| Summary: | 碩士 === 國立臺灣大學 === 資訊工程學研究所 === 96 === Financial engineering and financial innovation flourished in last decades. We have developed many new financial products to provide hedge instruments for risk management, and promoted market efficiency and completeness. The pricing problems of this financial field will try to build mathematical models and derive analytic pricing formulas. But most exotic derivatives are too complicated to derive formulas. We must use computers to handle numerical methods and simulations, so computer science can give them a favor. This thesis discusses pricing of Parisian options and includes a lot of subjects: financial theory, probability & statistics, discrete mathematics, computational complexity, design & analysis of algorithms, and parallel processing.
Parisian options are path-dependent options and their closed-form solutions are not available up to now. We propose two fast financial algorithms to solve it. First we price Parisian options based on a combinatorial approach in binomial tree by Costabile in 2002. To refine Costabile’s algorithm, time complexity O(n^3) can be reduced to O(n^2); If binomial coefficients are given, the space complexity O(n^2) could be reduced to O(n). Second on Monte Carlo simulation, we introduce the inverse Gaussian distribution and its sampling method. To combine simulations and the inverse Gaussian distribution sampling, it can reduce divided time intervals to save computational time. Because the paths generated by Monte Carlo simulation are independent, it is easy to apply parallel processing. Nowadays multi-core processors are very popular, it is also a good idea to enhance computational efficiency. We give some descriptions and applications on it.
All financial algorithms in this thesis are implemented in the C programming language. We execute the programs on our high-performance computing clustered platform, and deal with simulation jobs synchronously. Then the system can be fully exploited.
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