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Larsson, Karl. "Pricing American Options using Simulation." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-51341.
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American options are financial contracts that allow exercise at any time until ex- piration. While the pricing of standard American option contracts has been well researched, with a few exceptions no analytical solutions exist. Valuation of more in- volved American option contracts, which include multiple underlying assets or path- dependent payoff, is still to a high degree an uncharted area. Most numerical methods work badly for such options as their time complexity scales exponentially with the number of dimensions. In this Master’s thesis we study valuation methods based on Monte Carlo sim- ulations. Monte Carlo methods don’t suffer from exponential time complexity, but have been known to be difficult to use for American option pricing due to the forward nature of simulations and the backward nature of American option valuation. The studied methods are: Parametrization of exercise rule, Random Tree, Stochastic Mesh and Regression based method with a dual approach. These methods are evaluated and compared for the standard American put option and for the American maximum call option. Where applicable the values are compared with those from deterministic reference methods. The strengths and weaknesses of each method is discussed. The Regression based method essentially reduces the problem to one of selecting suitable basis functions. This choice is empirically evaluated for the following Amer- ican option contracts; standard put, maximum call, basket call, Asian call and Asian call on a basket. The set of basis functions considered include polynomials in the underlying assets, the payoff, the price of the corresponding European contract as well as certain analytic approximation of the latter. Results from the empirical studies show that the regression based method is the best choice when pricing exotic American options. Furthermore, using available analytical approximations for the corresponding European option values as a basis function seems to improve the performance of the method in most cases.
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Davenport, John D. "Analysis of American options." Connect to online resource, 2007. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3284479.
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Firth, Neil Powell. "High dimensional American options." Thesis, University of Oxford, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.427867.
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Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes–Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options.
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Richards, Darren Glyn. "Pricing American exotic options." Thesis, University of Cambridge, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.624594.
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Alobaidi, Ghada. "American options and their strategies." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/NQ58393.pdf.
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Duvel, Heimo. "Pricing methods for American options." Master's thesis, University of Cape Town, 2003. http://hdl.handle.net/11427/6903.
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Bibliography: leaves 89-94.This thesis is about the comparison of Pricing models for the valuation of American Options. Three classes of numerical approaches are considered. These are Lattice Methods, Analytic Approximations and Monte Carlo Simulation. Methods will be contrasted in terms of accuracy and speed of the computed American option price. One particular method utilises regression when estimating the American option price. For this approach the impact of outliers and multicollinearity is examined and alternative regression models fitted. Monte Carlo Simulation is implemented to calculate early exercise probabilities of American options in the South African market. Results are compared for both call and put options. A test set of 3550 options is simulated with parameters mirroring the South African economy. On this set, the accuracy of all methods is assessed relative to a benchmark price, which is computed by a convergent lattice approach. Finally, American Symmetry is used to evaluate both put and call options.
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Aguilar, Erick Trevino. "American options in incomplete markets." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2008. http://dx.doi.org/10.18452/15820.
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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.
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Lee, Jongwoo. "Analytic approximations for the valuation of American options : extensions and application to real American exotic options." Thesis, University of Manchester, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.629934.
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Determining the early exercise premiums for American options has provided a challenging task for option pricing problems. In the absence of an explicit expression for the optimal exercise bOWldary, solutions have tended to be computationally expensive numerical methods. In response, a variety of analytic approximations have been suggested. There is considerable interest in an analytic model, which retains a simple valuation framework with time-efficiency while maintaining its accuracy. In this study, based on the Geske and Johnson compoWld option approach, we propose an alternative analytic approximation for the American option by introducing a confined exponential extrapolation scheme. Establishing a specific restriction on the upper boWld between the American put price and the number of exercise points provides highly accurate estimations for short and even long-term options with computational efficiency. Numerical results show that our approximations clearly overcome the deficiencies of the existing two-point extrapolation schemes. Valuing American options has important implications for real options, since many investment opportWlities often have a finite-time horizon, which can be phrased in terms of American options. We extend the American option model to the American exotic options, which incorporate multiple-price operating options and sequential strategic options. As an application, we consider a hypothetical project, where the value of the option to invest and the value of the project are explicitly presented using real American exotic options. We further derive a closed-form approximation for the two-factor American option by incorporating the stochastic behaviour of the convenience yield as a mean-reverting process, which can be applied to commodityrelated real options as well as to American commodity futures options. There are limitations in deriving and applying analytical solutions for the valuation of real American exotic options associated with complex real investment situations. However, our proposed analytic approximations give a useful decision framework and the valuation tool without losing accuracy, while reducing the complexity of the valuation problem where no exact closed form solutions are known
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Han, Jun. "Pricing Some American Multi-Asset Options." Thesis, Uppsala University, Department of Mathematics, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-119966.
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Glover, Elistan Nicholas. "Analytic pricing of American put options." Thesis, Rhodes University, 2009. http://hdl.handle.net/10962/d1002804.
