Dissertations / Theses: 'Black Scholes'

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Cantaloni, Francesco. "Formula di Black-Scholes comportamentale." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2019.

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In questa trattazione verranno descritte alcune espressioni generali di valutazione per le opzioni che estendono i risultati classici di Black-Scholes e Merton all'ambito comportamentale. Dapprima mostreremo come estendere la nozione di agente rappresentativo al caso in cui i partecipanti al mercato siano caratterizzati da una funzione di utilità con avversione al rischio relativa costante ed eterogenea. Dopo aver presentato il teorema di esistenza dell'agente rappresentativo, deriveremo le principali implicazioni in termini di struttura a termine dei tassi di sconto e di valutazione degli strumenti derivati. In conclusione, discuteremo due esempi che illustrano le implicazioni empiriche dell'estensione comportamentale della formula di valutazione.

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del, Campo Daniel, and Fredrik Söderström. "Black & Scholes vs. Marknaden." Thesis, Södertörn University College, School of Business Studies, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:sh:diva-165.

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Chávez, Fuentes Jorge Richard. "El modelo de Black-Scholes." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96422.

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Se presenta el modelo de Black-Scholes, a través del más popular de los contratos financieros, esto es, la opción de compra europea. Se establece la fórmula de valuación martingala para reclamos contingentes en general y se muestra una aplicación de ella mediante la obtención del precio del contrato call. Al final se establece también la ecuación de Black-Scholes, que es una ecuación diferencial parcial no lineal de segundo orden, y que constituye una forma alternativa para la preciación de activos derivados.

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Pavlou, Petro. "KVA in Black Scholes Pricing." Master's thesis, Faculty of Commerce, 2019. http://hdl.handle.net/11427/30880.

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The post 2007-financial crisis era has led to renewed zeal in quantifying market incompleteness when pricing contingent claims. This quantification exercise is necessary in maintaining a stable and sustainable banking operation and thus the XVAs have emerged as the metrics for market incompleteness. This dissertation focuses solely on the capital valuation adjustment (KVA) and aims to use the definition of KVA as set out by Albanese et al. (2016) in an investigation of different numerical techniques for calculating KVA. A single equity forward is considered first, followed by an equity option and then portfolios of options on two underlying assets, with the dissertation ending by considering a practical example on discrete delta and vega-delta hedging an index option. The numerical approaches explored are the binomial tree method and a combination of the crude and quasi-Monte Carlo method.

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Lindström, Linnea. "Black-Scholes : En prissättningsmodell för optioner." Thesis, Umeå University, Department of Mathematics and Mathematical Statistics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-35084.

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This paper aims to derive the Black-Scholes equation for readers without advanced knowledge in finance and mathematics. To succeed, this paper contains a theoretical chapter in which concepts such as options, interest rate, differential equations and stochastic variable are explained. This paper also presents the theory of stochastic processes such as the Wiener process and Ito process. In the chapter on the Black-Scholes model the Ito process is used to describe price of shares and with the help of Ito's lemma Black-Scholes equation can be derived. In the paper, assumptions are listed that apply to the Black-Scholes model and then uses the Black-Scholes equation to calculate the price of a European call option. Finally, exotic options are described and also how options can be used to reduce risks.

Uppsatsens mål är att härleda Black-Scholes ekvation för läsare utan avancerade kunskaper inom finansiering och matematik. För att lyckas med detta innehåller uppsatsen ett teorikapitel där begrepp så som optioner, ränta, differentialekvation och stokastisk variabel förklaras. Där presenteras även teorier för stokastiska processer så som Wienerprocessen och Itoprocessen. I kapitlet om Black-Scholes modell används Itoprocessen för att beskriva aktiepriset och med hjälp av Itos lemma härleds Black-Scholes ekvation. Uppsatsen ställer upp antaganden som gäller för Black-Scholes modell och använder sedan Black-Scholes ekvation för att beräkna priset på en europeisk köpoption. Avslutningsvis beskrivs exotiska optioner samt hur optioner kan användas för att reducera risker.

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Karlsson, Olle. "The Black-Scholes Equation and Formula." Thesis, Uppsala universitet, Analys och tillämpad matematik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-200441.

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Durrell, Fernando. "Alternatives to the Black-Scholes model." Master's thesis, University of Cape Town, 2001. http://hdl.handle.net/11427/4881.

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Bibliography: leaves 44-45.
In this paper, I consider alternative models to the one posited by Black and Scholes. I consider discontinuous security price movements, non-constant volatility, and models very different from the Black-Scholes model. I found that most of the model prices for the close to at-the-money options are very different from the market prices. In general, the models did poorly in producing similar prices as the market.

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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 Finance
Os 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.
info:eu-repo/semantics/publishedVersion

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Bucic, Ida. "Heston vs Black Scholes stock price modelling." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-105614.

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In this thesis the Black Scholes and the Heston stock prices are investigated and the models are compared. The Black Scholes model assumes that the volatility is constant, while the Heston model allows stochastic volatility which is more flexible and can perform better with empirical data. Both models are analysed and simulated, and the parameters are estimated based on empirical data of S&P 500. Results are based on simulations and characteristic functions which are presented with figures of probability density functions.

