In this thesis, we derive a closed-form approximation to the implied volatility for a European option, assuming that the underlying asset follows the generalized Heston model. A new para- meter is added to the Heston model which constructed the generalized Heston model. Based on the results in Lorig, Pagliarani and Pascucci [11], we obtain implied volatility expansions up to third-order. We conduct numerical studies to check the accuracy of our expansions. More specifically we compare the implied volatilities computed using our expansions to the results by Monte Carlo simulation method. Our numerical results show that the third-order implied volatility expansion provides a very good approximation to the true value.

Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

The fast Fourier transform (FFT), even though it has been widely applicable in Physics and Engineering, it has become attractive in Finance as well for it’s enhancement of computational speed. Carr and Madan succeeded in implementing the FFT for pricing of an option. This project, inspired by Carr and Madan’s paper, attempts to elaborate and connect the various mathematical and theoretical concepts that are helpful in understanding of the derivation. Further, we derive the characteristic function of the risk neutral probability for the logarithmic terminal stock price. The Black-Scholes-Merton (BSM) model is also revised including derivation of the partial deferential equation and the formula. Finally, comparison of the BSM numerical implementation with and without the FFT method is done using MATLAB.

The famous Black-Scholes partial differential equation is one of the most widely used and researched equations in modern financial engineering to address the complex evaluations in the financial markets. This thesis investigates a numerical technique, using a fourth-order discretization in time and space, to solve a generalized version of the classical Black-Scholes partial differential equation. The numerical discretization in space consists of a fourth order centered difference approximation in the interior points of the spatial domain along with a fourth order left and right sided approximation for the points near the boundary. On the other hand, the temporal discretization is made by implementing a Runge-Kutta order four (RK4) method. The designed approximations are analyzed numerically with respect to stability and convergence properties.

This paper considers dynamic optimal portfolio strategies of utility maximizing investors in the presence of risk constraints. In particular, we investigate the optimization problem with an additional constraint modeling bounded shortfall risk measured by Value at Risk or Expected Loss. Using the Black-Scholes model of a complete financial market and applying martingale methods we give analytic expressions for the optimal terminal wealth and the optimal portfolio strategies and present some numerical results.

Employee share options and Black Economic Empowerment deals are financial instruments found in the South African market. Employee share options (ESOs) are issued as a form of non-cash compensation to the employees of the company in addition to their salaries or bonuses. Its value is linked to the share price and since there is no downside risk for the employee his share option is similar to owning a call option on the stock of his employer. Black economic empowerment (BEE) deals in this report refer to those types of transactions structured by listed South African companies to facilitate the transfer of a portion of their ordinary issued share capital to South African individuals or groups who qualify under the Broad-Based Black Economic Empowerment Act of 2003 (“the Act”). This Act requires a minimum percentage of the company to be black-owned in order to address the disproportionate distribution of wealth amongst racial groups in South Africa due to the legacy of Apartheid. These transactions are usually structured in such a way to allow the BEE partner to participate in the upside of the share price beyond a certain level but not in the downside which replicates a call option on the share price of the issuing company. The cost of both ESOs and BEE deals has to be accounted for on the balance sheet of the issuing company at its fair-value. Neither of these instruments can be traded and their extended option lifetimes are features that distinguish these deals significantly from regular traded options for which liquid markets exist. This makes pricing them a non-trivial exercise. A number of types of mathematical models have been developed to take the unique structure features into account to price them as accurately as possible. Research by Huddart&Lang (1995&1996) has shown that option holders often exercise their vested options long before the maturity of the transactions but are unable to quantify a measure that can be used. The wide variety of factors influencing option holders (recent stock price movements, market-to-strike ratio, proximity of vesting dates, time to maturity, share price volatility and wealth of option holder) as well as little exercise data publicly available prevents the options from being priced in a consistent manner. Various assumptions regarding the exercise behaviour of option holders are used that are not based on empirical observations even though the option prices are sensitive to this input. This dissertation provides an overview of the models, inputs and exercise behaviour assumptions that are recognized in pricing both ESOs and BEE deals under IFRS 2 in South Africa. This puts the reader in a position to evaluate all pricing aspects of these deals. Furthermore, their structuring are also analysed in order to identify the general issues related to them. A number of methods to manage the pricing issue surrounding exercise behaviour on ESOs have been considered for the South African market. The ESO Upper Bound-methodology showed that for each strike there is a threshold at which exercise will occur and the employee can invest the after-tax proceeds in a diversified portfolio with a higher expected return than that of the single equity option. This approach reduces the standard Black-Scholes option value without relying on assumptions about the employee’s exercise behaviour and is a viable alternative for the South African market. The derived option value represents the cost of the option. Seven large listed companies’ BEE transactions are dissected and compared against one another using the fair-value of the transaction as a percentage of the market capitalization of the company. The author shows how this measure is a more equitable way of assigning BEE credits to companies than the current practice which is shareholding-based. The current approach does not reward the effort (read cost) that a company has undertaken to transfer shares to black South Africans but only focuses on the amount that is finally owned by the BEE participants. This leaves the transaction vulnerable to a volatile share price and leads to transactions with extended lock-in periods that do not provide much economic benefit to the BEE participants for many years. Other inefficiencies in the type of BEE transactions that have emerged in reaction to the BEE codes that have been published by the South African government are also considered. Finally the funding model that is often used to facilitate these deals is assessed and the risks involved for the funder (bank) is reflected on. / Dissertation (MSc)--University of Pretoria, 2011. / Mathematics and Applied Mathematics / unrestricted

