Doctor of Philosophy (PhD) / For many practitioners and market participants, the valuation of financial derivatives is considered of very high importance as its uses range from a risk management tool, to a speculative investment strategy or capital enhancement. A developing market requires efficient but accurate methods for valuing financial derivatives such as American options. A closed form analytical solution for American options has been very difficult to obtain due to the different boundary conditions imposed on the valuation problem. Following the method of solving the American option as a free boundary problem in the spirit of the "no-arbitrage" pricing framework of Black-Scholes, the option price and hedging parameters can be represented as an integral equation consisting of the European option value and an early exercise value dependent upon the optimal free boundary. Such methods exist in the literature and along with risk-neutral pricing methods have been implemented in practice. Yet existing methods are accurate but inefficient, or accuracy has been compensated for computational speed. A new numerical approach to the valuation of American options by cubic splines is proposed which is proven to be accurate and efficient when compared to existing option pricing methods. Further comparison is made to the behaviour of the American option's early exercise boundary with other pricing models.

In this study, we use the McKean's integral equation to evaluate the American option price for constant jump di

The purpose of this thesis is to look into the difficulty of valuing American options, put as well as call, on an asset that pays continuous dividends. The authors are willing to demonstrate how mentioned above securities can be priced using a simple approximation of the Kim integral equations by quadrature formulas. This approach is compared with closed form American Option price formula proposed by Bjerksund-Stenslands in 2002. The results obtained by Bjerksund-Stenslands method are numerically compared by authors to the Kim’s. In Joon Kim’s approximation seems to be more accurate and closer to the chosen “true” value of an American option, however, Bjerksund-Stenslands model is demonstrating a higher speed in calculations.

Doctor of Philosophy (PhD) / For many practitioners and market participants, the valuation of financial derivatives is considered of very high importance as its uses range from a risk management tool, to a speculative investment strategy or capital enhancement. A developing market requires efficient but accurate methods for valuing financial derivatives such as American options. A closed form analytical solution for American options has been very difficult to obtain due to the different boundary conditions imposed on the valuation problem. Following the method of solving the American option as a free boundary problem in the spirit of the "no-arbitrage" pricing framework of Black-Scholes, the option price and hedging parameters can be represented as an integral equation consisting of the European option value and an early exercise value dependent upon the optimal free boundary. Such methods exist in the literature and along with risk-neutral pricing methods have been implemented in practice. Yet existing methods are accurate but inefficient, or accuracy has been compensated for computational speed. A new numerical approach to the valuation of American options by cubic splines is proposed which is proven to be accurate and efficient when compared to existing option pricing methods. Further comparison is made to the behaviour of the American option's early exercise boundary with other pricing models.

Pricing American options has always been problematic due to its early exercise characteristic. As no closed-form analytical solution for any of the widely used models exists, many numerical approximation methods have been proposed and studied. In this thesis, we investigate the Least-Square Monte Carlo Simulation (LSMC) method of Longstaff & Schwartz for pricing American options under the two-dimensional Heston model. By conducting extensive numerical experimentation, we put the LSMC to test and investigate four different continuation functions for the LSMC. In addition, we consider investigating seven different combination of Heston model parameters. We analyse the results and select the optimal continuation function according to our criteria. Then we uncover and study the early exercise boundary foran American put option upon changing initial volatility and other parameters of the Heston model.