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Andrén, August, and Patrik Hagernäs. "Data-parallel Acceleration of PARSEC Black-Scholes Benchmark." Thesis, KTH, Skolan för informations- och kommunikationsteknik (ICT), 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-128607.

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The way programmers has been relying on processor improvements to gain speedup in their applications is no longer applicable in the same fashion. Programmers usually have to parallelize their code to utilize the CPU cores in the system to gain a signicant speedup. To accelerate parallel applications furthermore there are a couple of techniques available. One technique is to vectorize some of the parallel code. Another technique is to move parts of the parallel code to the GPGPU and utilize this very good multithreading unit of the system. The main focus of this report is to accelerate the data-parallel workload Black-Scholes of PARSEC benchmark suite. We are going to compare three accelerations of this workload, using vector instructions in the CPU, using the GPGPU and using a combination of them both. The two fundamental aspects are to look at the speedup and determine which technique requires more or less programming eort. To accelerate with vectorization in the CPU we use SSE & AVX techniques and to accelerate the workload in the GPGPU we use OpenACC.

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Teka, Kubrom Hisho. "Parameter estimation of the Black-Scholes-Merton model." Kansas State University, 2013. http://hdl.handle.net/2097/15669.

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Master of Science
Department of Statistics
James Neill
In financial mathematics, asset prices for European options are often modeled according to the Black-Scholes-Merton (BSM) model, a stochastic differential equation (SDE) depending on unknown parameters. A derivation of the solution to this SDE is reviewed, resulting in a stochastic process called geometric Brownian motion (GBM) which depends on two unknown real parameters referred to as the drift and volatility. For additional insight, the BSM equation is expressed as a heat equation, which is a partial differential equation (PDE) with well-known properties. For American options, it is established that asset value can be characterized as the solution to an obstacle problem, which is an example of a free boundary PDE problem. One approach for estimating the parameters in the GBM solution to the BSM model can be based on the method of maximum likelihood. This approach is discussed and applied to a dataset involving the weekly closing prices for the Dow Jones Industrial Average between January 2012 and December 2012.

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Bonotto, Everaldo de Mello. "A equação de Black-Scholes com ação impulsiva." Universidade de São Paulo, 2008. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-02072008-101527/.

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Impulsos são perturbações abruptas que ocorrem em curto espaço de tempo e podem ser consideradas instantâneas. E os mercados financeiros estão sujeitos a choques bruscos como mudanças de governos, quebra de empresas, entre outros. Assim, é natural considerarmos a ação de tais eventos na precificação de ativos financeiros. Nosso objetivo neste trabalho é obtermos uma formulação para a equação diferencial parcial de Black-Scholes com ação impulsiva de modo que os impulsos representem estes choques. Utilizaremos a teoria de integração não-absoluta em espaço de funções para obtenção desta formulação
Impulses describe the evolution of systems where the continuous development of a process is interrupted by abrupt changes of state. Financial markets are subject to extreme events or shocks as government changes, companies colapse, etc. Thus it seems natural to consider the action of these events in the valuation of derivative securities. The aim of this work is to obtain a formulation for the Black-Scholes equation with impulse action where the impulses can represent these shocks. We use the non-absolute integration theory in functional spaces to obtain such formulation

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Saleemi, Asima Parveen. "Finite Difference Methods for the Black-Scholes Equation." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-48660.

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Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

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Saadat, Sajedeh, and Timo Kudljakov. "Deterministic Quadrature Formulae for the Black–Scholes Model." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-54612.

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There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

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Ryhed, Erik, Per Thornadsson, and Gunnar Holm. "Warrantvärdering : En jämförelse mellan Monte-Carlo och Black-Scholes." Thesis, Örebro University, Department of Business, Economics, Statistics and Informatics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:oru:diva-1044.

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Syftet med denna uppsats är att med tre GARCH-modeller skatta volatiliteten för fjorton aktier med t- och normalfördelade slumptermer. Dessa volatiliteter implementeras sedan i Black-Scholes modell samt i Monte-Carlo simuleringar och utfallen av dessa två värderingsmetoder jämförs.

Författarna har kommit fram till att GARCH-modeller behövs för att skatta volatiliteten för de aktier som ingår i arbetet då modellerna tar hänsyn till den föreliggande heteroskedasticiteten.

De skillnader som uppstår mellan Monte-Carlo simuleringar och Black-Scholes modell beror främst på skillnader mellan normal- och t-fördelningen samt att volatiliteten ger större effekt i Monte-Carlo simuleringarna. Författarna kan inte uttala sig om huruvida Monte-Carlo skattningarna ger bättre resultat än den vedertagna Black-Scholes modell, däremot är Monte-Carlo mer teoretiskt korrekt.

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Villeneuve, Stéphane. "Options américaines dans un modèle de Black-Scholes multidimensionnel." Marne-la-Vallée, 1999. http://www.theses.fr/1999MARN0043.