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American options are the most commonly traded financial derivatives in the market. Pricing these options fairly, so as to avoid arbitrage, is of paramount importance. Closed form solutions for American put options cannot be utilised in practice and so numerical techniques are employed. This thesis looks at the work done by other researchers to find an analytic solution to the American put option pricing problem and suggests a practical method, that uses Monte Carlo simulation, to approximate the American put option price. The theory behind option pricing is first discussed using a discrete model. Once the concepts of arbitrage-free pricing and hedging have been dealt with, this model is extended to a continuous-time setting. Martingale theory is introduced to put the option pricing theory in a more formal framework. The construction of a hedging portfolio is discussed in detail and it is shown how financial derivatives are priced according to a unique riskneutral probability measure. Black-Scholes model is discussed and utilised to find closed form solutions to European style options. American options are discussed in detail and it is shown that under certain conditions, American style options can be solved according to closed form solutions. Various numerical techniques are presented to approximate the true American put option price. Chief among these methods is the Richardson extrapolation on a sequence of Bermudan options method that was developed by Geske and Johnson. This model is extended to a Repeated-Richardson extrapolation technique. Finally, a Monte Carlo simulation is used to approximate Bermudan put options. These values are then extrapolated to approximate the price of an American put option. The use of extrapolation techniques was hampered by the presence of non-uniform convergence of the Bermudan put option sequence. When convergence was uniform, the approximations were accurate up to a few cents difference.
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Plavsic, Milos. "Pricing American options : aspects of computation." Thesis, Imperial College London, 2012. http://hdl.handle.net/10044/1/9245.
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An American option is a type of option that can be exercised at any time up to its expiration. American options are generally hard to value, as there is no closed-form solution for the price of an American option. When there are multiple stochastic factors in the equation, the usual solution methods – binomial trees and finite difference approaches – become infeasible. Therefore, only estimators based on Monte Carlo simulation can provide good quality results. The Least-square Monte Carlo method (LSM) is the most widely used Monte Carlo-based algorithm in the financial industry. In this thesis, the LSM algorithm and associated literature are reviewed and analysed. The first major contribution is the identification of the basic powers polynomial of 4th order as the most efficient basis polynomial for the least-squares regression within the LSM simulation. The conclusion is also drawn that the performance of LSM depends on both the number of time-steps and the number of simulated paths. Another significant finding in this thesis is that, for every option being valued with a predetermined number of paths, an 'optimal' number of time-steps exists for which the estimator's mean is closest to the exact value of the option. It is proved that, in the case of the LSM algorithm, the general belief that Monte Carlo simulations become more and more efficient with the increase in the number of iterations within the simulation does not necessarily hold. The proposed Average of Batch of LSM Estimates (ABO-LSME) approach calculates the average of multiple optimal LSM estimates within the same or less time than needed for the original LSM estimate and, surprisingly, yields more precise results than the original LSM approach. The basis of the newly introduced Bundled LSM (BLSM) algorithm is an LSM algorithm in which all of the in-the-money paths at each time-step are sorted (similar to Tilley's bundling algorithm, except only in-the-money paths are sorted) and divided into a predetermined number of bundles, to which separate least-squares regressions are applied. This method provides much more stable and precise results than the original LSM algorithm. When optimal BLSM is compared to the optimal LSM algorithm, the superiority of the BLSM estimator becomes clear. BLSM provides results with lower relative errors and RMSEs, around two times faster than optimal LSM.
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Calvo, Diego R., and Michail Musatov. "Pricing American Style Asian OptionsUsing Dynamic Programming." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-9880.
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The objective of this study is to implement a Java applet for calculating Bermudan/American-Asian call option prices and to obtain their respective optimal exercise strategies. Additionally, the study presents a computational time analysis and the effect of the variables on the option price.
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Hooks, Elizabeth R. "Kurdish nationalism : American interests and policy options /." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 1996. http://handle.dtic.mil/100.2/ADA327350.
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Thesis (M.A. in National Security Affairs) Naval Postgraduate School, December 1996.Thesis advisor(s): Daniel Moran. "December 1996." Includes bibliographical references (p. 115). Also available online.
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Jia, Quiyi. "Pricing American options using Monte Carlo methods." Thesis, Uppsala University, Department of Mathematics, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-119854.
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Linell, Mattias. "Pricing American Put Options using Numerical Methods." Thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120038.
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FROTA, ALEXANDRE ELISIO FARIAS. "VALUATION OF ORDINARY AND COMPLEX AMERICAN OPTIONS." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2003. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=4330@1.
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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOA maioria das opções negociadas atualmente é do estilo americano, no entanto sua avaliação continua sendo uma tarefa bastante difícil, constituindo-se numa das áreas mais desafiadoras no campo de derivativos financeiros, particularmente quando existem vários fatores afetando o preço da opção. Isso ocorre basicamente porque os métodos de árvores binomiais e diferenças finitas tornam-se impraticáveis na avaliação de opções com mais de três fatores de incerteza. No presente trabalho, faz-se um estudo prévio dos modelos de precificação tradicionais, para posteriormente nos estendermos a modelos mais flexíveis desenvolvidos recentemente baseados em simulações de Monte Carlo e Quase-Monte Carlo, até então considerados inaplicáveis na avaliação de opções americanas. Nesse sentido, pretendemos comprovar a aplicabilidade e versatilidade dos modelos baseados em simulação na avaliação de opções americanas tradicionais ou complexas. Nossa análise baseia-se, sobretudo na ilustração de exemplos práticos, dando especial ênfase à implementação computacional e precisão dos modelos.