Dans cette thèse, nous traitons deux problèmes de contrôle optimal stochastique. Chaque problème correspond à une Partie de ce document. Le premier problème traité est très précis, il s'agit de la valorisation des contrats optionnels de vente de type Américain (dit Put Américain) en présence de dividendes discrets (Partie I). Le deuxième est plus général, puisqu'il s'agit dans un cadre discret en temps de prouver l'existence d'un principe de programmation dynamique sous des contraintes en probabilités (Partie II). Bien que les deux problèmes soient assez distincts, le principe de programmation dynamique est au coeur de ces deux problèmes. La relation entre la valorisation d'un Put Américain et un problème de frontière libre a été prouvée par McKean. La frontière de ce problème a une signification économique claire puisqu'elle correspond à tout instant à la borne supérieure de l'ensemble des prix d'actifs pour lesquels il est préférable d'exercer tout de suite son droit de vente. La forme de cette frontière en présence de dividendes discrets n'avait pas été résolue à notre connaissance. Sous l'hypothèse que le dividende est une fonction déterministe du prix de l'actif à l'instant précédant son versement, nous étudions donc comment la frontière est modifiée. Au voisinage des dates de dividende, et dans le modèle du Chapitre 3, nous savons qualifier la monotonie de la frontière, et dans certains cas quantifier son comportement local. Dans le Chapitre 3, nous montrons que la propriété du smooth-fit est satisfaite à toute date sauf celles de versement des dividendes. Dans les deux Chapitres 3 et 4, nous donnons des conditions pour garantir la continuité de cette frontière en dehors des dates de dividende. La Partie II est originellement motivée par la gestion optimale de la production d'une centrale hydro-electrique avec une contrainte en probabilité sur le niveau d'eau du barrage à certaines dates. En utilisant les travaux de Balder sur la relaxation de Young des problèmes de commande optimale, nous nous intéressons plus spécifiquement à leur résolution par programmation dynamique. Dans le Chapitre 5, nous étendons au cadre des mesures de Young des résultats dûs à Evstigneev. Nous établissons alors qu'il est possible de résoudre par programmation dynamique certains problèmes avec des contraintes en espérances conditionnelles. Grâce aux travaux de Bouchard, Elie, Soner et Touzi sur les problèmes de cible stochastique avec perte contrôlée, nous montrons dans le Chapitre 6 qu'un problème avec contrainte en espérance peut se ramener à un problème avec des contraintes en espérances conditionnelles. Comme cas particulier, nous prouvons ainsi que le problème initial de la gestion du barrage peut se résoudre par programmation dynamique / In this thesis, we address two problems of stochastic optimal control. Each problem constitutes a different Part in this document. The first problem addressed is very precise, it is the valuation of American contingent claims and more specifically the American Put in the presence of discrete dividends (Part I). The second one is more general, since it is the proof of the existence of a dynamic programming principle under expectation constraints in a discrete time framework (Part II). Although the two problems are quite distinct, the dynamic programming principle is at the heart of these two problems. The relationship between the value of an American Put and a free boundary problem has been proved by McKean. The boundary of this problem has a clear economic meaning since it corresponds at all times to the upper limit of the asset price above which the holder of such an option would exercise immediately his right to sell. The shape of the boundary in the presence of discrete dividends has not been solved to the best of our knowledge. Under the assumption that the dividend is a deterministic function of asset prices at the date just before the dividend payment, we investigate how the boundary is modified. In the neighborhood of dividend dates and in the model of Chapter 3, we know what the monotonicity of the border is, and we quantify its local behavior. In Chapter 3, we show that the smooth-fit property is satisfied at any date except for those of the payment of dividends. In both Chapters 3 and 4, we are able to give conditions to guarantee the continuity of the border outside dates of dividend. Part II was originally motivated by the optimal management of the production of an hydro-electric power plant with a probability constraint on the reservoir level on certain dates. Using Balder'sworks on Young's relaxation of optimal control problems, we focus more specifically on their resolution by dynamic programming. In Chapter 5, we extend results of Evstigneev to the framework of Young measures. We show that dynamic programming can be used to solve some problems with conditional expectations constraints. Through the ideas of Bouchard, Elie, Soner and Touzi on stochastic target problems with controlled loss, we show in Chapter 6 that a problem with expectation constraints can be reduced to a problem with conditional expectation constraints. Finally, as a special case, we show that the initial problem of dam management can be solved by dynamic programming

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

In this thesis, important theories in financial mathematics will be explained and derived. These theories will later be used to value financial derivatives. An analytical formula for valuing European call and put option will be derived and European call options will be valued under the Black-Scholes partial differential equation using three different finite difference methods. The Crank-Nicholson method will then be used to value American call options and solve their corresponding free boundary value problem. The optimal exercise boundary can then be plotted from the solution of the free boundary value problem. The algorithm for valuing American call options will then be further developed to solve the stock loan problem. This will be achieved by exploiting a link that exists between American call options and stock loans. The Crank-Nicholson method will be used to value stock loans and their corresponding free boundary value problem. The optimal exit boundary can then be plotted from the solution of the free boundary value problem. The results that are obtained from the numerical calculations will finally be used to discuss how different parameters affect the valuation of American call options and the valuation of stock loans. In the end of the thesis, conclusions about the effect of the different parameters on the optimal prices will be presented. / I det här kandidatexamensarbetet kommer fundamentala teorier inom finansiell matematik förklaras och härledas. Dessa teorier kommer lägga grunden för värderingen av finansiella derivat i detta arbete. En analytisk formel för att värdera europeiska köp- och säljoptioner kommer att härledas. Dessutom kommer europeiska köpoptioner att värderas numeriskt med tre olika finita differensmetoder. Den finita differensmetoden Crank-Nicholson kommer sedan användas för att värdera amerikanska köpoptioner och lösa det fria gränsvärdesproblemet (free boundary value problem). Den optimala omvandlingsgränsen (Optimal Exercise Boundary) kan därefter härledas från det fria gränsvärdesproblemet. Algoritmen för att värdera amerikanska köpoptioner utökas därefter till att värdera lån med aktier som säkerhet. Detta kan åstadkommas genom att utnyttja ett samband mellan amerikanska köpoptioner med lån där aktier används som säkerhet. Den finita differensmetoden Crank-Nicholson kommer dessutom att användas för att värdera lån med aktier som säkerhet. Den optimala avyttringsgränsen (Optimal Exit Boundary) kan därefter härledas från det fria gränsvärdesproblemet. Resultaten från de numeriska beräkningarna kommer slutligen att användas för att diskutera hur olika parametrar påverkar värderingen av amerikanska köpoptioner, samt värdering av lån med aktier som säkerhet. Avslutningsvis kommer slutsatser om effekterna av dessa parametrar att presenteras.