Financial derivatives play significant role in an investor's success. Financial option is one form of derivatives. Option pricing is one of the challenging and fundamental problems of computational finance. Due to highly volatile and dynamic market conditions, there are no closed form solutions available except for simple styles of options such as, European options. Due to the complex nature of the governing mathematics, several numerical approaches have been proposed in the past to price American style and other complex options approximately. Bio-inspired and nature-inspired algorithms have been considered for solving large, dynamic and complex scientific and engineering problems. These algorithms are inspired by techniques developed by the insect societies for their own survival. Nature-inspired algorithms, in particular, have gained prominence in real world optimization problems such as in mobile ad hoc networks. The option pricing problem fits very well into this category of problems due to the ad hoc nature of the market. Particle swarm optimization (PSO) is one of the novel global search algorithms based on a class of nature-inspired techniques known as swarm intelligence. In this research, we have designed a sequential PSO based option pricing algorithm using basic principles of PSO. The algorithm is applicable for both European and American options, and handles both constant and variable volatility. We show that our results for European options compare well with Black-Scholes-Merton formula. Since it is very important and critical to lock-in profit making opportunities in the real market, we have also designed and developed parallel algorithm to expedite the computing process. We evaluate the performance of our algorithm on a cluster of multicore machines that supports three different architectures: shared memory, distributed memory, and a hybrid architectures. We conclude that for a shared memory architecture or a hybrid architecture, one-to-one mapping of particles to processors is recommended for performance speedup. We get a speedup of 20 on a cluster of four nodes with 8 dual-core processors per node.

Financial derivatives play significant role in an investor's success. Financial option is one form of derivatives. Option pricing is one of the challenging and fundamental problems of computational finance. Due to highly volatile and dynamic market conditions, there are no closed form solutions available except for simple styles of options such as, European options. Due to the complex nature of the governing mathematics, several numerical approaches have been proposed in the past to price American style and other complex options approximately. Bio-inspired and nature-inspired algorithms have been considered for solving large, dynamic and complex scientific and engineering problems. These algorithms are inspired by techniques developed by the insect societies for their own survival. Nature-inspired algorithms, in particular, have gained prominence in real world optimization problems such as in mobile ad hoc networks. The option pricing problem fits very well into this category of problems due to the ad hoc nature of the market. Particle swarm optimization (PSO) is one of the novel global search algorithms based on a class of nature-inspired techniques known as swarm intelligence. In this research, we have designed a sequential PSO based option pricing algorithm using basic principles of PSO. The algorithm is applicable for both European and American options, and handles both constant and variable volatility. We show that our results for European options compare well with Black-Scholes-Merton formula. Since it is very important and critical to lock-in profit making opportunities in the real market, we have also designed and developed parallel algorithm to expedite the computing process. We evaluate the performance of our algorithm on a cluster of multicore machines that supports three different architectures: shared memory, distributed memory, and a hybrid architectures. We conclude that for a shared memory architecture or a hybrid architecture, one-to-one mapping of particles to processors is recommended for performance speedup. We get a speedup of 20 on a cluster of four nodes with 8 dual-core processors per node.

Hitherto the diffusion of Electronic Data Interchange (EDI) has been limited due to high implementation and operational costs. On the other hand, the Extensible Markup Language (XML) has quickly become a generally accepted standard for integrating processing of formatted data - the literature is virtually unanimous that an integration of EDI into XML would make EDI more accessible and implementation faster and cheaper. The process of standardization of various EDI standard formats over XML is still underway and the question arises whether an early adoption of the technology would pay off. This thesis investigates the issue using real-options methodology. Starting from the well-known Black-Scholes model the parameters of the model are operationalized to decide about the best adoption timing: (i) project costs of implementation, (ii) value of savings of the project (substitutional, complementary, and strategic benefits), and (iii) project risk, expressed as the variance used in Black-Scholes. The latter considers both the external autonomy of the player in implementing new technology and internal properties in technology adoption. Discussing the technological properties of XML/EDI above parameters are operationalized step by step and integrated into a decision model to help each individual firm put the XML/EDI investment decision into real numbers. In order to better visualize the parameters of this decision framework, four company profiles, based on the theory of technology diffusion, will be introduced and mapped against the parameters of the Black-Scholes formula. (author's abstract)

The Black-Scholes option pricing model has several well recognized deficiencies, one of which is its assumption of a constant and time-homogeneous stock return volatility term. The implied volatility smile has been studied by subsequent researchers and various models have been developed in an attempt to reproduce this phenomenon from within the models. However, few of these models yield closed-form pricing formulas that are easy to implement in practice. In this thesis, we study a Mixture Lognormal model (MLN) for European option pricing, which assumes that future stock prices are conditionally described by a mixture of lognormal distributions. The ability of mixture models in generating volatility smiles as well as delivering pricing improvement over the traditional Black-Scholes framework have been much researched under multi-component mixtures for many derivatives and high-volatility individual stock options. In this thesis, we investigate the performance of the model under the simplest two-component mixture in a market characterized by relative tranquillity and over a relatively stable period for broad-based index options. A careful interpretation is given to the model and the results obtained in the thesis. This di erentiates our study from many previous studies on this subject. Throughout the thesis, we establish the unique advantage of the MLN model, which is having closed-form option pricing formulas equal to the weighted mixture of Black-Scholes option prices. We also propose a robust calibration methodology to fit the model to market data. Extreme market states, in particular the so-called crash-o-phobia effect, are shown to be well captured by the calibrated model, albeit small pricing improvements are made over a relatively stable period of index option market. As a major contribution of this thesis, we extend the MLN model to price exotic options including binary, Asian, and barrier options. Closed-form formulas are derived for binary and continuously monitored barrier options and simulation-based pricing techniques are proposed for Asian and discretely monitored barrier options. Lastly, comparative results are analysed for various strike-maturity combinations, which provides insights into the formulation of hedging and risk management strategies.