The majority of the options negotiated nowadays are of the american style, however its valuation goes on being a very hard job, constituting themselves in one of the most challenging areas in the financial derivative field, particularly when there are several factors affecting the price of the option. It happens basically because the binominal trees and finite differences methods become impracticable in the valuation of options with more than three factors of uncertainty. In this work we are doing a previous study of the traditional methods of american option valuation for later extending this study to more flexible and newly developed models based on simulations of Monte Carlo and Quase-Monte Carlo, which up to the present have been considered inapplicable in the valuation of the american style options. In this sense we intend to prove the applicability and versatility of the models based on simulation in the valuation of traditional and complex american options. Our analysis is, above all based on the illustration of practical examples giving special emphasis to the computational implementation and accuracy of the methods.
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Coelho, Afonso Valente Ricardo de Seabra. "American options and the Black-Scholes Model." Master's thesis, Instituto Superior de Economia e Gestão, 2020. http://hdl.handle.net/10400.5/20735.
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Mestrado em Mathematical FinanceOs problemas de apreçamento de opções têm sido um dos principais assuntos de em Matemática Financeira, desde a criação desse conceito nos anos 70. Mais especificamente, as opções americanas são de grande interesse nesta área do conhecimento porque são matematicamente muito mais complexas do que as opções europeias padrão e o modelo de Black-Scholes não fornece, na maioria dos casos, uma fórmula explícita para a determinação do preço deste tipo de opções. Nesta dissertação, mostramos como o estudo de opções americanas conduz à análise de problemas de fronteira livre devido à possibilidade de exercício antecipado, onde nosso principal objetivo é encontrar o preço de exercício ótimo. Também apresentamos a reformulação do problema em termos de um problema de complementaridade linear e de desigualdade variacional parabólica. Além disso, também abordamos a caracterização probabilística das opções americanas com base no conceito de tempos de paragem ótima. Essas formulações, aqui tratadas em termos analíticos ou probabilísticos, podem ser muito úteis na aplicação de métodos numéricos ao problema de precificação de opções do estilo americano, uma vez que, na maioria dos casos, é quase impossível encontrar soluções explícitas. Além disso, utilizamos o Método da Árvore Binomial, que é um método numérico muito simples do ponto de vista matemático, para ilustrar alguns aspectos da teoria estudada ao longo desta tese e para comparar as opções americanas com as opções europeias e bermudas, por meio de alguns exemplos numéricos.
Option pricing problems have been one of the main focuses in the field of Mathematical Finance since the creation of this concept in the 1970s. More specifically, American options are of great interest in this area of knowledge because they are much more complex mathematically than the standard European options and the Black-Scholes model cannot give an explicit formula to value this style options in most cases. In this dissertation, we show how pricing American options leads to free boundary problems because of the possibility of early exercise, where our main goal is to find the optimal exercise price. We also present how to reformulate the problem into a linear complementarity problem and a parabolic variational inequality. Moreover, we also address the probabilistic characterization of American options based on the concept of stopping times. These formulations, here viewed from the analytical and probabilistic point of view, can be very useful for applying numerical methods to the problem of pricing American style options since, in most cases, it is almost impossible to find explicit solutions. Furthermore, we use the Binomial Tree Method, which is a very simple numerical method from the mathematical point of view, to illustrate some aspects of the theory studied throughout this thesis and to compare American options with European and Bermudan Options, by means of a few numerical examples.
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Chirayukool, Pokpong. "The valuation of exotic barrier options and American options using Monte Carlo simulation." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/45027/.
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Monte Carlo simulation is a widely used numerical method for valuing financial derivatives. It can be used to value high-dimensional options or complex path-dependent options. Part one of the thesis is concerned with the valuation of barrier options with complex time-varying barriers. In Part one, a novel simulation method, the contour bridge method, is proposed to value exotic time-varying barrier options. The new method is applied to value several exotic barrier options, including those with quadratic and trigonometric barriers. Part two of this thesis is concerned with the valuation of American options using the Monte Carlo simulation method. Since the Monte Carlo simulation can be computationally expensive, variance reduction methods must be used in order to implement Monte Carlo simulation efficiently. Chapter 5 proposes a new control variate method, based on the use of Bermudan put options, to value standard American options. It is shown that this new control variate method achieves significant gains over previous methods. Chapter 6 focuses on the extension and the generalisation of the standard regression method for valuing American options. The proposed method, the sequential contour Monte Carlo (SCMC) method, is based on hitting time simulation to a fixed set of contours. The SCMC method values American put options without bias and achieves marginal gains over the standard method. Lastly, in Part three, the SCMC method is combined with the contour bridge method to value American knock-in options with a linear barrier. The method can value American barrier options very well and efficiency gains are observed.
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Djindja, Domingos. "Valuation of American put options with exercise restrictions." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-224246.