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Le but de cette these est l'etude des options americaines dans un modele de diffusion multi-dimensionnel. Mathematiquement, cette etude est liee a un probleme d'arret optimal a horizon fini ou non. La premiere partie s'interesse a la description du modele a la valorisation des options americaines comme solution d'une inequation variationnelle parabolique et a l'existence ou non d'une region d'arret plus communement appele region d'exercice en finance. Le premier chapitre fournit une condition necessaire et suffisante portant sur le generateur infinitesimal de la diffusion pour que la region d'arret soit non vide. Les chapitres suivants etudient les proprietes des regions d'exercice associees a certains types d'options couramment negociees sur les marches : convexite, regularite et comportement asymptotique pour des horizons infinis ou proches de zero. La deuxieme partie concerne l'analyse numerique des options americaines en dimension deux. Apres avoir rappele les differentes formulations utilisant des equations aux derivees partielles (solution dans des espaces de sobolev ou solution de viscosite), deux methodes d'approximation du type directions alternees sont proposees et deux theoremes de convergence sont etablis. Un resultat de comparaison entre ces methodes termine cette partie. La derniere partie etudie le prix critique du put americain au voisinage de l'echeance lorsque l'action verse des dividendes. Un resultat concernant la stricte monotonie du prix critique est prouve ainsi qu'un encadrement de ce prix au voisinage de l'echeance

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Nuugulu, Samuel Megameno. "Fractional black-scholes equations and their robust numerical simulations." University of the Western Cape, 2020. http://hdl.handle.net/11394/7612.

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Philosophiae Doctor - PhD
Conventional partial differential equations under the classical Black-Scholes approach have been extensively explored over the past few decades in solving option pricing problems. However, the underlying Efficient Market Hypothesis (EMH) of classical economic theory neglects the effects of memory in asset return series, though memory has long been observed in a number financial data. With advancements in computational methodologies, it has now become possible to model different real life physical phenomenons using complex approaches such as, fractional differential equations (FDEs). Fractional models are generalised models which based on literature have been found appropriate for explaining memory effects observed in a number of financial markets including the stock market. The use of fractional model has thus recently taken over the context of academic literatures and debates on financial modelling.
2023-12-02

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Karlén, Anne, and Hossein Nohrouzian. "Lattice approximations for Black-Scholes type models in Option Pricing." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-21951.

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This thesis studies binomial and trinomial lattice approximations in Black-Scholes type option pricing models. Also, it covers the basics of these models, derivations of model parameters by several methods under different kinds of distributions. Furthermore, the convergence of binomial model to normal distribution, Geometric Brownian Motion and Black-Scholes model isdiscussed. Finally, the connections and interrelations between discrete random variables under the Lattice approach and continuous random variables under models which follow Geometric Brownian Motion are discussed, compared and contrasted.

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Huaringa, Mosquera Luis Zacarías. "Ecuaciones diferenciales parciales aplicado a finanzas: modelo de black-scholes." Bachelor's thesis, Universidad Nacional Mayor de San Marcos, 2018. https://hdl.handle.net/20.500.12672/8372.

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Publicación a texto completo no autorizada por el autor
Desde su publicacion, el modelo Black – Scholes ha tenido un uso satisfactorio que ayuda en la toma de decisiones en sistemas financieros y empresas. Dicho modelo sirve para estimar el valor de las acciones a tiempo futuro, tanto en compra como venta, resolviendo una igualdad que sigue un movimiento browniano. Se busca resolver la ecuación en derivadas parciales de Black-Scholes, reduciéndola a través de un cambio de variables a la forma de una ecuación de calor la cual facilitará su desarrollo. Se pasará a resolver dicha ecuación usando transformada de Fourier obteniendo así su solución. Por último, la solución de la ecuación podrá pasar a ser estudiada y aplicada en un caso real en el cual se podría escoger cualquier acción que cotice la bolsa de valores como activo. Una vez resuelta la ecuación se plantearan formas en las cuales se pueden aplicar en la acciones de las principales empresas que coticen en Perú y a través de estos datos se calcularan los valores para la call europea. Se concluirá teniendo en cuenta el beneficio que nos otorga el modelo en la predicción de estas opciones, y que tan preciso es. A su vez se encontrará sus posibles aplicaciones y usos en la bolsa de valores.
Trabajo de suficiencia profesional

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Vakaloudis, Dmitri. "Application of Volatility Targeting Strategies within a Black-Scholes Framework." Master's thesis, Faculty of Commerce, 2019. http://hdl.handle.net/11427/31319.

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The traditional Black-Scholes (BS) model relies heavily on the assumption that underlying returns are normally distributed. In reality however there is a large amount of evidence to suggest that this assumption is weak and that actual return distributions are non-Gaussian. This dissertation looks at algorithmically generating a Volatility Targeting Strategy (VTS) which can be used as an underlying asset. The rationale here is that since the VTS has a constant prespecified level of volatility, its returns should be normally distributed, thus tending closer to an underlying that adheres to the assumptions of BS.

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Au, Chi Yan. "Numerical methods for solving Markov chain driven Black-Scholes model." HKBU Institutional Repository, 2010. http://repository.hkbu.edu.hk/etd_ra/1154.

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Nohrouzian, Hossein, and Anne Karlén. "Lattice Approximations for Black-Scholes type models in Option Pricing." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-23511.

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This thesis studies binomial and trinomial lattice approximations in Black-Scholes type option pricing models. Also, it covers the basics of these models, derivations of model parameters by several methods under different kinds of distributions. Furthermore, the convergence of the binomial model to normal distribution, Geometric Brownian Motion and Black-Scholes model is discussed. Finally, the connections and interrelations between discrete random variables under the Lattice approach and continuous random variables under models which follow Geometric Brownian Motion are discussed, compared and contrasted.

