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Title: Numerical analysis and multi-precision computational methods applied to the extant problems of Asian option pricing and simulating stable distributions and unit root densities
Author: Cao, Liang
ISNI:       0000 0004 5358 5508
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 2014
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This thesis considers new methods that exploit recent developments in computer technology to address three extant problems in the area of Finance and Econometrics. The problem of Asian option pricing has endured for the last two decades in spite of many attempts to find a robust solution across all parameter values. All recently proposed methods are shown to fail when computations are conducted using standard machine precision because as more and more accuracy is forced upon the problem, round-off error begins to propagate. Using recent methods from numerical analysis based on multi-precision arithmetic, we show using the Mathematica platform that all extant methods have efficacy when computations use sufficient arithmetic precision. This creates the proper framework to compare and contrast the methods based on criteria such as computational speed for a given accuracy. Numerical methods based on a deformation of the Bromwich contour in the Geman-Yor Laplace transform are found to perform best provided the normalized strike price is above a given threshold; otherwise methods based on Euler approximation are preferred. The same methods are applied in two other contexts: the simulation of stable distributions and the computation of unit root densities in Econometrics. The stable densities are all nested in a general function called a Fox H function. The same computational difficulties as above apply when using only double-precision arithmetic but are again solved using higher arithmetic precision. We also consider simulating the densities of infinitely divisible distributions associated with hyperbolic functions. Finally, our methods are applied to unit root densities. Focusing on the two fundamental densities, we show our methods perform favorably against the extant methods of Monte Carlo simulation, the Imhof algorithm and some analytical expressions derived principally by Abadir. Using Mathematica, the main two-dimensional Laplace transform in this context is reduced to a one-dimensional problem.
Supervisor: McCrorie, Roderick Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Laplace transforms ; Numerical inversion ; Multi-precision arithmetic ; Asian options ; Infinitely divisible distributions ; Stable distributions ; Unit root distributions ; Characteristic functions ; Generalized hypergeometric function ; Meijer G function ; Fox H function ; Euler method ; Post-Widder method ; Bromwich integral ; Gaver-Wynn-Rho algorithm ; Fixed Talbot method ; Unified Gaver-Stehfest algorithm ; Unified Euler algorithm ; Unified Talbot algorithm ; Laguerre method ; Spectral series expansion ; Constructive complex analysis ; Asymptotic method ; PDE method ; Monte Carlo simulation ; Turnbull and Wakeman approximation ; Milevsky and Posner approximation ; Joint densities ; Joint distribution functions ; Transformation of joint density ; Mathematica