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Roux, Alet. "European and American options under proportional transaction costs." Thesis, University of York, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.434154.
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EBRAICO, PAULA RUBEA BRETANHA MENDONCA. "AMERICAN GEOPOLITICAL OPTIONS: THE CASE OF PERSIAN GULF." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2005. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8064@1.
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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOO Golfo Pérsico é responsável por aproximadamente trinta por cento da produção mundial de petróleo e detém mais da metade das reservas petrolíferas mundiais. A concentração geográfica do principal recurso enérgico, que alimenta o atual padrão tecnológico mundial, eleva essa região a um ponto de passagem obrigatório nas opções de geopolítica de todos os países do Sistema Internacional. O Golfo Pérsico é uma região de grande instabilidade política, e em menos de trinta anos, enfrentou três guerras internacionais: nos anos oitenta a Guerra Irã- Iraque, nos anos noventa a Guerra do Golfo e, mais recentemente a Invasão Americana ao Iraque. Tais conflitos foram marcados pelo uso, ou pela ameaça de uso, de armas de destruição em massa, e pelas perdas de um contingente imenso das populações dos países em conflito. Esta dissertação analisa a participação americana nestes três conflitos, tomando como referenciais conceitos de geopolítica, uma vez que a especificidade da região exige a retomada dessa disciplina que anda esquecida nas análises internacionais. A geopolítica procura enfatizar o impacto da geografia sobre a política; desta forma, a presença do petróleo no território do Golfo Pérsico, entendido como o Coração Energético Mundial, vai influir decisivamente nas suas relações com os outros Estados do Sistema Internacional. Este estudo analisa as opções de geopolítica dos EUA para a região durante os três conflitos, uma vez que assegurar o acesso às fontes de suprimento energético do Golfo Pérsico é um interesse nacional vital americano.
The Persian Gulf produces about thirty per cent of the world's oil, while holding more than a half of the world's crude oil reserves. The geographical concentration of the most important energy resource that holds the world's contemporary technological standard, puts this region in a very important place for the geopolitical options for all countries in the International System. However, the Persian Gulf is a political unstable region in the world, in less than thirty years was involved in three international wars: in the eighties The Iran-Iraq War, in the nineties The Gulf War and recently The American Invasion of Iraq. These conflicts were known by the use or by the threat of use weapons of mass destruction (WMD), and by the heavy casualties in the countries involved in the war. This dissertation analyses the American participation in these three conflicts taking as referential geopolitical concepts, once the specificity of the region demands the rebirth of this discipline that was so often forgotten in the international analyses. The geopolitics emphasize the geographical impact over politics, so the oil reserves in the territory of the Persian Gulf, the energy heartland, will influence the relationship with the others States in the International System. This study examines the American geopolitical options for the region, once a secure access to Persian Gulf is America's national vital interest.
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Höggerl, Christoph. "Model-independent arbitrage bounds on American put options." Thesis, University of Bath, 2015. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.665399.
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The standard approach to pricing financial derivatives is to determine the discounted, risk-neutral expected payoff under a model. This model-based approach leaves us prone to model risk, as no model can fully capture the complex behaviour of asset prices in the real world. Alternatively, we could use the prices of some liquidly traded options to deduce no-arbitrage conditions on the contingent claim in question. Since the reference prices are taken from the market, we are not required to postulate a model and thus the conditions found have to hold under any model. In this thesis we are interested in the pricing of American put options using the latter approach. To this end, we will assume that European options on the same underlying and with the same maturity are liquidly traded in the market. We can then use the market information incorporated into these prices to derive a set of no-arbitrage conditions that are valid under any model. Furthermore, we will show that in a market trading only finitely many American and co-terminal European options it is always possible to decide whether the prices are consistent with a model or there has to exist arbitrage in the market.
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Jankelow, Adam. "Pricing American/Bermudan-style Options under Stochastic Volatility." Master's thesis, Faculty of Commerce, 2021. http://hdl.handle.net/11427/32755.
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A method to price American options under a stochastic volatility framework is introduced which is based on Rambharat and Brockwell (2010). We price American options under the Heston and Bates stochastic volatility models where volatility is assumed to be a latent process. The pricing algorithm is based on the least-squares Monte Carlo approach made popular by Longstaff and Schwartz (2001). Information about the volatility of the underlying asset is used to assist in solving the pricing problem. Since volatility is assumed to be a latent, a particle filter is used to estimate the filtering distribution of volatility. A summary vector is constructed which captures the essential features of the filtering distribution. At each time step before maturity, the elements of the summary vector and the current share price are used as explanatory variables in a regression function which estimates the continuation value of the option. Estimating the continuation value assists in finding the optimal time to exercise the option. This pricing approach is benchmarked against a method which assumes volatility is observable. Furthermore, our pricing approach is compared to simpler methods which do not use particle filtering. Results from our numerical experiments suggest the proposed approach produces accurate option prices.
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Zhang, Jin. "Some innovative numerical approaches for pricing American options." Access electronically, 2007. http://www.library.uow.edu.au/adt-NWU/public/adt-NWU20080915.125545/index.html.