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Mautner, Karin. "Numerical treatment of the Black-Scholes variational inequality in computational finance." [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=983505780.

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Gustafsson, Jonas. "Värdering av Företags Totala Aktiekapital utifrån Black-Scholes Modell och Redovisningsdata." Thesis, Uppsala University, Department of Economics, 2006. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7046.

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I denna uppsats undersöks hur väl det går att värdera marknadsvärdet av företagets aktiekapital med hjälp av en modifierad Black-Scholes köpoptionsmodell. Tidigare undersökningar har gjorts bland annat i USA där man gjorde en studie på S & P100 och fick starka indikationer på att modellen väl förklarar marknadsvärdet av aktiekapitalet. Denna uppsats följer metoden som användes vid studien på S & P100 för att se om den även ger en hög förklaringsgrad på den svenska aktiemarknaden. Undersökningen genomförs med hjälp av redovisningsdata från samtliga bolag listade på A-listan vid Stockholmsbörsen. Redovisningsdata används för att få fram de variabler som används i den Black-Scholes modell som modifierats från att värdera köpoptioner på aktier till att värdera marknadsvärdet av företags aktiekapital. Dessutom undersöks vilken förklaringsgrad de variabler som används i den modifierade Black-Scholes modellen har på P/e-värdet för företagens aktier. Resultaten från undersökning med den modifierade Black-Scholes modellen visar att den ger en bra estimering för företags totala aktiekapital. Dessutom visar resultaten att undersökningen av variablernas förklaringsgrad på P/e värdet har ett signifikant värde för vissa av variablerna.

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Furtado, Susana Margarida Figueiredo de Sousa Borges. "Avaliação de opções : fundamentos e análise do modelo de black-scholes." Master's thesis, Instituto Superior de Economia e Gestão, 1994. http://hdl.handle.net/10400.5/19656.

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Saleh, Ali, and Ahmad Al-Kadri. "Option pricing under Black-Scholes model using stochastic Runge-Kutta method." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-53783.

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The purpose of this paper is solving the European option pricing problem under the Black–Scholes model. Our approach is to use the so-called stochastic Runge–Kutta (SRK) numericalscheme to find the corresponding expectation of the functional to the stochastic differentialequation under the Black–Scholes model. Several numerical solutions were made to study howquickly the result converges to the theoretical value. Then, we study the order of convergenceof the SRK method with the help of MATLAB.

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Banfi, del Río Pablo, Finsterbusch Maximiliano Correa, Diez Ricardo Romero, and Jensen Ricardo Wechsler. "Análisis de la capacidad predictiva del modelo de Black and Scholes." Tesis, Universidad de Chile, 2003. http://repositorio.uchile.cl/handle/2250/108217.

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Seminario para optar al título de Ingeniero Comercial
El objetivo de este trabajo es analizar la capacidad predictiva del modelo desarrollado por Black y Scholes. Para este propósito se testeó la capacidad de diversos modelos autoregresivos, además de un modelo multivariable y otro ingenuo. Las conclusiones se detallan a nivel de acción, modelo, sector y vencimiento. El modelo que obtuvo los mejores resultados a nivel promedio fue el Arima recursivo seguido por el Arima rolling. Durante el desarrollo del trabajo surgieron objetivos secundarios, uno de estos fue verificar si realmente el mercado sigue o no a B and S. Para esto se compararon los precios calculados por la fórmula con los precios observados en el mercado y se encontró que en un 95% de los casos, el mercado transó los activos a un precio superior al arrojado por la formula. Por esto no somos capaces de afirmar si el mercado sigue o no a B and S, pero es muy probable que su valor sea tomado en cuenta pero complementado por otras variables que el modelo no considera, como podrían ser las expectativas, el valor de la cobertura por riesgo, o la estabilidad económica y política del país. Por último buscamos un modelo que tuviera resultados probados para una determinada acción y encontramos que el modelo multivariado con las variables Dow (-1), Dow (-4), Dow (-5), Nasdaq (-1), Klac (-1), Klac (-2) y Error (-2) es el que había arrojado la mejor predicción para la acción Klac el mes anterior. Entonces aplicamos los modelos anteriores en conjunto con este a Klac y comparamos los resultados. La idea fue comprobar si es que modelos que fueron precisos en un pasado cercano, lo siguen siendo con el transcurso del tiempo. Los nuevos resultados indicaron que a medida que pasa el tiempo y cambian las condiciones económicas del mercado, los modelos pierden precisión. Finalmente el modelo con mejores resultados fue el modelo Multivariado rolling. Corroboramos que el mercado es complejo, dinámico y que aprende. En ocasiones modelos tan simples como el ingenuo obtuvieron mejores resultados que los modelos más complejos, lo que indica que hay cuotas de azar, expectativas y otros factores aleatorios que hacen muy difícil predecir el comportamiento del mercado y superar su rendimiento.

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Mautner, Karin. "Numerical treatment of the Black-Scholes variational inequality in computational finance." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät II, 2007. http://dx.doi.org/10.18452/15595.

