This paper examines different combinations of close-field and far-field boundary conditions for solving the 2D Black-Scholes model using finite difference methods in space. The combinations were also tested for different parameter settings. The research showed that in the area close to the strike price, the error was not particularly affected by the boundary conditions but rather by the characteristics of the problem itself. The main differences in error for the combinations of conditions are located close to the boundaries. However, if the computational domain for some reason has to be reduced, e.g. to save computational time, the boundary conditions will play an important role on the error in the area close to the strike price. Based on the findings presented in this report, Dirichlet boundary condition on the far- field boundary together with no boundary condition on the close-field is the best combination. If any of those are not applicable, the linearity condition should be used on that boundary instead.

This work describes stochastic volatility models and application of such models for option pricing. Models for underlying asset and then pricing models for options with stochastic volatility are derived. Black-Scholes and Heston-Nandi models are compared in empirical part of this work.

The purpose of this paper is solving the European option pricing problem under the Black–Scholes model. Our approach is to use the so-called stochastic Runge–Kutta (SRK) numericalscheme to find the corresponding expectation of the functional to the stochastic differentialequation under the Black–Scholes model. Several numerical solutions were made to study howquickly the result converges to the theoretical value. Then, we study the order of convergenceof the SRK method with the help of MATLAB.

Asset pricing models are well established and have been used extensively by practitioners both for pricing options as well as for hedging them. Though Black-Scholes is the original and most commonly communicated asset pricing model, alternative asset pricing models which incorporate additional features have since been developed. We present three asset pricing models here - the Black-Scholes model, the Heston model and the Merton (1976) model. For each asset pricing model we test the hedge effectiveness of delta hedging, minimum variance hedging and static hedging, where appropriate. The options hedged under the aforementioned techniques and asset pricing models are down-and-out call options, lookback options and cliquet options. The hedges are performed over three strikes, which represent At-the-money, Out-the-money and In-the-money options. Stock prices are simulated under the stochastic-volatility double jump diffusion (SVJJ) model, which incorporates stochastic volatility as well as jumps in the stock and volatility process. Simulation is performed under two ’Worlds’. World 1 is set under normal market conditions, whereas World 2 represents stressed market conditions. Calibrating each asset pricing model to observed option prices is performed via the use of a least squares optimisation routine. We find that there is not an asset pricing model which consistently provides a better hedge in World 1. In World 2, however, the Heston model marginally outperforms the Black-Scholes model overall. This can be explained through the higher volatility under World 2, which the Heston model can more accurately describe given the stochastic volatility component. Calibration difficulties are experienced with the Merton model. These difficulties lead to larger errors when minimum variance hedging and alternative calibration techniques should be considered for future users of the optimiser.

The objective of this thesis is to review the two popular mathematical models of the financialderivatives market. The models are the classical Black–Scholes–Merton and the Continuoustime Markov chain (CTMC) model. We study the CTMC model which is illustrated by themathematician Ragnar Norberg. The thesis demonstrates how the fundamental results ofFinancial Engineering work in both models.The construction of the main financial market components and the approach used for pricingthe contingent claims were considered in order to review the two models. In addition, the stepsused in solving the first–order partial differential equations in both models are explained.The main similarity between the models are that the financial market components are thesame. Their contingent claim is similar and the driving processes for both models utilizeMarkov property.One of the differences observed is that the driving process in the BSM model is the Brownianmotion and Markov chain in the CTMC model.We believe that the thesis can motivate other students and researchers to do a deeper andadvanced comparative study between the two models.

There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

In this thesis the Black Scholes and the Heston stock prices are investigated and the models are compared. The Black Scholes model assumes that the volatility is constant, while the Heston model allows stochastic volatility which is more flexible and can perform better with empirical data. Both models are analysed and simulated, and the parameters are estimated based on empirical data of S&P 500. Results are based on simulations and characteristic functions which are presented with figures of probability density functions.

