Pan, Tang.
2010. Exotic Interest Rate Options in
Quantum Finance. Doctoral Dissertation, NUS.
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.
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