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In der Finanzmathematik hat der Besitzer einer amerikanische Option das Recht aber nicht die Pflicht, eine Aktie innerhalb eines bestimmten Zeitraums, für einen bestimmten Preis zu kaufen oder zu verkaufen. Die Bewertung einer amerikanische Option wird als so genanntes optimale stopping Problem formuliert. Erfolgt die Modellierung des Aktienkurses durch eine geometrische Brownsche Bewegung, wird der Wert einer amerikanischen Option durch ein deterministisches freies Randwertproblem (FRWP), oder einer äquivalenten Variationsungleichung (VU) auf ganz R in gewichteten Sobolev Räumen gegeben. Um Standardmethoden der Numerischen Mathematik anzuwenden, wird das unbeschränkte Gebiet zu einem beschränkten Gebiet abgeschnitten. Mit Hilfe der Fourier-Transformation wird eine Integraldarstellung der Lösung die den freien Rand explizit beinhaltet, hergeleitet. Mittels dieser Integraldarstellung werden Abschneidefehlerschranken bewiesen. Danach werden gewichtete Poincare Ungleichungen mit expliziten Konstanten bewiesen. Der Abschneidefehler und die gewichtete Poincare Ungleichung ermöglichen, einen zuverlässigen a posteriori Fehlerschätzer zwischen der exakten Lösung der VU und der semidiskreten Lösung des penalisierten Problems auf R herzuleiten. Eine hinreichend glatte Lösung der VU garantiert die Konvergenz der Lösung des penaltisierten Problems zur Lösung der VU. Ein a priori Fehlerschätzer für den Fehler zwischen der exakten Lösung der VU und der semidiskreten Lösung des penaltisierten Problems beendet die numerische Analysis. Die eingeführten aposteriori Fehlerschätzer motivieren einen Algorithmus für adaptive Netzverfeinerung. Numerische Experimente zeigen die verbesserte Konvergenz des adaptiven Verfahrens gegenüber der uniformen Verfeinerung. Der zuverlässige a posteriori Fehlerschätzer ermöglicht es, den Abschneidepunkt so zu wählen, dass der Gesamtfehler (Diskretisierungsfehler plus Abschneidefehler) kleiner als eine gegebenen Toleranz ist.
Among the central concerns in mathematical finance is the evaluation of American options. An American option gives the holder the right but not the obligation to buy or sell a certain financial asset within a certain time-frame, for a certain strike price. The valuation of American options is formulated as an optimal stopping problem. If the stock price is modelled by a geometric Brownian motion, the value of an American option is given by a deterministic parabolic free boundary value problem (FBVP) or equivalently a non-symmetric variational inequality (VI) on weighted Sobolev spaces on R. To apply standard numerical methods, the unbounded domain R is truncated to a bounded one. Applying the Fourier transform to the FBVP yields an integral representation of the solution including the free boundary explicitely. This integral representation allows to prove explicit truncation errors. Since the VI is formulated within the framework of weighted Sobolev spaces, we establish a weighted Poincare inequality with explicit determined constants. The truncation error estimate and the weighted Poncare inequality enable a reliable a posteriori error estimate between the exact solution of the VI and the semi-discrete solution of the penalised problem on R. A sufficient regular solution provides the convergence of the solution of the penalised problem to the solution of the VI. An a priori error estimate for the error between the exact solution of the VI and the semi-discrete solution of the penalised problem concludes the numerical analysis. The established a posteriori error estimates motivates an algorithm for adaptive mesh refinement. Numerical experiments show the improved convergence of the adaptive algorithm compared to uniform mesh refinement. The reliable a posteriori error estimate including explicit truncation errors allows to determine a truncation point such that the total error (discretisation and truncation error) is below a given error tolerance.

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Mawah, Bernard. "Option pricing with transaction costs and a non-linear Black-Scholes equation." Thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120920.

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Österholm, Göran. "EMPIRISK STUDIE AV BLACK-SCHOLES PRISSÄTTNINGSMODELL : OMXS30-OPTIONERS PRISRÖRELSER OCH DELTA HEDGING." Thesis, Uppsala University, Department of Economics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-7665.

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I uppsatsen studeras diskrepansen mellan Black-Scholes prissättningsmodell och prissättningen på marknaden för OMXS30-optioner – i ett delta hedging-perspektiv.

Resultaten i uppsatsen antyder att Black-Scholes modell ger för höga D -värden för OMXS30 köpoptioner och likaså ger modellen för höga D -värden för OMXS30 säljoptioner gentemot verkligheten, under testperioden.

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Hussain, Zahid, Muhammad Sulaiman, and Edward K. E. Sackey. "Optimal System of Subalgebras and Invariant Solutions for the Black-Scholes Equation." Thesis, Blekinge Tekniska Högskola, Sektionen för ingenjörsvetenskap, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-2817.

