Credit risk modelling in Markovian HJM term structure class of models with stochastic volatility
- Publication Type:
- Thesis
- Issue Date:
- 2011
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Empirical evidence strongly suggests that interest rate volatility is stochastic and correlated
to changes in interest rates. In addition, the intensity process has been shown to generate
heavy-tailed behavior and this has been attributed to stochastic volatility. A good credit risk
model should incorporate the correlation between the short rate and credit spread or indirectly
influence the market's perception of default risk which has an impact on credit spreads.
The objective of this thesis is to model credit risk within a Markovian Heath, Jarrow, and
Morton [1992] (hereafter HJM) term structure model with stochastic volatility by extending
the defaultable framework developed in Schonbucher [1998]. Adapting the HJM framework to
including default risk in a generalised framework that incorporates all the information on
the current risk free term structure as well as the credit spread curve. Under some conditions
on the specification of the volatility function, the model admits finite dimensional Markovian
realisations and as a result, the default-free yield curve as well as the credit spread curves
can be calculated with low computational cost at any given time.
The main contributions of this thesis are:
Markovian Defaultable HJM Term Structure Models with Unspanned Stochastic Volatility -
Chapter 2. Stochastic volatility is introduced into the Schonbucher [1998] model
and we generalise it to allow for a correlation structure between the default-free forward
rate, the forward credit spread and stochastic volatility. Under certain level dependent
volatility specifications, we derive a Markovian representation of the default able short
rate in terms of a finite number of state variables which we then express in terms of
economic qualities observed in the market, specifically in terms if discrete tenor forward
rates. A numerical experiment is then conducted to investigate the distributional
properties of the defaultable bond price and bond returns which reveals the existence
of a left tail.
Credit Derivative Pricing under a Markovian HJM Term Structure Model with (Diffusion
Driven) Humped Volatility - Chapter 3. We verify that under the assumption
of a humped volatility specification, the defaultable forward rates admits finite
dimensional affine realisations. The default of the underlying reference entity is modelled
as a Cox process and we derive exponential affine bond price formulas in the presence
of stochastic volatility. We then investigate the pricing of single-name credit default
swaps both in the presence and absence of counterparty risk and derive formulas for the
valuation of credit default swaptions within the framework. On relaxing the level dependency
assumption within the humped volatility specification, we price knocked-out
put options on defaultable bonds using the Fourier transform approach.
Valuation of Bond Options under a Defaultable HJM Class of Models with Regime Switching
Volatility - Chapter 4. We allow the defaultable forward rate volatility to
depend on the current forward rate curve as well as on a modulating continuous time
Markov chain making use of the results in Valchev [2004] and Elhouar [2008]. Stochasticity
is then introduced to the volatility function by a separable volatility specification
which guarantees finite-dimensional Markovian realisations under regime switching. A
special case of the short rate class of models, the Hull-White-Extended-Vasicek type
of model is obtained in the defaultable setting from which an explicit bond pricing
formula is derived. We then apply finite difference methods to price European options
under two-state regimes.
We give a summary of all the thesis findings in Chapter 5 where we also present the concluding
remarks and directions for future work.
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