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Arotiba, Gbenga Joseph. "Pricing American Style Employee Stock Options having GARCH Effects." Thesis, University of the Western Cape, 2010. http://etd.uwc.ac.za/index.php?module=etd&action=viewtitle&id=gen8Srv25Nme4_3057_1298615964.
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We investigate some simulation-based approaches for the valuing of the employee stock options. The mathematical models that deal with valuation of such options include the work of Jennergren and Naeslund [L.P Jennergren and B. Naeslund, A comment on valuation of executive stock options and the FASB proposal, Accounting Review 68 (1993) 179-183]. They used the Black and Scholes [F. Black and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81(1973) 637-659] and extended partial differential equation for an option that includes the early exercise. Some other major relevant works to this mini thesis are Hemmer et al. [T Hemmer, S. Matsunaga and T Shevlin, The influence of risk diversification on the early exercise of employee stock options by executive officers, Journal of Accounting and Economics 21(1) (1996) 45-68] and Baril et al. [C. Baril, L. Betancourt, J. Briggs, Valuing employee stock options under SFAS 123 R using the Black-Scholes-Merton and lattice model approaches, Journal of Accounting Education 25 (1-2) (2007) 88-101]. The underlying assets are studied under the GARCH (generalized autoregressive conditional heteroskedasticity) effects. Particular emphasis is made on the American style employee stock options.
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Karlsson, Jesper. "Pricing of European- and American-style Asian Options using the Finite Element Method." Thesis, Umeå universitet, Institutionen för fysik, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-150290.
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An option is a contract between two parties where the holder has the option to buy or sell some underlying asset after a predefined exercise time. Options where the holder only has the right to buy or sell at the exercise time is said to be of European-style, while options that can be exercised any time before the exercise time is said to be of American-style. Asian options are options where the payoff is determined by some average value of the underlying asset, e.g., the arithmetic or the geometric average. For arithmetic Asian options, there are no closed-form pricing formulas, and one must apply numerical methods. Several methods have been proposed and tested for Asian options. For example, the Monte Carlo method isslowforEuropean-styleAsianoptionsandnotapplicableforAmerican-styleAsian options. In contrast, the finite difference method have successfully been applied to price both European- and American-style Asian options. But from a financial point of view, one is also interested in different measures of sensitivity, called the Greeks, which are hard approximate with the finite difference method. For more accurate approximations of the Greeks, researchers have turned to the finite element method with promising results for European-style Asian options. However, the finite element method has never been applied to American-style Asian options, which still lack accurate approximations of the Greeks. Here we present a study of pricing European- and American-style Asian options using the finite element method. For European-style options, we consider two different pricing PDEs. The first equation we consider is a convection-dominated problem, which we solve by applying the so-called streamline-diffusion method. The second equation comes from modelling Asian options as options on a traded account, which we solve by using the so-called cG(1)cG(1) method. For American-style options, the model based on options on a traded account is not applicable. Therefore, we must consider the first convection-dominated problem. To handle American-style options, we study two different methods, a penalty method and the projected successive over-relaxation method. For European-style Asian options, both approaches give good results, but the model based on options on a traded account show more accurate results. For American-style Asian options, the penalty method give accurate results. Meanwhile, the projected successive over-relaxation method does not converge properly for the tested parameters. Our result is a first step towards an accurate and fast method to calculate the price and the Greeks of both European- and American-style Asian options. Because good estimations of the Greeks are crucial when hedging and trading of options, we anticipate that the ideas presented in this work can lead to new ways of trading with Asian options.
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Fagerlund, Fredrik. "Pricing and Hedging American Options Using Monte Carlo Simulation." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-51320.
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This thesis is devoted to pricing and hedging of American style op- tions by the use of Monte Carlo simulation. We describe and implement numerous methods, developed for pricing American options by simula- tion. We show how Monte Carlo simulation can be used to achieve a plausible, and accurate, price approximation and illustrate this by nu- merical results. Both single asset, and multiple asset, contract structures have been applied to these methods. This study points out the strengths, and weaknesses, of Monte Carlo simulation when using it for pricing an American style option. Finally, we use Monte Carlo simulation to esti- mate the hedge ratios of American options and the result is illustrated in tables.
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Tomek, Michal. "A stochastic tree approach to pricing multidimensional American options." Thesis, Imperial College London, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.429210.
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Tien, Chih-Yuan. "Mixed stopping times and American options under transaction costs." Thesis, University of York, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.547377.
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Sheludchenko, Dmytro, and Daria Novoderezhkina. "Pricing American options using approximations by Kim integral equations." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-14366.
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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.
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Wolff, Patrick N. "How Tragedy Impacts American Market Returns and Options Volatility." Ohio University Honors Tutorial College / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=ouhonors1429892639.
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Chang, Chuang-Chang. "Efficient binomial methods for option valuation and hedging : the case of American currency options and warrants." Thesis, Lancaster University, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.260944.
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Tang, Yin Chiu. "Optimal entry and exit strategies of an investment project : compound American options /." View Abstract or Full-Text, 2002. http://library.ust.hk/cgi/db/thesis.pl?MATH%202002%20TANG.
