Using the dual Black-Scholes partial differential equation, Dupire derived an explicit formula, involving the ratio of partial derivatives of the evolving fair value of a European call option (ECO), for recovering information about its variable volatility. Because the prices, as a function of maturity and strike, are only available as discrete noisy observations, the evaluation of Dupire’s formula reduces to being an ill-posed numerical differentiation problem, complicated by the need to take the ratio of derivatives. In order to illustrate the nature of ill-posedness, a simple finite difference scheme is first used to approximate the partial derivatives. A new method is then proposed which reformulates the determination of the volatility, from the partial differential equation defining the fair value of the ECO, as a parameter identification activity. By using the weak formulation of this equation, the problem is localized to a subregion on which the volatility surface can be approximated by a constant or a constant multiplied by some known shape function which models the local shape of the volatility function. The essential regularization is achieved through the localization, the choice of the analytic weight function, and the application of integration-by-parts to the weak formulation to transfer the differentiation of the discrete data to the differentiation of the analytic weight function.

No description available.

Credit default swaptions (CDS options) are credit derivatives that are widely used by finan-cial institutions such as banks and hedging companies to manage their credit risk. These options are usually priced using Black-Scholes model, but the assumptions underlying this model do not always hold especially when solving complex financial problems. The proposed solution is to use numerical methods such as finite difference method (FDM) to approximate the solution of the Black-Scholes PDE in cases where closed form solutions cannot be obtained. The pricing of swaptions are important in financial markets, hence we specifically discuss the pricing of interest rate swaptions, CDS options, commodity swaptions and energy swap-tions using Black-Scholes model. Simple parabolic PDE known as heat equation given at (Higham, 2004) forms a foundations to understand the application of FDM when solving a PDE. Since, Black-Scholes PDE is also a parabolic equation it is transformed to a form of a heat equation (diffusion equation) by applying change of variables technique. FDM, specifically Crank-Nicolson method can be applied to the heat equation but in this dissertation it is applied directly to the Black-Scholes PDE to approximate its solution. Therefore, it is preferable to use Crank-Nicolson method because it is known to be second- order accurate, unconditionally stable, very flexible, suitable and can accommodate varia- tions in financial problems, (Duffy, 2008). The stability of this method is investigated using a matrix approach because it accommodates the effect of boundary conditions. To test the convergence of Crank-Nicolson method, it is compared with the Black-Scholes method used in (Tucker and Wei, 2005) to price CDS options. Conclusively the results obtained by Crank-Nicolson method to price CDS options are similar to those obtained using Black-Scholes method. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Credit default swaptions (CDS options) are credit derivatives that are widely used by finan-cial institutions such as banks and hedging companies to manage their credit risk. These options are usually priced using Black-Scholes model, but the assumptions underlying this model do not always hold especially when solving complex financial problems. The proposed solution is to use numerical methods such as finite difference method (FDM) to approximate the solution of the Black-Scholes PDE in cases where closed form solutions cannot be obtained. The pricing of swaptions are important in financial markets, hence we specifically discuss the pricing of interest rate swaptions, CDS options, commodity swaptions and energy swap-tions using Black-Scholes model. Simple parabolic PDE known as heat equation given at (Higham, 2004) forms a foundations to understand the application of FDM when solving a PDE. Since, Black-Scholes PDE is also a parabolic equation it is transformed to a form of a heat equation (diffusion equation) by applying change of variables technique. FDM, specifically Crank-Nicolson method can be applied to the heat equation but in this dissertation it is applied directly to the Black-Scholes PDE to approximate its solution. Therefore, it is preferable to use Crank-Nicolson method because it is known to be second- order accurate, unconditionally stable, very flexible, suitable and can accommodate varia- tions in financial problems, (Duffy, 2008). The stability of this method is investigated using a matrix approach because it accommodates the effect of boundary conditions. To test the convergence of Crank-Nicolson method, it is compared with the Black-Scholes method used in (Tucker and Wei, 2005) to price CDS options. Conclusively the results obtained by Crank-Nicolson method to price CDS options are similar to those obtained using Black-Scholes method. / Thesis (MSc (Risk Analysis))--North-West University, Potchefstroom Campus, 2013.

Pricing options quickly and accurately is a well known problem in finance. Quantum computing is being researched with the hope that quantum computers will be able to price options more efficiently than classical computers. This research extends the quantum binomial option pricing model proposed by Zeqian Chen to European put options and to Barrier options and develops a quantum algorithm to price them. This research produced three key results. First, when Maxwell-Boltzmann statistics are assumed, the quantum binomial model option prices are equivalent to the classical binomial model. Second, options can be priced efficiently on a quantum computer after the circuit has been built. The time complexity is O((N − τ)log(N − τ)) and it is in the BQP quantum computational complexity class. Finally, challenges extending the quantum binomial model to American, Asian and Bermudan options exist as the quantum binomial model does not take early exercise into account.

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

In this master thesis, we compare three different types of funding alternatives from a Swedish municipality's point of view, with the main focus on analysing a Nikkei-linked loan. We do this by analysing the resulting interest rate and the expected exposures, taking collateral into consideration. We conclude, with certainty, that there are many alternatives for funding and that they each need to be analysed and compared on many levels to be able to make a correct decision as to which ones to choose. An important part of this is to consider the implications of the newest regulations and risk exposure, as it might greatly influence the final price for contracts. Between the cases that we considered, the SEK bond was the one with the lowest resulting spread, and the one which is the simplest considering the collateral involved. While other alternatives might be better depending on how profitable it is for the municipality to receive collateral, the SEK bond is the most transparent one and with least risk involved.

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