| Summary: | 碩士 === 國立臺灣大學 === 資訊工程學研究所 === 87 === A financial derivative is a financial instrument whose payoff is
based on other elementary financial instruments, such as bonds or stocks. With the rapid growth and deregulation of many economies, more derivatives are being designed by the financial institutions to satisfy the needs of their clients. This also gives rise to new problems in pricing and hedging.
It has been proved that pricing arbitrary European options is a
#P-hard problem. Even now, there are some notorious derivatives, such as Asian options, that can not be efficiently priced. These sophisticated derivatives are playing
important roles in financial markets.
Pricing path-dependent derivatives with tree models combined with state variables is a standard numerical approach, especially when we can not get proper closed form. Monte Carlo simulation is also a good alternative, but it is less efficient than the tree methods in general.
A systematic approach to constructing data structures and
algorithms for pricing is the first goal of this thesis. I will
first introduce how the idea works by illustrating the underlying ideas. Then I will apply the ideas to European-style
path-dependent options, such as barrier options, geometric
average-rate option and Asian-like interest rate options. For
American-style options, the early exercise property of these
options is critical, and the numerical data from Asian-like
derivatives also suggests that it is a monotone curve rather than an oscillated one. The second goal of this thesis is therefore about demonstrating these properties of the pricing data.
Pricing the arithmetic average rate options is a hard problem.
This is because we can''t derive a proper formula for describing
the distribution of the sums of log-normal random variables. A new lattice model is designed for pricing the arithmetic average-rate options. This efficient approach can give a more reliable answer than other approaches.
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