The thesis is an investigation into the pricing of credit risk under the intensity framework
with a copula generating default dependence between obligors. The challenge of quantifying
credit risk and the derivatives that are associated with the asset class has seen an
explosion of mathematical research into the topic. As credit markets developed the modelling
of credit risk on a portfolio level, under the intensity framework, was unsatisfactory
in that either:
1. The state variables of the intensities were driven by diffusion processes and so could
not generate the observed level of default correlation (see Schönbucher (2003a)) or,
2. When a jump component was added to the state variables, it solved the problem of
low default correlation, but the model became intractable with a high number of parameters
to calibrate to (see Chapovsky and Tevaras (2006)) or,
3. Use was made of the conditional independence framework (see Duffie and Garleanu
(2001)). Here, conditional on a common factor, obligors’ intensities are independent.
However the framework does not produce the observed level of default correlation,
especially for portfolios with obligors that are dispersed in terms of credit quality.
Practitioners seeking to have interpretable parameters, tractability and to reproduce observed
default correlations shifted away from generating default dependence with intensities
and applied copula technology to credit portfolio pricing. The one factor Gaussian
copula and some natural extensions, all falling under the factor framework, became standard
approaches. The factor framework is an efficient means of generating dependence
between obligors. The problem with the factor framework is that it does not give a representation
to the dynamics of credit risk, which arise because credit spreads evolve with
time.
A comprehensive framework which seeks to address these issues is developed in the thesis.
The framework has four stages:
1. Choose an intensity model and calibrate the initial term structure.
2. Calibrate the variance parameter of the chosen state variable of the intensity model.
3. When extended to a portfolio of obligors choose a copula and calibrate to standard
market portfolio products.
4. Combine the two modelling frameworks, copula and intensity, to produce a dynamic
model that generates dependence amongst obligors.
The thesis contributes to the literature in the following way:
• It finds explicit analytical formula for the pricing of credit default swaptions with an
intensity process that is driven by the extended Vasicek model. From this an efficient
calibration routine is developed.
Many works (Jamshidian (2002), Morini and Brigo (2007) and Schönbucher (2003b))
have focused on modelling credit swap spreads directly with modified versions of
the Black and Scholes option formula. The drawback of using a modified Black and
Scholes approach is that pricing of more exotic structures whose value depend on
the term structure of credit spreads is not feasible. In addition, directly modelling
credit spreads, which is required under these approaches, offers no explicit way of
simulating default times.
In contrast, with intensity models, there is a direct mechanism to simulate default
times and a representation of the term structure of credit spreads is given.
Brigo and Alfonsi (2005) and Bielecki et al. (2008) also consider intensity modelling
for the purposes of pricing credit default swaptions. In their works the dynamics of
the intensity process is driven by the Cox Ingersoll and Ross (CIR) model. Both works
are constrained because the parameters of the CIR model they consider are constant.
This means that when there is more than one tradeable credit default swaption exact
calibration of the model is usually not possible. This restriction is not in place in our
methodology.
• The thesis develops a new method, called the loss algorithm, in order to construct the
loss distribution of a portfolio of obligors. The current standard approach developed
by Turc et al. (2004) requires differentiation of an interpolated curve (see Hagan and
West (2006) for the difficulties of such an approach) and assumes the existence of a
base correlation curve. The loss algorithm does not require the existence of a base
correlation curve or differentiation of an interpolated curve to imply the portfolio loss
distribution.
• Schubert and Schönbucher (2001) show theoretically how to combine copula models
and stochastic intensity models. In the thesis the Schubert and Schönbucher (2001)framework is implemented by combining the extended Vasicek model and the Gaussian
copula model. An analysis of the impact of the parameters of the combined
models and how they interact is given. This is as follows:
– The analysis is performed by considering two products, securitised loans with
embedded triggers and leverage credit linked notes with recourse. The two
products both have dependence on two obligors, a counterparty and a reference
obligor.
– Default correlation is shown to impact significantly on pricing.
– We establish that having large volatilities in the spread dynamics of the reference
obligor or counterparty creates a de-correlating impact: the higher the volatility
the lower the impact of default correlation.
– The analysis is new because, classically, spread dynamics are not considered
when modelling dependence between obligors.
• The thesis introduces a notion called the stochastic liquidity threshold which illustrates
a new way to induce intensity dynamics into the factor framework.
• Finally the thesis shows that the valuation results for single obligor credit default
swaptions can be extended to portfolio index swaptions after assuming losses on the
portfolio occur on a discretised set and independently to the index spread level
