Valuation of default -risky securities under a mixed diffusion -jump process

• Main Author: Aina, Victor Olugbenga
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-1999
• Subjects: Assets; Bankruptcy; Corporate bonds; Default; Finance; Interest rates; Regulation of financial institutions; Securities; Stock exchanges; Studies; Valuation; Volatility
• ISBN: 0612518337; 9780612518339

This thesis studies the valuation of default-risky interest rate and firm value contingent securities under a mixed diffusion-jump process. One of the contributions is the explicit modelling of the evolution of the value of the assets of a default-risky firm as a convolution of Gaussian diffusion and Poisson driven jump processes. The approach allows us to bring two strands of the extant credit risk literature-structural and the reduced form-together. The jump process enables the model developed to capture the impact of rare but very significant information on the fortunes of the firm; and hence its default probability. Standard techniques lead to a two-state variable plus time valuation Partial Differential-Difference Equation (PDDE) for the value of the securities examined. The PDDE derived admits of no known closed form formula, necessitating the use of numerical methods for approximating the solution. Specifically, finite difference techniques are employed. For the single-state variable (firm value and rare innovations to the value of the firm) PDDE, a mixed implicit-explicit variety of the Crank-Nicholson method is employed; while the two-state variable plus time PDDE is solved by the Alternating Direction Implicit (ADI) scheme of McKee and Mitchell. Among the interesting results obtained is the relative importance of innovations. For example, in an application to default-risky bonds, it was found that the Poisson jump process is better suited to capturing spreads on bonds of shorter maturities, while the Gaussian pure diffusion has an advantage with respect to characterising the spread on bonds of longer term maturities. Furthermore, the correlation between interest rate and default risk was found to significantly influence the spread on corporate bonds. In general, there is a tendency for model generated credit spread to revert towards the mean; a feature consistent with empirical regularity. Taken together, it is demonstrated that a structural model that recognizes rare but important information and therefore incorporates both the diffusion and jump components is much richer than any of the other models alone. This result is particularly true for a variety of shapes of the term structure of credit spreads between treasury and risky corporate bonds including flat, downward sloping, upward sloping and hump-shaped commonly observed in the market.