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Gustafsson, William. "Evaluating the Longstaff-Schwartz method for pricing of American options." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-254406.
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Bergström, Jonas. "Pricing American Options using Lévy Processes and Monte Carlo Simulations." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-254547.
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Simkus, Darius. "Monte Carlo Pricing of American Style Options under Stochastic Volatility." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-224882.
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Lundgren, Jacob. "Pricing of American Options by Adaptive Tree Methods on GPUs." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-265257.
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An assembled algorithm for pricing American options with absolute, discrete dividends using adaptive lattice methods is described. Considerations for hardware-conscious programming on both CPU and GPU platforms are discussed, to provide a foundation for the investigation of several approaches for deploying the program onto GPU architectures. The performance results of the approaches are compared to that of a central processing unit reference implementation, and to each other. In particular, an approach of designating subtrees to be calculated in parallel by allowing multiple calculation of overlapping elements is described. Among the examined methods, this attains the best performance results in a "realistic" region of calculation parameters. A fifteen- to thirty-fold improvement in performance over the CPU reference implementation is observed as the problem size grows sufficiently large.
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Fouse, Bradley Warren. "Pricing American options with jump-diffusion by Monte Carlo simulation." Thesis, Manhattan, Kan. : Kansas State University, 2009. http://hdl.handle.net/2097/1505.
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Zhou, Zhenhao. "From valuing equity-linked death benefits to pricing American options." Diss., University of Iowa, 2017. https://ir.uiowa.edu/etd/5690.
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Motivated by the Guaranteed Minimum Death Benefits (GMDB) in variable annuities, we are interested in valuing equity-linked options whose expiry date is the time of the death of the policyholder. Because the time-until-death distribution can be approximated by linear combinations of exponential distributions or mixtures of Erlang distributions, the analysis can be reduced to the case where the time-until-death distribution is exponential or Erlang.
We present two probability methods to price American options with an exponential expiry date. Both methods give the same results. An American option with Erlang expiry date can be seen as an extension of the exponential expiry date case. We calculate its price as the sum of the price of the corresponding European option and the early exercise premium. Because the optimal exercise boundary takes the form of a staircase, the pricing formula is a triple sum. We determine the optimal exercise boundary recursively by imposing the “smooth pasting” condition. The examples of the put option, the exchange option, and the maximum option are provided to illustrate how the methods work.
Another issue related to variable annuities is the surrender behavior of the policyholders. To model this behavior, we suggest using barrier options. We generalize the reflection principle and use it to derive explicit formulas for outside barrier options, double barrier options with constant barriers, and double barrier options with time varying exponential barriers.
Finally, we provide a method to approximate the distribution of the time-until-death random variable by combinations of exponential distributions or mixtures of Erlang distributions. Compared to directly fitting the distributions, my method has two advantages: 1) It is more robust to the initial guess. 2) It is more likely to obtain the global minimizer.
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Tiplea, Ana Camelia. "Super-replication of American Options in an Uncertain Volatility Model." Thesis, The University of Sydney, 2019. http://hdl.handle.net/2123/20815.
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We study the pricing of multi-asset American derivatives in an Uncertain Volatility model for general payoffs. We apply stochastic optimal control techniques and viscosity theory to characterize the pricing function of such derivatives as unique solutions of a nonlinear parabolic PDE with obstacles. We further prove sufficient regularity of the solution and we find the optimal super-replicating strategy for the American claim.
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Laminou, Abdou Souleymane. "Optimality of the Financial Decision and the Theory of American and Exotic Options." Thesis, Rennes 1, 2016. http://www.theses.fr/2016REN1G016/document.
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Cette thèse examine les décisions financières à travers la théorie des options Américaines et Exotiques. Dans un premier temps, nous avons présenté une revue de la littérature sur les options de type Américain. La tarification de l’option Américaine standard d’achat est revisitée en vue de fournir les pré-requis. Dans l’étape suivante, un nouveau type de contrat d’option, appelé Strangle Euro-American ou Strangle Hybride, a été introduit. Des formules analytiques ont été fournies pour leurs prix ainsi que leurs paramètres de gestion. Une nouvelle méthode est proposée pour calculer les intégrales qui définissent les bornes d’exercice anticipé. Il a été démontré que cette méthode est efficiente, précise et rapide pour la tarification de tous les types de Strangle voir au delà. Puis, nous avons examiné les options Step de type Américain. Nous avons démontré que les propriétés des options d’achat "vanille" ne s’appliquent pas aux Step dans certaines situations. Les formules d’évaluation et des paramètres de gestion ont été déterminés. Et enfin, nous avons considéré l’évaluation d’une firme détenant simultanément une option d’abandon et une option d’expansion de ses activités selon des conditions du marché (favorables ou défavorables). Les seuils critiques de décision ont été obtenus. Des formules analytiques pour la valeur de la firme ont été obtenues. Des simulations illustrent le comportement de ces seuils critiques de décisions anticipéesThis thesis investigates the financial decisions through the theory of American and Exotic options. First, the literature on American-style derivatives is surveyed. The pricing of standard American call option in the early exercise premium representation is addressed in order to provide prerequisites for what follows. Second, a new variant of Strangle contracts, called Euro-American or Hybrid Strangles, is introduced and priced. Analytical formulas are provided for the prices of all these option contracts as well as their hedging parameters. A new quadrature is proposed to account for the systems of coupled integral equations that locate the early exercise boundaries. It is shown to be efficient, accurate, and fast for pricing all types of early exercisable strangles and more. Third, we examines the valuation of American Step options contract. The structures of the immediate exercise regions of the various contracts are identified. Typical properties of American vanilla calls are shown to fail in some cases. Formulas for prices and hedging parameters, for the American Step options, are derived. Finally, we consider the valuation of a firm holding simultaneously an option to expand and to abandon productions depending on the state of the market (good or bad) in a real option framework. Optimal decision levels are obtained. Analytical formulas for the firm’s value are provided. Numerical results document the behavior of the firm’s value and optimal exercise boundaries levels
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Gao, Min. "Early exercise options with discontinuous payoff." Thesis, University of Manchester, 2018. https://www.research.manchester.ac.uk/portal/en/theses/early-exercise-options-with-discontinuous-payoff(83d6dee7-dbdd-4f42-b350-48f973594feb).html.
