Essays on pricing fixed income derivatives and risk management
This dissertation consists of four essays on pricing fixed income derivatives and risk management. The first essay presents pricing and duration formulas for floating rate bonds and interest rate swaps with embedded options. It combines Briys et al.'s approximation with the extended Vasicek term structure model to value caps and floors. Using this approach, it computes the durations of caps, floors, collars, floating rate bonds with collars and interest rate swaps with collars, and provides comparative statics analyses of these durations with respect to the underlying variables such as the cap rate, the floor rate, the interest rate volatility, and the level of interest rates.
The second essay explores a class of polynomial Taylor series expansions for approximating the bond return function, and examines its implication for managing interest rate risk. The generalized duration vector models derived from alternative Taylor series expansion extend Fong and Fabozzi's M-square model and Nawalkha and Chambers' M-vector model, and the empirical tests show that immunization results can be improved for models g( t) = tα with α less than 1 when higher order generalized duration vectors are used.
The third essay develops a methodology to build recombining trees for pricing American options on bonds under deterministic volatility HJM models. Without imposing the HJM drift restriction, our approach uses the Nelson-Ramaswamy transformation to generate recombining forward rate trees. We show that the option prices obtained from our recombining trees satisfy Merton's bond option PDE when step size approaches zero. Numerical simulations provide evidence that this approach is efficient in pricing both European and American contingent claims.
The fourth essay obtains computationally efficient trees for pricing European options under two types of proportional volatility HJM models. We construct a numeraire economy in which European options are priced using a maturity-specific equivalent martingale measure. We then show that for the two types of proportional volatility models, European option prices are independent of the forward rate drift under this maturity-specific equivalent martingale measure. Our method is particularly beneficial when used to price long-dated caps, floors and collars because these instruments involve a large number of long-dated puts and calls.