Pricing swaptions and credit default swaptions in the quadratic Gaussian factor model
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In this thesis we show how the multi-factor quadratic Gaussian model can be used to price default free and defaultable securities. The mathematical tools used include the theory of stochastic processes, the theory of matrix Riccati equations, the change of measure technique, Ito's formula, use of Fourier Transforms in swaption valuation and approximation methods based on replacing the values of some stochastic processes by their time zero values. The first chapter of the thesis deals with the derivation of efficient closed form formulas for the price of zero coupon bonds in the multi-factor quadratic Gaussian model and the calibration of the multi-factor quadratic Gaussian model to the domestic and foreign forward rate term structures through closed form formulas. In the second chapter of the thesis, we derive approximations for the price of default free swaptions which are based on log-quadratic Gaussian processes. Using numerical experiments, we show the limitations of these approximations. We also give some numerical results for the pricing of a default free swaption using moment-based density approximants of the probability density function of the swaption's payoff. The third chapter of the thesis deals with the calibration of a quadratic Gaussian reduced form model of credit risk to the default free forward rate curve and to the survival probability of an obligor. We also consider different approximations for the price of credit default swaptions. Using numerical experiments, we show the limitations of the approximations. The final chapter of this thesis considers a two country reduced form model of credit risk. We examine the relationship between the domestic forward credit spread and the foreign forward credit spread of an obligor and provide quanto adjustment formulas for the probability of survival of an obligor. In the final part of this chapter, we show that the valuation of a quanto default swap is tractable in a contagion type reduced form model of credit risk which assumes that underlying processes are modelled by quadratic Gaussian processes.
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