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The main purpose of this thesis is to use modern goal-oriented adaptive methods of Lie group analysis to construct the optimal sys- tem of Black-Scholes equation. We will show in this thesis how to obtain all invariant solutions by constructing what has now become so popular, optimal system of sub-algebras, the main Lie algebra admit- ted by the Black-Scholes equation. First, we obtain the commutator table of already calculated symmetries of the Black-Scholes equation. We then followed with the calculations of transformation of the gen- erators with the Lie algebra L6 which provides one-parameter group of linear transformations for the operators. Here we make use of the method of Lie equations to solve the partial di®erential equations. Next, we consider the construction of optimal systems of the Black- Scholes equation where the method requires a simpli¯cation of a vector to a general form to each of the transformations of the generators. Further, we construct the invariant solutions for each of the op- timal system. This study is motivated by the analysis of Lie groups which is being taken to another level by ALGA here in Blekinge In- stitute Technology, Sweden. We give a practical and in-depth steps and explanation of how to construct the commutator table, the calcu- lation of the transformation of the generators and the construction of the optimal system as well as their invariant solutions. Keywords: Black-Scholes Equation, commutators, commutator table, Lie equa- tions, invariant solution, optimal system, generators, Airy equation, structure constant,
It was an accolade for us to work with Professor Nail.H. Ibrgimov. +46762600953

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Oga, Luis Fernando. "A teoria da ciência no modelo Black-Scholes de apreçamento de opções." Universidade de São Paulo, 2007. http://www.teses.usp.br/teses/disponiveis/8/8133/tde-18032008-132755/.

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O presente trabalho pretende introduzir uma visão das Finanças sob o aspecto da Filosofia da Ciência. Para permitir um estudo mais detalhado, optou-se por utilizar um dos modelos mais utilizados em Finanças, o modelo Black-Scholes de apreçamento de opções, e situá-lo dentro do campo de aplicação da Filosofia da Ciência. Primeiramente buscou-se, antes de entrar numa análise do texto original que apresentou o modelo, contextualizá-lo no campo da Economia e das Finanças e reconstruir historicamente suas bases conceituais. Em seguida são apresentados alguns dos elementos principais que caracterizam os modelos filosóficos de mudança científica posteriores à posição definida pelo positivismo lógico. Especial atenção é dada às concepções Realista e Anti-Realista da Ciência. Ao final, é feita uma descrição de algumas peculiaridades empíricas do modelo Black-Scholes e é analisada a função do modelo dentro do campo da Economia e das Finanças.
This work is an introduction of a Philosophy of Science view of the Finance. We choose the Black-Scholes option valuation model, one of the most famous models of finance, and we submet it of an analysis in the Philosophy of Science point of view. At first, we present an historical reconstruction of Black-Scholes model conceptual basis, using the original text of 1973. After this, we show some aspects of philosophical models of scientific change after the position defined by Positivism. Special attention is given to Realism and Anti-Realismo conception of science. At the end, we describe some empirical aspects of Black-Scholes model and its correlation inside the Economy and Modern Theory of Finance.

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Uhliarik, Marek. "Operator Splitting Methods and Artificial Boundary Conditions for a nonlinear Black-Scholes equation." Thesis, Högskolan i Halmstad, Sektionen för Informationsvetenskap, Data– och Elektroteknik (IDE), 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-6111.

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There are some nonlinear models for pricing financial derivatives which can improve the linear Black-Scholes model introduced by Black, Scholes and Merton. In these models volatility is not constant anymore, but depends on some extra variables. It can be, for example, transaction costs, a risk from a portfolio, preferences of a large trader, etc. In this thesis we focus on these models. In the first chapter we introduce some important theory of financial derivatives. The second chapter is devoted to the volatility models. We derive three models concerning transaction costs (RAPM, Leland's and Barles-Soner's model) and Frey's model which assumes a large (dominant) trader on the market. In the third and in the forth chapter we derive portfolio and make numerical experiments with a free boundary. We use the first order additive and the second order Strang splitting methods. We also use approximations of Barles-Soner's model using the identity function and introduce an approximation with the logarithm function of Barles-Soner's model. These models we finally compare with models where the volatility includes constant transaction costs.

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Lee, Chi-ming Simon, and 李志明. "A study of Hong Kong foreign exchange warrants pricing using black-scholes formula." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1992. http://hub.hku.hk/bib/B3126542X.

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Núñez, Vargas Sandra Isabel. "Adaptación del modelo Black-Scholes en la simulación de un portafolio de acciones." Bachelor's thesis, Pontificia Universidad Católica del Perú, 2009. http://tesis.pucp.edu.pe/repositorio/handle/123456789/298.

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El modelo de Black-Scholes fue publicado en 1973. Para este modelo, el movimiento browniano geométrico está asociado a la dinámica de los precios de las acciones, la cual está descrita por una ecuación diferencial estocástica. Este modelo tiene debilidades que están relacionadas a la inexactitud de sus presunciones con respecto a lo que sucede en el mercado de valores y a los factores externos que son incontrolables. Por ello, se ha realizado un estudio de mejora del mismo, para lo cual se ha escogido cuatro empresas que son representativas del mercado de valores del país. Se ha planteado cuatro propuestas en las que se modifica el valor de la volatilidad y mediante el software SciLab se simulan los valores de las acciones para cada una de las empresas y luego se comparan los resultados y se escoge la que estima mejor el precio de las acciones y con la que se obtiene un error menor.
Tesis

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Hassan, Shakill. "The Black-Scholes model and the pricing of stock options in South Africa." Master's thesis, University of Cape Town, 1999. http://hdl.handle.net/11427/14302.