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The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee. This includes British cash-or-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves, we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called `one-touch' binary options. Then we move on to examine the British binary options. Chapter~2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the American-type (one-touch) binary options using the risk-neutral probability method. In Chapters~3 ~4 and ~5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the `best prediction' of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters~3 and~4 focus on the British binary put options and Chapter~5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter~6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in \cite{dai2004knock} in the application to the knock-in binary options. For the American knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function.
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Saize, Stefane. "Analytical Valuation of American-Style Asian Options under Jump-Diffusion Processes." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-224885.
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Zhou, Tingwen. "Arbitrage-Free Pricing of XVA for American Options in Discrete Time." Digital WPI, 2017. https://digitalcommons.wpi.edu/etd-theses/348.
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Total valuation adjustment (XVA) is a new technique which takes multiple material financial factors into consideration when pricing derivatives. This paper explores how funding costs and counterparty credit risk affect pricing the American option based on no-arbitrage analysis. We review previous studies of European option pricing with different funding costs. The conclusions help to compute the no- arbitrage price of the American option in the model with different borrowing and lending rates. Another model with counterparty credit risk is set up, and this pricing approach is referred to as credit valuation adjustment (CVA). A defaultable bond issued by the counterparty is used to hedge the loss from the option's default. We incorporate these two models to assess the XVA of an American option. The collateral, which protects the option investors from default, is considered in our benchmark model. To illustrate our results, numerical experiments are designed to demonstrate the relationship between XVA and parameters, which include the funding rates, bond's rate of return, and number of periods.
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Zamrik, Tamim. "Pricing American exotic options under levy processes a direct resolvent approach." Thesis, Imperial College London, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.530470.
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Andersson, Niklas. "Regression-Based Monte Carlo For Pricing High-Dimensional American-Style Options." Thesis, Umeå universitet, Institutionen för fysik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-119013.
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Pricing different financial derivatives is an essential part of the financial industry. For some derivatives there exists a closed form solution, however the pricing of high-dimensional American-style derivatives is still today a challenging problem. This project focuses on the derivative called option and especially pricing of American-style basket options, i.e. options with both an early exercise feature and multiple underlying assets. In high-dimensional problems, which is definitely the case for American-style options, Monte Carlo methods is advantageous. Therefore, in this thesis, regression-based Monte Carlo has been used to determine early exercise strategies for the option. The well known Least Squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001) has been implemented and compared to Robust Regression Monte Carlo (RRM) by C.Jonen (2011). The difference between these methods is that robust regression is used instead of least square regression to calculate continuation values of American style options. Since robust regression is more stable against outliers the result using this approach is claimed by C.Jonen to give better estimations of the option price. It was hard to compare the techniques without the duality approach of Andersen and Broadie (2004) therefore this method was added. The numerical tests then indicate that the exercise strategy determined using RRM produces a higher lower bound and a tighter upper bound compared to LSM. The difference between upper and lower bound could be up to 4 times smaller using RRM. Importance sampling and Quasi Monte Carlo have also been used to reduce the variance in the estimation of the option price and to speed up the convergence rate.Prissättning av olika finansiella derivat är en viktig del av den finansiella sektorn. För vissa derivat existerar en sluten lösning, men prissättningen av derivat med hög dimensionalitet och av amerikansk stil är fortfarande ett utmanande problem. Detta projekt fokuserar på derivatet som kallas option och särskilt prissättningen av amerikanska korg optioner, dvs optioner som både kan avslutas i förtid och som bygger på flera underliggande tillgångar. För problem med hög dimensionalitet, vilket definitivt är fallet för optioner av amerikansk stil, är Monte Carlo metoder fördelaktiga. I detta examensarbete har därför regressions baserad Monte Carlo använts för att bestämma avslutningsstrategier för optionen. Den välkända minsta kvadrat Monte Carlo (LSM) algoritmen av Longstaff och Schwartz (2001) har implementerats och jämförts med Robust Regression Monte Carlo (RRM) av C.Jonen (2011). Skillnaden mellan metoderna är att robust regression används istället för minsta kvadratmetoden för att beräkna fortsättningsvärden för optioner av amerikansk stil. Eftersom robust regression är mer stabil mot avvikande värden påstår C.Jonen att denna metod ger bättre skattingar av optionspriset. Det var svårt att jämföra teknikerna utan tillvägagångssättet med dualitet av Andersen och Broadie (2004) därför lades denna metod till. De numeriska testerna indikerar då att avslutningsstrategin som bestämts med RRM producerar en högre undre gräns och en snävare övre gräns jämfört med LSM. Skillnaden mellan övre och undre gränsen kunde vara upp till 4 gånger mindre med RRM. Importance sampling och Quasi Monte Carlo har också använts för att reducera variansen i skattningen av optionspriset och för att påskynda konvergenshastigheten.
