Portfolio selection has always been a fundamental challenge in the field of finance and captured the attention of researchers in the financial area. Merton's portfolio problem is an optimization problem in finance and aims to maximize an investor's portfolio. This thesis studies Merton's Optimal Investment-Consumption Problem under the Jourdain--Sbai stochastic volatility model and seeks to maximize the expected discounted utility of consumption and terminal wealth. The results of our study can be split into three main parts. First, we derived the Hamilton--Jacobi--Bellman equation related to our stochastic optimal control problem.  Second, we simulated the optimal controls, which are the weight of the risky asset and consumption. This has been done for all the three models within the scope of the Jourdain--Sbai model: Quadratic Gaussian, Stein & Stein, and Scott's model. Finally, we developed the system of equations after applying the Crank-Nicolson numerical scheme when solving our HJB partial differential equation.

Empirical evidence indicates that the volatility in asset prices is not constant, but varies over time. However, many simple models for asset pricing rest on an assumption of constancy. In this thesis we analyse the zero-coupon bond price under a two-factor Vasicek model, where both the short rate and its volatility follow Ornstein-Uhlenbeck processes. Yield curves based on the two-factor model are then compared to those obtained from the standard Vasicek model with constant volatility. The simulated yield curves from the two-factor model exhibit "humps" that can be observed in the market, but which cannot be obtained from the standard model. / Det finns empiriska bevis som indikerar att volatiliteten i finansiella marknader inte är konstant, utan varierar över tiden. Dock så utgår många enkla modeller för tillgångsprisättning från ett antagande om konstans. I det här examensarbetet analyserar vi priset på nollkupongobligationer under en stokastisk Vasicekmodell, där både den korta räntan och dess volatilitet följer Ornstein-Uhlenbeck processer. De räntekurvor som tas fram genom två-faktormodellen jämförs sedan med de kurvor som erhålls genom den enkla Vasicekmodellen med konstant volatilitet. De simulerade räntekurvorna från två-faktormodellen uppvisar "pucklar" som kan urskiljas i marknaden, men som inte kan erhållas genom standardmodellen.

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

Les titres financiers sont souvent modélisés par des équations différentielles stochastiques (ÉDS). Ces équations peuvent décrire le comportement de l'actif, et aussi parfois certains paramètres du modèle. Par exemple, le modèle de Heston (1993), qui s'inscrit dans la catégorie des modèles à volatilité stochastique, décrit le comportement de l'actif et de la variance de ce dernier. Le modèle de Heston est très intéressant puisqu'il admet des formules semi-analytiques pour certains produits dérivés, ainsi qu'un certain réalisme. Cependant, la plupart des algorithmes de simulation pour ce modèle font face à quelques problèmes lorsque la condition de Feller (1951) n'est pas respectée. Dans ce mémoire, nous introduisons trois nouveaux algorithmes de simulation pour le modèle de Heston. Ces nouveaux algorithmes visent à accélérer le célèbre algorithme de Broadie et Kaya (2006); pour ce faire, nous utiliserons, entre autres, des méthodes de Monte Carlo par chaînes de Markov (MCMC) et des approximations. Dans le premier algorithme, nous modifions la seconde étape de la méthode de Broadie et Kaya afin de l'accélérer. Alors, au lieu d'utiliser la méthode de Newton du second ordre et l'approche d'inversion, nous utilisons l'algorithme de Metropolis-Hastings (voir Hastings (1970)). Le second algorithme est une amélioration du premier. Au lieu d'utiliser la vraie densité de la variance intégrée, nous utilisons l'approximation de Smith (2007). Cette amélioration diminue la dimension de l'équation caractéristique et accélère l'algorithme. Notre dernier algorithme n'est pas basé sur une méthode MCMC. Cependant, nous essayons toujours d'accélérer la seconde étape de la méthode de Broadie et Kaya (2006). Afin de réussir ceci, nous utilisons une variable aléatoire gamma dont les moments sont appariés à la vraie variable aléatoire de la variance intégrée par rapport au temps. Selon Stewart et al. (2007), il est possible d'approximer une convolution de variables aléatoires gamma (qui ressemble beaucoup à la représentation donnée par Glasserman et Kim (2008) si le pas de temps est petit) par une simple variable aléatoire gamma. / Financial stocks are often modeled by stochastic differential equations (SDEs). These equations could describe the behavior of the underlying asset as well as some of the model's parameters. For example, the Heston (1993) model, which is a stochastic volatility model, describes the behavior of the stock and the variance of the latter. The Heston model is very interesting since it has semi-closed formulas for some derivatives, and it is quite realistic. However, many simulation schemes for this model have problems when the Feller (1951) condition is violated. In this thesis, we introduce new simulation schemes to simulate price paths using the Heston model. These new algorithms are based on Broadie and Kaya's (2006) method. In order to increase the speed of the exact scheme of Broadie and Kaya, we use, among other things, Markov chains Monte Carlo (MCMC) algorithms and some well-chosen approximations. In our first algorithm, we modify the second step of the Broadie and Kaya's method in order to get faster schemes. Instead of using the second-order Newton method coupled with the inversion approach, we use a Metropolis-Hastings algorithm. The second algorithm is a small improvement of our latter scheme. Instead of using the real integrated variance over time p.d.f., we use Smith's (2007) approximation. This helps us decrease the dimension of our problem (from three to two). Our last algorithm is not based on MCMC methods. However, we still try to speed up the second step of Broadie and Kaya. In order to achieve this, we use a moment-matched gamma random variable. According to Stewart et al. (2007), it is possible to approximate a complex gamma convolution (somewhat near the representation given by Glasserman and Kim (2008) when T-t is close to zero) by a gamma distribution.