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Bibliography: leaves 52-54.
Option Pricing Theory (OPT), along with the Capital Asset Pricing Model, the Theory of Capital Structure, and the Efficient Markets Hypothesis, form one of the pillars of modem finance theory. Central to OPT is the Black-Scholes model, the first option pricing model derived within a general equilibrium framework, and therefore consistent with all arbitrage conditions an asset pricing model must satisfy. An attempt is made at explaining this model, and the first part of the paper is devoted to this objective. The appreciation of the theoretical elegance of the Black-Scholes model can be considerably enhanced through the understanding of the issues that made (and still make in the case of American put options) the derivation of an equilibrium model of option pricing such an immense task. With the intention of emphasising such issues, the first section of Part One covers the option pricing models that had been suggested before Black and Scholes (and Merton). This helps to put the Black-Scholes model in context, as well as facilitate an understanding of the approach Black and Scholes adopted in developing their model. Its derivation is the central focus of section 3, the second section of Part One. The second part of the paper contains an attempt at testing the Black-Scholes model, first in its "pure form," and then adjusted to account for the possibility of early exercise. Simple regression tests are performed, where daily prices of a sample of stock options traded on the Johannesburg Stock Exchange are used as dependent variables in regression equations. Black-Scholes model prices are computed, and used as the explanatory variables in these equations. But before the tests could be conducted, the distributions of the underlying assets' returns had to be examined and due consideration had to be given to the estimation of the volatility parameters. Part Two starts with a very brief overview of the South African exchange-traded stock options market. This is followed by a description of the data used in the tests, and discussions on the statistical behaviour of the underlying assets. A discussion on volatility estimating follows, and the test results are then presented.

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Lee, Chi-ming Simon. "A study of Hong Kong foreign exchange warrants pricing using black-scholes formula /." [Hong Kong] : University of Hong Kong, 1992. http://sunzi.lib.hku.hk/hkuto/record.jsp?B13302838.

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Costa, Thadeu Antonio Ferreira de Melo. "Coprojeto hardware/software das equações de Black-Scholes para precificação de opções no mercado financeiro." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-19102018-102741/.

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Este trabalho apresenta a implementação em hardware das Equações de Black-Scholes para precificação de opções usando Método de Monte Carlo. A implementação foi feita em OpenCL compatível com FPGAs recentes da Altera/Intel. Essa implementação é modular e permite a utilização de diferentes geradores de números aleatórios em configurações diferentes de software e hardware. A proposta é que essas implementações possam aproveitar as vantagens de cada componente, resultando em uma maior quantidade de simulações e por consequência melhorando a precisão dos resultados.
This paper presents the hardware implementation of Black-Scholes Equations for pricing options using Monte Carlo Method. The implementation was made in OpenCL compatible with recent Altera / Intel FPGAs. This implementation is modular and allows the use of different random number generators in different software and hardware configurations. The proposal is that these implementations can take advantage of each component, resulting in a greater number of simulations and consequently improving the accuracy of the results.

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Kolesnichenko, Anna, and Galina Shopina. "Valuation of portfolios under uncertain volatility : Black-Scholes-Barenblatt equations and the static hedging." Thesis, Halmstad University, School of Information Science, Computer and Electrical Engineering (IDE), 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-1634.

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The famous Black-Scholes (BS) model used in the option pricing theory

contains two parameters - a volatility and an interest rate. Both

parameters should be determined before the price evaluation procedure

starts. Usually one use the historical data to guess the value of these

parameters. For short lifetime options the interest rate can be estimated

in proper way, but the volatility estimation is, as well in this case,

more demanding. It turns out that the volatility should be considered

as a function of the asset prices and time to make the valuation self

consistent. One of the approaches to this problem is the method of

uncertain volatility and the static hedging. In this case the envelopes

for the maximal and minimal estimated option price will be introduced.

The envelopes will be described by the Black - Scholes - Barenblatt

(BSB) equations. The existence of the upper and lower bounds for the

option price makes it possible to develop the worse and the best cases

scenario for the given portfolio. These estimations will be financially

relevant if the upper and lower envelopes lie relatively narrow to each

other. One of the ideas to converge envelopes to an unknown solution

is the possibility to introduce an optimal static hedged portfolio.

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Prudente, Leandro da Fonseca 1985. "Estimação da superficie de volatilidade dos ativos atraves da equação de Black-Scholes generalizada." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307442.

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Orientador: Jose Mario Martinez
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: Nesta dissertação expomos algumas propriedades das opções e desenvolvemos a teoria clássica que resulta na Equação de Black-Scholes Generalizada, o modelo utilizado no mercado para precificar uma opção. Neste cenário apresentamos o Princípio da Retrodifusão. A ideia de obtermos a Equação de Black-Scholes por meios mais simples e que possibilitem uma interpretação intuitiva desta equação. Mostramos uma maneira numérica para resolver a Equação de Black-Scholes Generalizada e por fim desenvolvemos um método para estimar a superfície de volatilidade de um ativo usando como ferramenta conhecidas opções comercializadas.
Abstract: In this work expose some properties of the options and developed the classical theory which results in the Generalized Black-Scholes equation, the model used in the market for pricing an option. In this context we present the Princípio da Retrodifusão. The idea is to get the Black-Scholes equation by simpler means and enabling an intuitive interpretation of this equation. We show a numerical way to solve the Generalized Black-Scholes equation and finally developed a method to estimate the volatility surface of an asset using as a tool known options traded.
Mestrado
Mestre em Matemática Aplicada

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Ferreira, Fausto Junior Martins. "Fatores de risco adaptados de taxa de câmbio no modelo de Black e Scholes." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/3/3142/tde-12072016-113756/.

