Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

Made available in DSpace on 2010-04-20T20:20:31Z (GMT). No. of bitstreams: 1 129707.pdf: 13779695 bytes, checksum: 46556262df0871aa08d9eac2d7e18db9 (MD5) Previous issue date: 2006-04-17T00:00:00Z / Foram selecionados três modelos de apreçamento dos warrtants implícitos nas debêntures conversíveis que foram usados nos estudos de quatro casos selecionados de emissão de debêntures conversíveis neste período. Os modelos de apreçamento usados para a verificação do valor justo de lançamento das debêntures selecionadas foram: Modelo de projeção de resultados futuros, Modelo de Black-Scholes e Modelo de Avaliação Binomial

Magister Scientiae - MSc / The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant. / South Africa

Philosophiae Doctor - PhD / Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature. / South Africa

>Magister Scientiae - MSc / This dissertation studies the computation methods of pricing of Asian options. Asian options are options in which the underlying variable is the average price over a period of time. Because of this, Asian options have a lower volatility and this render them cheaper relative to their European counterparts. Asian options belong to the so-called path-dependent derivatives; they are among the most difficult to price and hedge both analytically and numerically. In practice, it is only discrete Asian options that are traded, however continuous Asian options are used for studying purposes. Several approaches have been proposed in the literature, including Monte Carlo simulations, tree-based methods, Taylor’s expansion, partial differential equations, and analytical ap- proximations among others. When using partial differential equations for pricing of continuous time Asian options, the high dimensionality is problematic. In this dissertation we focus on the discrete time methods. We start off by explaining the binomial tree method, and our last chapter presents the very exciting and relatively simple method of Tsao and Huang, using Taylor approximations. The main papers that are used in this dissertation are articles by Jan Vecer (2001); LCG Rogers (1995); Eric Benhamou (2001); Gianluca Fusai (2007); Kamizono, Kariya and Nakatsuma (2006) and Tsao and Huang (2007). The author has provided computations, including graphs and tables dispersed over the different chapters, to demonstrate the utility of the methods. We observe various parameters of influence such as correlation, volatility, strike, etc. A further contribution by the author of this dissertation is, in particular, in Chapter 5, in the presentation of the work of Tsao et al. Here we have provided slightly more detailed explanations and again some further computational tables.

Financial derivatives have been constituting one of the most dynamic fields in the mathematical finance. The main task is represented by the valuation or pricing of these instruments. This theses deals with standard models and their limits, tries to explore advanced methods of continuous martingale measures and on their bases proposes numerical methods applicable to derivatives valuation. Some procedures leading to elimination of certain simplifying assumptions are presented as well.

In the present work, we study the topic of stochastic differential equations, their numerical solution and solution of models for pricing of options which follow from stochastic differential equations using the Itô calculus. We present several numerical methods for solving stochastic differential equations. These methods are then implemented in MATLAB and we investigate their properties, especially their convergence characteristics. Furthermore, we formulate two models for pricing of European call options. We solve these models using a variant of the spectral collocation method, again in MATLAB.

Over 22 billion derivative contracts were traded on different stock exchanges globally during the year 2010 of which almost 50% were futures while the remaining 50% were options. An overall 25% increase in such contracts was registered as compared to those traded in the year 2009 (International Options Market Association (IOMA) Report, 2011).Investors often use a wide array of trading tools, market indicators and market trading strategies to get the best possible returns for the money that was invested. The main objective of this paper is to focus on the use of market sentiment indicators, specifically the Put-Call Ratio (PCR) as a predictor of returns for an investor.The Put-Call Ratio is defined as a ratio of the trading volume of put options to call options. It is called a sentiment indicator because it measures the “feelings” of option traders. Additionally, it has longed been viewed as an indicator of investors’ sentiment in the market (Put-Call Ratio, 2012) and is possibly the most favoured description of market psychology (James, 2011). / Dissertation (MBA)--University of Pretoria, 2012. / Gordon Institute of Business Science (GIBS) / unrestricted

The presentation shows sample paths of stochastic processes in form of animations. Those stochastic procsses are usually used to model financial quantities like exchange rates, interest rates and stock prices. In the second part the solution of the Black-Scholes PDE using the finite difference method is illustrated. / Der Vortrag zeigt Animationen von Realisierungen stochstischer Prozesse, die zur Modellierung von Groessen im Finanzbereich haeufig verwendet werden (z.B. Wechselkurse, Zinskurse, Aktienkurse). Im zweiten Teil wird die Loesung der Black-Scholes Partiellen Differentialgleichung mittels Finitem Differenzenverfahren graphisch veranschaulicht.

The paper investigates dynamic optimal portfolio strategies of utility maximizing portfolio managers in the presence of risk constraints. Especially we consider the risk, that the terminal wealth of the portfolio falls short of a certain benchmark level which is proportional to the stock price. This risk is measured by the Expected Utility Loss. We generalize the findings our previous papers to this case. Using the Black-Scholes model of a complete financial market and applying martingale methods, analytic expressions for the optimal terminal wealth and the optimal portfolio strategies are given. Numerical examples illustrate the analytic results.