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The main aim of this thesis is in analyzing and empirically testing the various valuation models and hedging schemes of foreign exchange barrier options and their robustness with respect to changing of market conditions. The purpose of the main empirical section is to get a detailed understanding of the static and dynamic performance of the analyzed models for the barrier options payoff mainly in the extreme market conditions, where we performed a benchmarking of the various hedging schemes. As a by-product, we analyzed the accomplishment of some of the model assumptions in real world setting, and the model dependency of the barrier options.

Die vorliegende Arbeit beschäftigt sich mit verschiedenen Aspekten der Finanzmathematik. Ein erweitertes Libor Markt Modell wird betrachtet, welches genug Flexibilität bietet, um akkurat an Caplets und Swaptions zu kalibrieren. Weiterhin wird die Bewertung komplexerer Finanzderivate, zum Beispiel durch Simulation, behandelt. In hohen Dimensionen können solche Simulationen sehr zeitaufwendig sein. Es werden mögliche Verbesserungen bezüglich der Komplexität aufgezeigt, z.B. durch Faktorreduktion. Zusätzlich wird das sogenannte Andersen-Simulationsschema von einer auf mehrere Dimensionen erweitert, wobei das Konzept des „Momentmatchings“ zur Approximation des Volaprozesses in einem Heston Modell genutzt wird. Die daraus resultierende verbesserten Konvergenz des Gesamtprozesses führt zu einer verringerten Komplexität. Des Weiteren wird die Bewertung Amerikanischer Optionen als optimales Stoppproblem betrachtet. In höheren Dimensionen ist die simulationsbasierte Bewertung meist die einzig praktikable Lösung, da diese eine dimensionsunabhängige Konvergenz gewährleistet. Eine neue Methode der Varianzreduktion, die Multilevel-Idee, wird hier angewandt. Es wird eine untere Preisschranke unter zu Hilfenahme der Methode der „policy iteration“ hergeleitet. Dafür werden Konvergenzraten für die Simulation des Optionspreises erarbeitet und eine detaillierte Komplexitätsanalyse dargestellt. Abschließend wird das Preisen von Amerikanischen Optionen unter Modellunsicherheit behandelt, wodurch die Restriktion, nur ein bestimmtes Wahrscheinlichkeitsmodell zu betrachten, entfällt. Verschiedene Modelle können plausibel sein und zu verschiedenen Optionswerten führen. Dieser Ansatz führt zu einem nichtlinearen, verallgemeinerten Erwartungsfunktional. Mit Hilfe einer verallgemeinerte Snell''sche Einhüllende wird das Bellman Prinzip hergeleitet. Dadurch kann eine Lösung durch Rückwärtsrekursion erhalten werden. Ein numerischer Algorithmus liefert untere und obere Preisschranken. / The work presented here deals with several aspects of financial mathematics. An extended Libor market model is considered offering enough flexibility to accurately calibrate to various market data for caplets and swaptions. Moreover the evaluation of more complex financial derivatives is considered, for instance by simulation. In high dimension such simulations can be very time consuming. Possible improvements regarding the complexity of the simulation are shown, e.g. factor reduction. In addition the well known Andersen simulation scheme is extended from one to multiple dimensions using the concept of moment matching for the approximation of the vola process in a Heston model. This results in an improved convergence of the whole process thus yielding a reduced complexity. Further the problem of evaluating so called American options as optimal stopping problem is considered. For an efficient evaluation of these options, particularly in high dimensions, a simulation based approach offering dimension independent convergence often happens to be the only practicable solution. A new method of variance reduction given by the multilevel idea is applied to this approach. A lower bound for the option price is obtained using “multilevel policy iteration” method. Convergence rates for the simulation of the option price are obtained and a detailed complexity analysis is presented. Finally the valuation of American options under model uncertainty is examined. This lifts the restriction of considering one particular probabilistic model only. Different models might be plausible and may lead to different option values. This approach leads to a non-linear expectation functional, calling for a generalization of the standard expectation case. A generalized Snell envelope is obtained, enabling a backward recursion via Bellman principle. A numerical algorithm to valuate American options under ambiguity provides lower and upper price bounds.

