This dissertation empirically examines refunding decisions employed by issuers of tax-exempt bonds. Callable bonds contain embedded call options by virtue of provisions in bond indentures that permit the issuing firm to buy back the bond at a predetermined strike price. Such an embedded American call option has two components to its value, the intrinsic value and the time value. The issuer can realize at least as much as the intrinsic value by exercising immediately, when the option is in-the-money. Usually it is optimal for the holder of an in-the money American option to wait rather than exercise immediately, because the option has time value. It is rational for the holder to exercise the option when the total value of the option is no more than the intrinsic value. Option pricing theory can be used to identify two sub-optimal refunding strategies: those that refund too early, and those that refund too late. In such cases the holder incurs losses. I analyze the refunding decisions for two different samples of tax-exempt bonds issued between 1986 and 1993: the first consists of 2,620 bonds that are called, and the second contains 23,976 bonds that are never called. The generalized Vasicek (1977) model in the Heath, Jarrow, and Morton (1992) framework is used to construct binomial trees for interest rates, bond prices, and call option prices. The option pricing lattice is then used to compute the loss in value from sub-optimal refunding strategies, refunding efficiency, and months from optimal time for bonds in these two samples. Results suggest that sub-optimal refunding decisions cause losses to the issuers, which are present across bond and issuer characteristics. For the pooled sample of 26,596 bonds, the loss in value from sub-optimal refunding decisions totaled $7.2 billion, amounting to a loss of about 1.75% of total principal amount. Results indicate that issuers either wait too long to refund or never refund and cannot realize the present value saving of switching a high coupon bond with a low coupon bond, over a longer period of time. These results critically depend on the assumptions of underlying term structure model and are sensitive to model calibrated parameter values. / Ph. D.

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.

In dieser Dissertation werden Amerikanischen Optionen in einem unvollst¨andigen Markt und in stetiger Zeit untersucht. Die Dissertation besteht aus zwei Teilen. Im ersten Teil untersuchen wir ein stochastisches Optimierungsproblem, in dem ein konvexes robustes Verlustfunktional ueber einer Menge von stochastichen Integralen minimiert wird. Dies Problem tritt auf, wenn der Verkaeufer einer Amerikanischen Option sein Ausfallsrisiko kontrollieren will, indem er eine Strategie der partiellen Absicherung benutzt. Hier quantifizieren wir das Ausfallsrisiko durch ein robustes Verlustfunktional, welches durch die Erweiterung der klassischen Theorie des erwarteten Nutzens durch Gilboa und Schmeidler motiviert ist. In einem allgemeinen Semimartingal-Modell beweisen wir die Existenz einer optimalen Strategie. Unter zusaetzlichen Kompaktheitsannahmen zeigen wir, wie das robuste Problem auf ein nicht-robustes Optimierungsproblem bezueglich einer unguenstigsten Wahrscheinlichkeitsverteilung reduziert werden kann. Im zweiten Teil untersuchen wir die obere und die untere Snellsche Einhuellende zu einer Amerikanischen Option. Wir konstruieren diese Einhuellenden fuer eine stabile Familie von aequivalenten Wahrscheinlichkeitsmassen; die Familie der aequivalentenMartingalmassen ist dabei der zentrale Spezialfall. Wir formulieren dann zwei Probleme des robusten optimalen Stoppens. Das Stopp-Problem fuer die obere Snellsche Einhuellende ist durch die Kontrolle des Risikos motiviert, welches sich aus der Wahl einer Ausuebungszeit durch den Kaeufer bezieht, wobei das Risiko durch ein kohaerentes Risikomass bemessen wird. Das Stopp-Problem fuer die untere Snellsche Einhuellende wird durch eine auf Gilboa und Schmeidler zurueckgehende robuste Erweiterung der klassischen Nutzentheorie motiviert. Mithilfe von Martingalmethoden zeigen wir, wie sich optimale Loesungen in stetiger Zeit und fuer einen endlichen Horizont konstruieren lassen. / This thesis studies American options in an incomplete financial market and in continuous time. It is composed of two parts. In the first part we study a stochastic optimization problem in which a robust convex loss functional is minimized in a space of stochastic integrals. This problem arises when the seller of an American option aims to control the shortfall risk by using a partial hedge. We quantify the shortfall risk through a robust loss functional motivated by an extension of classical expected utility theory due to Gilboa and Schmeidler. In a general semimartingale model we prove the existence of an optimal strategy. Under additional compactness assumptions we show how the robust problem can be reduced to a non-robust optimization problem with respect to a worst-case probability measure. In the second part, we study the notions of the upper and the lower Snell envelope associated to an American option. We construct the envelopes for stable families of equivalent probability measures, the family of local martingale measures being an important special case. We then formulate two robust optimal stopping problems. The stopping problem related to the upper Snell envelope is motivated by the problem of monitoring the risk associated to the buyer’s choice of an exercise time, where the risk is specified by a coherent risk measure. The stopping problem related to the lower Snell envelope is motivated by a robust extension of classical expected utility theory due to Gilboa and Schmeidler. Using martingale methods we show how to construct optimal solutions in continuous time and for a finite horizon.

[pt] Foi implementado um algoritmo capaz de avaliar opções americanas de compra usando a técnica de Simulação de Monte Carlo aliada à Programação Dinâmica. O objetivo é avaliar opções de expansão na produção de campos de Petróleo já desenvolvidos sob condições de incerteza técnica e de mercado, tipicamente presentes em projetos de exploração e produção dessa commodity. Para modelar os movimentos do preço do Petróleo foram utilizados os modelos de Reversão à Média de Schwartz e de Dias. Explorando o algoritmo, foram estudados os efeitos da taxa de juro livre de risco, do preço de exercício da opção, da velocidade de reversão à média, da depleção secundária do campo, da volatilidade e do preço de longo prazo do Petróleo no valor da opção real. Além disso foram obtidas curvas de gatilho interpretadas como ferramentas gerenciais capazes de fornecer uma regra de decisão sobre quando investir. Os resultados foram comparados com o esperado pela teoria tradicional de opções, a qual utiliza o movimento Geométrico Browniano para modelar os movimentos do preço do ativo objeto. / [en] A program in C++ was used to evaluate American calls using both the approaches of Monte Carlo Simulation and Dynamic Programming. Its main purpose is to evaluate expanding options in a developed oil field. The investment is considered under both economic and technical uncertainties, what is common place in petroleum exploration and production problems. In order to model the commodity price, the Dias and Schwartzs mean reversion processes were used. The influences of the risk free rate, the exercise price, the mean reversion speed, the field depletion, the volatility and the oil long term price in the value of the real option were studied. Besides that, optimal exercise frontiers were obtained as a rule for the early exercise. The results obtained in this dissertation were compared to the results expected by the financial option theory, which models stocks prices using the Geometric Brownian Motion.

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.