Exotic Interest Rate Options in Quantum Finance

Abstract

A major subject matter of this thesis is focused on studying the generalized forward interest rate model and the Libor Market Model in Quantum Finance. Compared to the stochastic interest rate models, the imperfectly correlated interest rates are modeling as a Gaussian field. The feature of the Gaussian field is that it contains much more information than the one-dimensional stochastic processes, which drive the entire evolution of interest rates in traditional financial theory. The simulation algorithm for modeling interest rates is extensively studied. Due to the complex structure of interest rate instruments, the approximate price only can be derived based on the perturbation expansion for small value of volatility. The comparison between simulation results and analytical formula is studied for many instruments and shows the flexible and potential of simulation method in pricing interest rate derivatives. In particular, it is shown that the simulation method provides a powerful tool in studying any kind of interest rate instruments without limitation.

Another part of this thesis is studying the Constant Elasticity of Variance (CEV) process. A recursion equation of CEV process is developed and used to calibrate the value of beta, which is the key term in CEV model. The value of beta for market observed Equity Default Swaps (EDS) spreads is obtained and agrees with the recent studies. However, the results for Credit Default Swaps (CDS) show that the market observed CDS spreads have no sensitivity to the implied volatility, which cannot be explained by CEV process. It is suggested that the EDS spreads with low barriers are more attractive to the market compared to CDS spreads.

In the third part, an unequal time Gaussian model is developed to calibrate the stock market data. The nontrivial Lagrangian is defined and the unequal time propagator is studied for fitting the correlation of different stocks on different time. Compared to modern portfolio theory, Gaussian model is more powerful in describing the behavior of unequal time correlation. Based on the nontrivial Lagrangian, Gaussian model is generally applicable to other liquid markets which have strong unequal time correlation.