In this dissertation we study new methods for pricing of portfolio credit derivatives such as $k$th-to-default swaps and single tranche synthetic CDOs.
First, we derive the distribution function of the $k$th default time in a semi-analytic and an analytic forms based on one factor contagion model,
which is a combination of a factor copula model and a contagion model.
The correlation between the default times are modeled by Marshall-Olkin copula and the individual default intensities of surviving firms jump at default times by contagion effect.
Using the distribution function, we compute premiums of portfolio credit derivatives
and compare the estimated price with both the existing result and result from Monte carlo method for accuracy and efficiency.
We also check how the correlation and contagion affect the premiums of portfolio credit derivatives.
We also propose a large homogeneous portfolio(LHP) approximation method with two-factor Gaussian copula and random recovery rate.
In addition, we assume that the earlier the default occurs, the less the asset recovers, in other words,
random recovery rate and individual default times have positive rank correlation.
Under the LHP assumption, the conditional cumulative loss of the reference portfolio is approximated by
the product of loss given default and conditional default probability.
In order to derive semi-analytic formulas for the loss distribution and the expected tranche loss,
we use Gaussian two-factor model and assume that the recovery rate depends on one systematic factor.
In addition, we also consider stochastic correlation for a better fit to CDS index tranches.
The derived semi-analytic formula only involves the integration with respect to the standard normal density and can be computed by Gauss-Hermite quadrature.
Numerical tests show that the two-factor model with stochastic correlation and random recovery fits better to iTraxx tranche premiums
than other correlation or recovery assumptions.