The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

The Black-Scholes model and its assumptions has endured its fair share of criticism. One problematic issue is the model’s assumption that market volatility is constant. The past decade has seen numerous publications addressing this issue by adapting the Black-Scholes model to incorporate stochastic volatility. In this dissertation, American put options are priced under the Heston stochastic volatility model using the Crank- Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). Due to the early exercise facility, the pricing of American put options is a challenging task, even under constant volatility. Therefore the pricing problem under constant volatility is also included in this dissertation. It involves transforming the Black-Scholes partial differential equation into the heat equation and re-writing the pricing problem as a linear complementary problem. This linear complimentary problem is solved using the Crank-Nicolson finite difference method in combination with the Projected Over-Relaxation method (PSOR). The basic principles to develop the methods necessary to price American put options are covered and the necessary numerical methods are derived. Detailed algorithms for both the constant and the stochastic volatility models, of which no real evidence could be found in literature, are also included in this dissertation. / MSc (Applied Mathematics), North-West University, Potchefstroom Campus, 2013

Este trabalho analisa três estratégias de hedge de opção, buscando identificar a importância da escolha da estratégia para a obtenção de um bom desempenho do hedge. O conceito de hedge é analisado de forma retrospectiva e uma teoria geral de hedge é apresentada. Em seguida são descritos alguns estudos comparativos de desempenho de estratégias de hedge de opção e suas metodologias de implementação. Para esta análise comparativa são selecionadas três estratégias de hedge de opção de compra do tipo européia: a primeira utiliza o modelo Black-Scholes-Merton de precificação de opções, a segunda utiliza uma solução de programação dinâmica para hedge dinâmico multiperiódico e a terceira utiliza um modelo GARCH para precificação de opções. As estratégias são comentadas e comparadas do ponto de vista de suas premissas teóricas e por meio de testes comparativos de desempenho. O desempenho das estratégias é comparado sob uma perspectiva dinâmicamente ajustada, multiperiódica e autofinanciável. Os dados para comparação de desempenho são gerados por simulação e o desempenho é avaliado pelos erros absolutos médios e erros quadráticos médios, resultantes na carteira de hedge. São feitas ainda considerações a respeito de alternativas de estimação e suas implicações no desempenho das estratégias. / This work analyzes three option hedging strategies, to identify the importance of choosing a strategy in order to achieve a good hedging performance. A retrospective analysis of the concept of hedging is conducted and a general hedging theory is presented. Following, some comparative papers of hedging performance and their implementation methodologies are described. For the present comparative analysis, three hedging strategies for European options have been selected: the first one based on the Black-Scholes-Merton model for option pricing, the second one based on a dynamic programming solution for dynamic multiperiod hedging and the third one based on a GARCH model for option pricing. The strategies are compared under their theoric premisses and through comparative performance testes. The performances of the strategies are compared under a dynamically adjusted multiperiodic and self-financing perspective. Data for performance comparison are generated by simulation and performance is evaluated by mean absolute errors and mean squared errors resulting on the hedging portfolio. An analysis is also done regarding estimation approaches and their implications over the performance of the strategies.

This thesis considers the price and characteristics of an exotic option called the Volatility-Cap-Target-Level(VCTL) option. The payoff function is a simple European option style but the underlying value is a dynamic portfolio which is comprised of two components: A risky asset and a non-risky asset. The non-risky asset is a bond and the risky asset can be a fund or an index related to any asset category such as equities, commodities, real estate, etc. The main purpose of using a dynamic portfolio is to keep the realized volatility of the portfolio under control and preferably below a certain maximum level, denoted as the Volatility-Cap-Target-Level (VCTL). This is attained by a variable allocation between the risky asset and the non-risky asset during the maturity of the VCTL-option. The allocation is reviewed and if necessary adjusted every 15th day. Adjustment depends entirely upon the realized historical volatility of the risky asset. Moreover, it is assumed that the risky asset is governed by a certain group of stochastic differential equations called affine jump-diffusion models. All models will be calibrated using out-of-the money European call options based on the Deutsche-Aktien-Index(DAX). The numerical implementation of the portfolio diffusions and the use of Monte Carlo methods will result in different VCTL-option prices. Thus, to price a nonstandard product and to comply with good risk management, it is advocated that the financial institution use several research models such as the SVSJ- and the Seppmodel in addition to the Black-Scholes model. Keywords: Exotic option, basket option, risk management, greeks, affine jumpdiffusions, the Black-Scholes model, the Heston model, Bates model with lognormal jumps, the Bates model with log-asymmetric double exponential jumps, the Stochastic-Volatility-Simultaneous-Jumps(SVSJ)-model, the Sepp-model.

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