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Shah, Premal (Premal Y. ). "No-arbitrage bounds on American Put Options with a single maturity." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/36232.
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Thesis (S.M.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2006.Includes bibliographical references (p. 63-64).
We consider in this thesis the problem of pricing American Put Options in a model-free framework where we do not make any assumptions about the price dynamics of the underlying except those implied by the no-arbitrage conditions. Our goal is to obtain bounds on the price of an American put option with a given strike and maturity directly from the prices of other American put options with the same maturity but different strikes and the current price of the underlying. We proceed by first investigating the structural properties of the price curve of American Put Options of a fixed maturity and derive necessary and sufficient conditions that strike - price pairs of these options must satisfy in order to exclude arbitrage. Using these conditions, we can find tight bounds on the price of the option of interest by solving a very tractable Linear Programming Problem. We then apply the methods developed to real market data. We observe that the quality of bounds that we obtain compares well with the quoted bid-ask spreads in most cases.
by Premal Shah.
S.M.
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Lin, Chia-Yu, and 林嘉祐. "Pricing American Rainbow Options." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/33904053622079603217.
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碩士國立臺灣大學
國際企業學研究所
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This paper extends the forward Monte Carlo (FMC) method, which have been developed for the basic types of American options, to the valuation of two-asset American rainbow options. The main advantage of this method is that it does not use backward induction as required by other methods. Instead, the proposed approach relies on a wise determination about whether a pair of simulated stock prices has entered the exercise region. A series of numerical experiments are provided to compare the performance with the binomial tree model and least squares method and demonstrate the efficiency of the forward methods.
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Liao, Shi-Hau. "American options pricing and Interpolation." 2004. http://www.cetd.com.tw/ec/thesisdetail.aspx?etdun=U0001-1607200400253000.
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"CEV asymptotics of American options." 2013. http://library.cuhk.edu.hk/record=b5549268.
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常方差彈性(CEV) 模型能夠刻畫波動率微笑的優點使之成為期權定價中的實用工具，然而它在應用到美式衍生工具時面臨分析上及計算上的挑戰。現行的解析方法是對代表著期權價格函數和其最佳履約曲線的自由邊界問題進行拉普拉斯卡森變換(LCT) ，繼而獲得在此變換下的解析解，可是此解含有合流超線幾何函數，使得它的數值計算在某些參數下顯得不穩定及低效。本文運用漸近法徹底解決美式期權在常方差彈性模型下的定價問題，並用永久性和限時性的美式看跌期權作為例子闡述所提出的方法。The constant elasticity of variance (CEV) model is a practical approach to option pricing by fitting to the implied volatility skew. Its application to American-style derivatives, however, poses analytical and numerical challenges. By taking the Laplace Carson transform (LCT) to the free-boundary value problem characterizing the option value function and the early exercise boundary, the analytical result involves confluent hyper-geometric functions. Thus, the numerical computation could be unstable and inefficient for certain set of parameter values. We solve this problem by an asymptotic approach to the American option pricing problem under the CEV model. We demonstrate the use of the proposed approach using perpetual and finite-time American puts.
Detailed summary in vernacular field only.
Pun, Chi Seng.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2013.
Includes bibliographical references (leaves 39-40).
Abstracts also in Chinese.
Chapter 1 --- Introduction --- p.1
Chapter 2 --- Problem Formulation --- p.4
Chapter 2.1 --- The CEV model --- p.4
Chapter 2.2 --- The free-boundary value problem --- p.5
Chapter 2.2.1 --- Perpetual American put --- p.5
Chapter 2.2.2 --- Finite-time American put --- p.6
Chapter 3 --- Asymptotic expansion of American put --- p.8
Chapter 3.1 --- Perpetual American put --- p.8
Chapter 3.2 --- Finite-time American put --- p.16
Chapter 4 --- Numerical examples --- p.24
Chapter 4.1 --- Perpetual American put --- p.24
Chapter 4.2 --- Finite-time American put --- p.26
Chapter 5 --- Conclusion --- p.29
Chapter A --- Proof of Lemma 3.1 --- p.30
Chapter B --- Property of ak --- p.32
Chapter C --- Explicit formulas for u₂(S) --- p.34
Chapter D --- Closed-form solutions --- p.37
Bibliography --- p.40