Financial securities under nonlinear diffusion asset pricing models.

摘要

In this thesis we investigate two pricing models for valuing financial derivatives. Both models are diffusion processes with a linear drift and nonlinear diffusion coefficient. The forward price process of these models is a martingale under an assumed risk-neutral measure and the transition probability densities are given in analytically closed form. Specifically, we study and calibrate two different families of models that are constructed based on a so-called diffusion canonical transformation. One family follows from the Ornstein-Uhlenbeck diffusion (the UOU family) and the other - from the Cox-Ingersoll-Ross process (the Confluent-U family).;In the second part of the thesis we examine the application of the Confluent-U model to the credit risk modeling. An equity-based structural first-passage time default model is constructed based on the Confluent-U model with efficient closed-form (i.e. spectral expansions) formulas for default probabilities. The model robustness is tested by its calibration to the credit default swap (CDS) spreads for companies with various credit ratings. It is shown that the model can be accurately calibrated to the credit spreads with a piecewise default barrier level. Finally, we investigate the linkage between CDS spreads and out-of-the-money put options.;The first part of the thesis considers single-asset and multi-asset modeling under the UOU model. By applying a Gaussian copula, a multivariate UOU model is constructed whereby all discounted asset (forward) prices are martingales. We succeed in calibrating the UOU model to market call option prices for various companies. Moreover, the multivariate UOU model is calibrated to historical return data and captures the correlations for a pool of 4 assets.