Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options

In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above

Function approximation for option pricing and risk management Methods, theory and applications.

PhD Thesis.This thesis investigates the application of function approximation techniques for computationally demanding problems in nance. We focus on the use of Chebyshev interpolation and its multivariate extensions. The main contribution of this thesis is the development of a new pricing method for path-dependent options. In each step of the dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. A key advantage of this approach is that it allows us to shift all modeldependent computations into a pre-computation step. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. We provide a theoretical error analysis and nd conditions that imply explicit error bounds. Numerical experiments con rm the fast convergence of prices and sensitivities. We use the new method to calculate credit exposures of European and path-dependent options for pricing and risk management. The simple structure of the Chebyshev interpolation allows for a highly e cient evaluation of the exposures. We validate the accuracy of the computed exposure pro les numerically for di erent equity products and a Bermudan swaption. Benchmarking against the least-squares Monte Carlo approach shows that our method delivers a higher accuracy in a faster runtime. We extend the method to e ciently price early-exercise options depending on several risk-factors. As an example, we consider the pricing of callable bonds in a hybrid twofactor model. We develop an e cient and stable calibration routine for the model based on our new pricing method. Moreover, we consider the pricing of early-exercise basket options in a multivariate Black-Scholes model. We propose a numerical smoothing in the dynamic programming time-stepping using the smoothing property of a Gaussian kernel. An extensive numerical convergence analysis con rms the e ciency

Maritime expressions:a corpus based exploration of maritime metaphors

This study uses a purpose-built corpus to explore the linguistic legacy of Britain’s maritime history found in the form of hundreds of specialised ‘Maritime Expressions’ (MEs), such as TAKEN ABACK, ANCHOR and ALOOF, that permeate modern English. Selecting just those expressions commencing with ’A’, it analyses 61 MEs in detail and describes the processes by which these technical expressions, from a highly specialised occupational discourse community, have made their way into modern English. The Maritime Text Corpus (MTC) comprises 8.8 million words, encompassing a range of text types and registers, selected to provide a cross-section of ‘maritime’ writing. It is analysed using WordSmith analytical software (Scott, 2010), with the 100 million-word British National Corpus (BNC) as a reference corpus. Using the MTC, a list of keywords of specific salience within the maritime discourse has been compiled and, using frequency data, concordances and collocations, these MEs are described in detail and their use and form in the MTC and the BNC is compared. The study examines the transformation from ME to figurative use in the general discourse, in terms of form and metaphoricity. MEs are classified according to their metaphorical strength and their transference from maritime usage into new registers and domains such as those of business, politics, sports and reportage etc. A revised model of metaphoricity is developed and a new category of figurative expression, the ‘resonator’, is proposed. Additionally, developing the work of Lakov and Johnson, Kovesces and others on Conceptual Metaphor Theory (CMT), a number of Maritime Conceptual Metaphors are identified and their cultural significance is discussed