| LIBOR market model is the benchmark model for interest-rate derivatives. It has been a challenge to improve the standard model to better capture market dynamics. In this thesis two extensions of standard LIBOR market model are presented, some applications of this model are also discussed.; Constant maturity instruments such as swaps, caps and floors have gained increasing popularity in recent years, a new approach which is motivated by traditional convexity adjustment and change of measure is adopted in this thesis to evaluate these constant maturity instruments, while closed-form solutions are given to apply this accurate and computational efficient technique.; Volatility skews are reflected in many markets. Generalized CEV market model is an extension to the standard market model. The great advantage of the CEV model is its capacity to produce the volatility skew that is pronounced in the swaption prices. As an application of the calibration of CEV model by Wu (2001), we use a practical problem to illustrate how to calibrate the volatility coefficients and elasticity of the CEV market model using the eigenvalue decomposition rank reduction technique and Hessian-based algorithm. This work can also be regarded as a numerical complement to the calibration of LIBOR market model.; The phenomena of non-monotonic volatility smiles is also persistent in many major markets, we extend the standard market model to cope with this feature by allowing stochastic volatility. To achieve analytical tractability, we adopt a multiplicative stochastic factor for the volatility functions of all relevant forward rates. The stochastic factor follows a squared-root diffusion process, and it can be correlated with the forward rate processes. Approximate processes for forward rates and swap rates are introduced and closed-form pricing formulae are derived. We develop a fast Fourier transform algorithm for the implementation of the formula. The approximations are well supported by pricing accuracy. By adjusting the correlation between the forward rates and the volatility in a way consistent with intuition, we can generate volatility smiles or skews of the swaption prices similar to those observed in the markets. |
