Application of Perturbation Methods to Credit and Equity Derivatives.

by Bo Yang

Institution: University of Michigan
Department: Mathematics
Degree: PhD
Year: 2008
Keywords: Multi-scale Perturbation Methods; Derivative Pricing; Credit Risk; Spectral Expansions; Implied Volatility; Mathematics; Science
Record ID: 1814625
Full text PDF: http://hdl.handle.net/2027.42/61625


This thesis studies the application of perturbation methods in developing and solving credit and equity derivative pricing models. Chapter II proposes a uni???ed framework for pricing credit and equity derivatives that incorporates stochastic volatility, default intensity, and interest rates. It is demonstrated that the model can be jointly calibrated to the bond and equity options of a same company. It is observed that the model implied CDS spread matches the market CDS spread. Chapter III studies the pricing of convertible bonds and barrier and lookback options in the framework of Chapter II. By applying perturbation methods, the author is able to reduce the dimension of the free-boundary problem for pricing convertible bonds and to solve the corresponding Dirichlet and mixed (Dirichlet and Neumann) boundary-value problems for approximate prices of barrier and lookback options. Chapter IV extends Linetsky???s negative-power intensity model [29] by introducing a fast evolving factor. It is shown that the resulting approximation for derivatives prices are Linetsky???s prices with a ???Greek??? correction term, and the approximations for the double barrier options prices are derived. Chapter V studies stochastic parameter extensions of a top-down model proposed in [14] for multi-name credit derivatives, where the default process is a time-changed birth process. The calibration exercise shows that the introduction of stochastic parameters brings in more ???exibility and improves ???tting the market data.