In this thesis, modelling with L&eacute / vy processes is considered in three parts. In the first part, the general geometric L&eacute / vy market model is examined in detail. As such markets are generally incomplete, it is shown that the market can be completed by enlarging with a series of new artificial assets called &ldquo / power-jump assets&rdquo / based on the power-jump processes of the underlying L&eacute / vy process. The second part of the thesis presents two different methods for pricing European options: the martingale pricing approach and the Fourier-based characteristic formula method which is performed via fast Fourier transform (FFT). Performance comparison of the pricing methods led to the fact that the fast Fourier transform produces very small pricing errors so the results of both methods are nearly identical. Throughout the pricing section jump sizes are assumed to have a particular distribution. The third part contributes to the empirical applications of L&eacute / vy processes. In this part, the stochastic volatility extension of the jump diffusion model is considered and calibration on Standard&amp / Poors (S&amp / P) 500 options data is executed for the jump-diffusion model, stochastic volatility jump-diffusion model of Bates and the Black-Scholes model. The model parameters are estimated by using an optimization algorithm. Next, the effect of additional stochastic volatility extension on explaining the implied volatility smile phenomenon is investigated and it is found that both jumps and stochastic volatility are required. Moreover, the data fitting performances of three models are compared and it is shown that stochastic volatility jump-diffusion model gives relatively better results.

本文採用 Chib， Nardari， 與 Shephard(2006) 的多變量動態因子隨機波動模型(MSV)， 來探討美、日、台三國的資訊、電腦類股股價報酬率波動的共同行為。 我們將模型中的因子解釋為產業的前景或信心，並藉由模擬的方式描繪出其樣貌，進而希望了解產業景氣循環在股價的波動行為中扮演什麼角色。 研究財務市場間的關聯性一值是一項重要的課題，也發展出各種的模型來描述既有的現象。 MSV 模型將看不到的解釋變量數量化，並將變數的波動行為切割為可由因子所解釋與不能解釋的部分。 且藉由將觀察值的誤差項以及單一因子的波動行為設定為隨機波動，放寬共變數變異數矩陣為定值的假設，讓每一時點都能依時變動，在同類的模型中對資料的設定是較少的。 在實證分析中我們有幾點發現：1. 因子能夠解釋資產間的波動行為，其反映在扣除因子波動之後的自有波動，其波動水準值的降低。 2. 在股價波動劇烈期間，因子解釋能力提高。 3. 因子的解釋能力在不同的國家中差異幅度很大，日本有超過一半的波動可以為因子的波動所解釋，而因子在台灣股價的波動行為只有兩成左右的解釋能力。