Credit risk modelling in Markovian HJM term structure class of models with stochastic volatility
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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  (hereafter HJM) term structure model with stochastic volatility by extending the defaultable framework developed in Schonbucher . 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  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  and Elhouar . 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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