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Yes / This paper studies the predictive ability of corridor implied volatility (CIV) measure. It is motivated by the fact that CIV is measured with better precision and reliability than the model-free implied volatility due to the lack of liquid options in the tails of the risk-neutral distribution. By adding CIV measures to the modified GARCH specifications, the out-of-sample predictive ability of CIV is measured by the forecast accuracy of conditional volatility. It finds that the narrowest CIV measure, covering about 10% of the RND, dominate the 1-day ahead conditional volatility forecasts regardless of the choice of GARCH models in high volatile period; as market moves to non volatile periods, the optimal width broadens. For multi-day ahead forecasts narrow and mid-range CIV measures are favoured in the full sample and high volatile period for all forecast horizons, depending on which loss functions are used; whereas in non turbulent markets, certain mid-range CIV measures are favoured, for rare instances, wide CIV measures dominate the performance. Regarding the comparisons between best performed CIV measures and two benchmark measures (market volatility index and at-the-money Black–Scholes implied volatility), it shows that under the EGARCH framework, none of the benchmark measures are found to outperform best performed CIV measures, whereas under the GARCH and NAGARCH models, best performed CIV measures are outperformed by benchmark measures for certain instances.

Made available in DSpace on 2016-04-25T18:40:44Z (GMT). No. of bitstreams: 1 Carl Douglas De Gennaro Oliveira.pdf: 1342470 bytes, checksum: 9868002de42872f20913eb856aa2b173 (MD5) Previous issue date: 2010-05-24 / The process of Brazil s compliance with the International Financial Reporting Standard (IFRS) took a big step forward, definitively getting on the agenda of regulatory agencies, companies and auditing firms, when Federal Law 11.638 was signed in December 2007, altering the accounting chapter of Brazilian Corporate Law, 6.404/76. This study contributes to Brazil s process of compliance with the IFRS, specifically regarding the applicability of IFRS 2 Share-based Payment, or its Brazilian corollary CPC 10 Pagamento Baseado em Ações, and the impact on accounting and on the disclosure of a long-term compensation program for executives, characterized as phantom stock options. IFRS 2 was published in February 2002 and was required internationally from January 2005, as an outcome of the growing use of commercial transaction payments based on shares, and also the IOSCO´s report that pointed out the lack of an accounting standard dealing with this kind of transaction. The study found that IFRS 2 or CPC 10 can be appropriately applied to guide the accounting treatment given to a phantom stock option program, and was a more informative accounting practice than that which had been used in Brazil, before 2008. The study also found a wide-spread need of financial knowledge regarding the valuation of stock options, such as the Black-Scholes-Merton model, as well as statistical methods for appropriately account and disclose the fair value of share-based incentive plans. Furthermore, in order to understand more fully the economic event which is being accounted, it is highly important to understand its essence. In the case of long-term share-based incentives for executives, the essence of their existence can be found in agency theory / O processo de convergência do Brasil às Normas Internacionais de Contabilidade (IFRSs) deu um grande salto e entrou definitivamente na agenda dos órgãos reguladores, empresas e auditorias, com a sanção da lei federal 11.638 em dezembro de 2007, que alterou o capítulo contábil da Lei das Sociedades Anônimas, 6.404/76. Este estudo contribui para o processo de convergência brasileiro às IFRSs, especificamente quanto à aplicabilidade do IFRS 2 Share Based Payment, ou sua correlação brasileira CPC 10 Pagamento Baseado em Ações, e dos impactos contábeis e de divulgação decorrentes de um programa de compensação de longo prazo a executivos, com as características de phantom stock options, ou opções fantasmas. O IFRS 2 foi publicado em fevereiro de 2002 e requerido internacionalmente a partir de janeiro de 2005, como uma decorrência do crescente uso de pagamento das transações comerciais com base em ações e também do relatório da IOSCO, que identificou como falha a lacuna de norma contábil que tratasse deste tipo de transação. O estudo identificou que o IFRS 2 ou CPC 10 aplica-se adequadamente para orientar o tratamento contábil de um programa de phantom stock option e representou uma prática contábil mais informativa que aquela até então adotada no Brasil, antes do ano de 2008. O estudo também identificou a grande necessidade de conhecimento de finanças relacionado à avaliação de opções, tal como o modelo Black-Scholes-Merton, bem como de métodos estatísticos, para uma apropriada contabilização e divulgação do valor justo dos planos de incentivo baseados em ações. Além disso, para que se entenda com profundidade o evento econômico que se contabiliza, é de suma importância a compreensão de sua essência. No caso de incentivos de longo prazo para executivos, baseados em ações, a essência de sua existência pode ser encontrada na Teoria de Agência