This work presents a class of models in asset pricing, whose underlying has dynamics of Affine jump diffusion type. We first present L´evy processes with their properties. We then introduce Affine jump diffusion processes which are basically a particular class of L´evy processes. Our motivation for these is driven by the fact that many financial models are built on them. Affine jump diffusion processes present good analytical properties that allow one to get close form formulas for a wide range of option pricing. The approach we use here is based on the paper by Duffie D, Pan J, and Singleton K. An example will show how incorporating parameters such as the volatility of the underlying asset in the model, can influence the resulting price of the financial instrument under consideration. We will also show how this class of models incorporate well known models, specially those used to model interest rates dynamics, like for instance the Vasicek model. / Dissertation (MSc (Mathematics of Finance))--University of Pretoria, 2008. / Mathematics and Applied Mathematics / unrestricted

In the mathematical modeling of the classical option pricing models it is assumed that the underlying stock price process follows a geometric Brownian motion, but through statistical analysis persistency was found in the log-returns of some South African stocks and Brownian motion does not have persistency. We suggest the replacement of Brownian motion with fractional Brownian motion which is a Gaussian process that depends on the Hurst parameter that allows for the modeling of autocorrelation in price returns. Three fractional Black-Scholes (Black) models were investigated where the underlying is assumed to follow a fractional Brownian motion. Using South African options on futures and warrant prices these models were compared to the classical models. / Dissertation (MSc)--University of Pretoria, 2012. / Mathematics and Applied Mathematics / unrestricted

Philosophiae Doctor - PhD / Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling. / 2023-12-02

In this paper, we obtain convergence of solutions of stochastic differential systems with memory gap to those with full finite memory. More specifically, solutions of stochastic differential systems with memory gap are processes in which the intrinsic dependence of the state on its history goes only up to a specific time in the past. As a consequence of this convergence, we obtain a new existence proof and approximation scheme for stochastic functional differential equations (SFDEs) whose coefficients have linear growth. In mathematical finance, an option pricing formula with full finite memory is obtained through convergence of stock dynamics with memory gap to stock dynamics with full finite memory.

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Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Barrier options are options whose payoff depends on whether or not the underlying asset price hits a certain level - the barrier - during the life of the option. Closed-form solutions for the prices of these path-dependent options are available in the Black-Scholes framework. It is well{known, however, that the Black-Scholes model does not price even the so-called vanilla options correctly. There are a number of popular asset price models based on exponential Lévy dynamics which are all able to capture the volatility smile, i.e. reproduce market-observed prices of vanilla options. This thesis investigates the potential model risk associated with the pricing of barrier options in several exponential Lévy models. First, the Variance Gamma, Normal Inverse Gaussian and CGMY models are calibrated to market-observed vanilla option prices. Barrier option prices are then evaluated in these models using Monte Carlo methods. The prices obtained are then compared to each other, as well as the Black-Scholes prices. It is observed that the different exponential Lévy models yield barrier option prices which are quite close to each other, though quite different from the Black-Scholes prices. This suggests that the associated model risk is low. / AFRIKAANSE OPSOMMING: Versperring opsies is opsies met 'n afbetaling wat afhanklik is daarvan of die onderliggende bateprys 'n bepaalde vlak - die versperring - bereik gedurende die lewe van die opsie, of nie. Formules vir die pryse van sulke opsies is beskikbaar binne die Black-Scholes raamwerk. Dit is egter welbekend dat die Black-Scholes model nie in staat is om selfs die sogenaamde vanilla opsies se pryse korrek te bepaal nie. Daar bestaan 'n aantal populêre bateprysmodelle gebaseer op eksponensiële Lévy-dinamika, wat almal in staat is om die mark-waarneembare vanilla opsie pryse te herproduseer. Hierdie tesis ondersoek die potensiële modelrisiko geassosieer met die prysbepaling van versperring opsies in verskeie eksponseniële Lévy-modelle. Eers word die Variance Gamma{, Normal Inverse Gaussian- en CGMY-modelle gekalibreer op mark-waarneembare vanilla opsiepryse. Die pryse van versperring opsies in hierdie modelle word dan bepaal deur middel van Monte Carlo metodes. Hierdie pryse word dan met mekaar vergelyk, asook met die Black-Scholespryse. Dit word waargeneem dat die versperring opsiepryse in die verskillende eksponensiële Lévymodelle redelik na aan mekaar is, maar redelik verskil van die Black-Scholespryse. Dit suggereer dat die geassosieerde modelrisiko laag is.