The main objective of this thesis is to implement stochastic correlation into the existing structural credit risk models. There are two stochastic models suggested for the covariance matrix of the assets' prices. In our first model, to induce the stochasticity into the structure of the correlation, we assume that the eigenvectors of the covariance matrix are
constant but the eigenvalues are driven by independent Cox-Ingersoll-Ross processes. To price equity options on this framework we first transform the calculations from the pricing domain to the frequency domain. Then we derive a closed formula for the Fourier transform of the Green's function of the pricing PDE. Finally we use the method of images to find the price of the equity options. The same method is used to find closed formulas for marginal probabilities of defaults and CDS prices. In our second model, the covariance of the assets follows a Wishart process, which is an extension of the CIR model to dimensions greater than one. The popularity of the Heston model, which uses the CIR process to model the stochastic volatility, could be a promising point for using Wishart process to model stochastic correlation. We give closed form solutions for equity options, marginal probabilities of defaults, and some other major financial derivatives. For the calculation of our pricing formulas we make a bridge between two recent trends in pricing theory; from one side, pricing of barrier options by Lipton (2001) and Sepp (2006) and from other side the development of Wishart processes by Bru (1991), Gourieroux (2005) and Fonseca et al. (2006, 2007a, 2007b). After obtaining the mathematical results above, we then estimate the parameters of the two models we have developed by an evolutionary algorithm. We prove a theorem which guarantees the convergence of the evolutionary algorithm to the set of optimizing parameters. After estimating the parameters of the two stochastic correlation models, we conduct a comparative analysis of our stochastic correlation models. We give an approximation formula for the joint and marginal probabilities of default for General Motors and Ford. For the marginal probabilities of default, a closed formula is given and for the joint probabilities of default an approximation formula is suggested. To show the convergence properties of this approximation method, we perform the Monte Carlo simulation in two forms: a full and a partial Monte Carlo simulation. At the end, we compare the marginal and joint probabilities with full and partial Monte Carlo simulations.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/33798 |
| Date | 05 December 2012 |
| Creators | Arian, Hamidreza |
| Contributors | Seco, Luis |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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