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Title: The generalised Fourier B-spline methodology and applications of the generalised Fourier transform
Author: Haslip, Gareth G.
Awarding Body: City University London
Current Institution: City, University of London
Date of Award: 2012
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This thesis develops a new efficient and robust approximation framework, the Generalised Fourier B-spline (GFBS) method, for option pricing in the setting of continuous-time asset models from the family of exponen- tial semi martingale processes. The GFBS method introduces B-spline in- terpolation theory to derivative pricing to provide an accurate closed-form representation of the option pricing under an inverse generalised Fourier transform. The GFBS methodology is developed through a series of ap- plications that explore the interplays between financial mathematics and actuarial science. The first application considered is the pricing of non-life reinsurance con- tracts in the presence catastrophe bonds. Using the generalised Fourier transform, an appropriate financial pricing formula is derived, and the fast- and fractional fast Fourier transforms are used to evaluate prices. This methodology is extended to a wider range of reinsurance contracts, in- cluding individual excess of loss reinsurance with reinstatements. The GFBS method is then derived in the context of pricing European op- tions. This is a new efficient and robust framework for option pricing under continuous-time asset models from the family of exponential semimartin- gale processes. The GFBS method introduces B-spline interpolation the- ory to derivative pricing to provide an accurate closed-form representa- tion of the option price under an inverse generalised Fourier transform. The GFBS method is extremely fast and accurate and is demonstrated to be more efficient than existing numerical methods. This suggests a wide range of applications, including the use of more realistic asset mod- els in high frequency trading. Examples considered include pricing under a range of asset models, including stochastic volatility and jump diffusion, computation of the Greeks, and the inverse problem of cross-sectional cal- ibration. Next, the GFBS method is shown to be applicable in the space of exotic derivatives for pricing discrete lookback options for which, in general, no closed-form pricing formula exist. Using B-spline interpolation, an accu- rate closed-form representation of the lookback option price is obtained under an inverse generalised Fourier transform. This provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. An explicit representation for the characteristic function of the maximum of a discretely observed stochastic process is derived, which provides a significant improvement in terms of numerical efficiency over the Spitzer- recurrence formula. This is of fundamental importance and could have a wide range of applications where the Spitzer formula is utilised. Several examples are considered covering a range of asset models that exhibit closed-form solutions for the price of a European option, which is required as an input to the GFBS pricing method as presented here. Finally, the GFBS method for discrete lookback options is extended to the case where the underlying asset model is less tractable and does not exhibit a closed-form solution for the price of European options. This is an important result that opens up the possibility of using more realistic asset models when pricing discrete lookback option.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available