This thesis aims to fill this gap between static and dynamic risk measures. It presents a theory of dynamic risk measures based directly on classical, static risk measures. This allows for a direct connection of the static, the discrete time as well as the continuous time setting. Unlike the existing literature this approach leads to a interpretable pendant to the well-understood static risk measures. As a key concept the notion of divisible families of risk measures is introduced. These families of risk measures admit a dynamic version in continuous time. Moreover, divisibility allows the definition of the risk generator, a nonlinear extension of the classical infinitesimal generator. Based on this extension we derive a nonlinear version of Dynkins lemma as well as risk-averse Hamilton–Jacobi–Bellman equations.

Future 450mm semiconductor wafer foundries are expected to produce billions of low cost, leading-edge processors, memories, and wireless sensors for Internet of Everything applications in smart cities, smart grids, and smart infrastructures. The problem has been a lack of wise investment decision making using traditional semiconductor industry models. The purpose of this study was to design decision-making models to conserve financial resources from conception to commercialization using real options to optimize production capacity, to defer an investment, and to abandon the project. The study consisted of 4 research questions that compared net present value from real option closed-form equations and binomial lattice models using the Black-Scholes option pricing theory. Three had focused on sensitivity parameters. Moore's second law was applied to find the total foundry cost. Data were collected using snowball sampling and face-to-face surveys. Original survey data from 46 Americans in the U.S.A. were compared to 46 Europeans in Germany. Data were analyzed with a paired-difference test and the Box-Behnken design was employed to create prediction models to support each hypothesis. Data from the real option models and survey findings indicate American 450mm foundries will likely capture greater value and will choose the differentiation strategy to produce premium chips, whereas higher capacity, cost leadership European foundries will produce commodity chips. Positive social change and global quality of life improvements are expected to occur by 2020 when semiconductors will be needed for the $14 trillion Internet of Everything market to create safe self-driving vehicles, autonomous robots, smart homes, novel medical electronics, wearable computers with streaming augmented reality information, and digital wallets for cashless societies.

This thesis investigates the relationship between implied volatility and dividend yield in the options market, focusing on testing the Bird-in-Hand theory versus the Dividend Irrelevancy theory. Utilizing panel data analysis and regression techniques, with both ordinary and lagged regressions, the study explores how dividend yield impacts European options implied volatility across European markets over ten years from February 2013 to February 2023. Employing the Hausman specification test, Breusch Pagan multiplier test, cluster standard errors, and heteroskedasticity for robustness. The analysis includes both call and put options, incorporating various control variables and market factors. The findings reveal that changes in dividend yield consistently impact call option implied volatility and also exhibit a stronger and more consistent negative relationship with put option implied volatility, overall, supporting the Bird-in-Hand theory. Furthermore, this thesis highlights the importance of considering alternative methodologies, expanding sample sizes, and exploring additional variables to enhance understanding of option pricing dynamics.

This thesis investigates the application of the Black-Scholes model for pricing long-maturity options, primarily utilizing historical data on S\&P500 options. It compares prices computed with the Black-Scholes formula to actual market prices and critically examines the validity of the Black-Scholes model assumptions over long time frames. The assumptions mainly focused on are the constant volatility assumption, the assumption of normally distributed returns, the constant interest rate assumption and the no transaction cost assumption. The results show that the differences between computed prices and actual prices decrease as options get closer to maturity. They also show that several of the Black-Scholes model assumptions are not entirely realistic over long time frames. The conclusion of the thesis is that there are several limitations to the Black-Scholes model when it comes to pricing long-maturity options.