolio credit risk modeling is that there are hundreds orthousands of assets in the portfolio, and time-consuming Monte Carlo simulation areneeded to price the derivatives and calibrate the model. In Chapters2and3, we giveflexible and computationally tractable portfolio credit risk models. We consider thepricing problem of portfolio credit derivatives in two ways: bottom-up approach andtop-down approach.In Chapter2, we study in the bottom-up way. We first use the model of [1] andpropose a new method to compute the prices of CDO tranches without using Fouriertransforms, inverse Fourier transforms or Monte Carlo simulation. We also proposea model different from [1] by using a stochastic central tendency factor. Our centraltendency model belongs to models of affine jump diffusion processes, which make themodel computationally tractable. Pricing methods of corporate bond, CDS and popularportfolio credit derivatives are given.In Chapter3, we study the portfolio credit risk in the top-down way. The numberof defaulted assets in the reference portfolio is modeled by a discrete time branchingprocess with immigration. The pricing method of CDO tranche is given, and the nu-merical experiment shows that our model is flexible enough to model price changes. The defect of the model is that the number of defaulted assets maybe larger than thenumber of assets in reference portfolio. If the number of assets in reference portfolio islarge enough, such as larger than1000, it would be OK. If the number is small, problemwould arise. Hence we improve the model by introducing a continuously rolled port-folio. We model the number of defaulted assets in rolled portfolio, and map it into thenumber of defaulted assets in original portfolio and fix the flaw of model.In Chapter4, we focus on the impact of liquidity risk on corporate bond price. Wegive an analytic bond pricing formula under the structural model. In classic structuralmodel of [2], the risk spread between corporate bond and treasury bond in short termis zero, which is inconsistent with empirical observations. By introducing liquidity riskfactor, the risk spread in short term is now a positive number.In Chapter5, we study the problem of hedging volatility swaps. We obtain theoptimal dynamic hedging strategy respectively by letting the dynamics of underlyingreferences follow independent, stationary increments processes under discrete and con-tinuous time settings. We implement numerical experiment with the jump diffusionprocess of [3], the hedging error ratios are very small (smaller than0.5%).