| Summary: | 博士 === 國立臺灣大學 === 資訊工程學研究所 === 99 === Derivatives are popular financial instruments whose values depend on
other more fundamental financial assets (called the underlying
assets). As they play essential roles in financial markets, evaluating
them efficiently and accurately is critical. In 1973, Black and Scholes arrived at their ground-breaking analytical pricing formula, which assumes that the underlying asset follows a lognormal diffusion process with a constant risk-free interest rate and a constant volatility of the underlying asset.
However, the lognormal diffusion process, which has been widely used
to model the underlying asset''s price dynamics, does not capture the
empirical findings satisfactorily. Therefore, many alternative processes have been proposed, and a very popular one is the jump-diffusion process.
Additionally, since interest rates do not stay constant in the
real world, many stochastic interest rate models are put forward.
Most derivatives have no analytical formulas once one goes beyond the
most basic setup; therefore, they must be priced by numerical methods
like lattices.
A lattice divides the time interval between the derivative''s initial
date and the maturity date into $n$ equal time steps.
The pricing results converge to the theoretical values when the number
of time steps increases.
Unfortunately, the nonlinearity error introduced by the nonlinearity
of the value function may cause the pricing results to converge slowly
or even oscillate significantly.
This dissertation first proposes an accurate and efficient lattice for
the jump-diffusion process.
The proposed lattice is accurate because its structure can suit the
derivatives’ specifications so that the pricing results converge
smoothly.
To our knowledge, no other lattices for the jump-diffusion process
have successfully solved the oscillation problem.
In addition, the time complexity of our lattice is lower than those of
existing lattice methods by at least half an order.
Numerous numerical calculations confirm the superior performance of
our lattice to existing methods in terms of accuracy, speed, and
generality.
As for the stochastic interest models, the second part of this
dissertation shows that, when the interest rate models allow rates to
grow without bounds in magnitude, previous work on the lattice methods
shares a fundamental flaw: invalid transition probabilities.
As the overwhelming majority of stochastic interest rate models share
this property, a solution to the problem becomes important.
This thesis presents the first bivariate lattice that guarantees valid
probabilities even when interest rates can go without bounds.
Also, we prove that the proposed bivariate lattice grows
(super)polynomially in size if the interest rate model allows rates to
grow (super)polynomially.
Finally, we show the optimality of our bivariate lattice.
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