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Este trabalho apresenta uma metodologia de cálculo de sensibilidades utilizando equa- ções analíticas, levando em a conta a correção de smile na superfície de volatilidade, que não é contemplada no modelo de Black e Scholes. Dada a diferença signicativa na mensura ção do risco as instituições nanceiras calculam suas sensibilidades incorporando esta correção, mas tal determinação tem sido realizada por métodos numéricos, que acabam sendo mais lentos que a abordagem aqui proposta. São apresentadas equações analíticas para as principais sensibilidades do modelo a partir de dados de mercado usados na constução da superfície de volatilidade implícita. Ilustramos a comparação da técnica proposta com o método numérico com base no mercado de opções sobre taxa de câmbio Brasileiro.
This work presents a study on how we should adapted the Greeks or risk factors of the Black and Scholes model. We can derive analytical equations for the main sensitivities of the model and using the market data to build an implied volatility surface and to get additional terms for the risk factors. We propose to implement this model in a scheme of analytic differential equations derived from the pricing model and from the implied volatility function. The building of this implied volatility and risk factors was based on the foreign exchange Brazilian market.

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Rich, Don R. "Incorporating default risk into the Black-Scholes model using stochastic barrier option pricing theory." Diss., This resource online, 1993. http://scholar.lib.vt.edu/theses/available/etd-06062008-171359/.

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Salomão, Martinho. "Precificação de opções financeiras: um estudo sobre os modelos de Black Scholes e Garch." Universidade Federal do Espírito Santo, 2011. http://repositorio.ufes.br/handle/10/2642.

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Precificação de opções financeiras: um estudo sobre os modelos de Black Scholes e Garch

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Salomão, Martinho de Freitas. "Precificação de opções financeiras: um estudo sobre os modelos de Black Scholes e Garch." Universidade Federal do Espírito Santo, 2011. http://repositorio.ufes.br/handle/10/6007.

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Neste trabalho são analisadas as propriedades teóricas e empíricas de três modelos de precificação de opções financeiras sobre ações: Black Scholes (1973), ad-hoc Black Scholes (Dumas, Fleming e Whaley, 1998), e o modelo GARCH assimétrico proposto por Heston e Nandi (2000), ou HN-GARCH. Os modelos são testados em opções de compra sobre ações preferenciais da Petrobras. É mostrado que o modelo Black Scholes (1973), por supor que a variância do ativo subjacente seja constante, apresentou o pior desempenho de predição comparativamente aos outros dois modelos, que consideram a volatilidade uma variável. Enquanto o modelo ad-hoc Black Scholes precificou melhor as opções muito dentro do dinheiro, dentro do dinheiro e muito fora do dinheiro, o modelo HN-GARCH obteve desempenho superior em opções no dinheiro e fora do dinheiro
This study analyzes the theoretical and empirical properties of three models for pricing options on financial stocks: Black Scholes (1973), ad-hoc Black Scholes (Dumas, Fleming and Whaley, 1998), and the asymmetric GARCH model proposed by Heston and Nandi (2000), or HN-GARCH. The models are tested in call s options on shares of Petrobras. It is shown that the Black Scholes model (1973), by assuming that the variance of the underlying asset is constant, showed the worst performance prediction compared to the other two models that consider volatility a variable. While the model adhoc Black Scholes priced much better options deep in the money, in the money and deep out of the money, the HN-GARCH model had superior performance for at the money and out of the money options

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Fischbach, Pascal. "Derivate für FX-Absicherungen." St. Gallen, 2008. http://www.biblio.unisg.ch/org/biblio/edoc.nsf/wwwDisplayIdentifier/05608120001/$FILE/05608120001.pdf.

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Rabeau, Nicholas Marc. "Probabilistic approach to contingent claims analysis." Thesis, Imperial College London, 1996. http://hdl.handle.net/10044/1/8195.

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Elder, John. "Hedging strategies for financial derivatives." Thesis, University of Oxford, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275325.

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Yang, Yuankai. "Pricing American and European options under the binomial tree model and its Black-Scholes limit model." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-68264.

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We consider the N step binomial tree model of stocks. Call options and put options of European and American type are computed explicitly. With appropriate scaling in time and jumps, convergence of the stock prices and the option prices are obtained as N-> infinite. The obtained convergence is the Black-Scholes model and, for the particular case of European call option, the Black-Scholes formula is obtained. Furthermore, the Black-Scholes partial differential equation is obtained as a limit from the N step binomial tree model. Pricing of American put option under the Black-Scholes model is obtained as a limit from the N step binomial tree model. With this thesis, option pricing under the Black-Scholes model is achieved not by advanced stochastic analysis but by elementary, easily understandable probability computation. Results which in elementary books on finance are mentioned briefly are here derived in more details. Some important Java codes for N step binomial tree option prices are constructed by the author of the thesis.

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Dremkova, Ekaterina. "A high order compact method for nonlinear Black-Scholes option pricing equations with transaction costs." Thesis, Halmstad University, School of Information Science, Computer and Electrical Engineering (IDE), 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-3198.

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In this work we consider the nonlinear case of Black-Scholes equation and apply it to American options. Also, method of Liao and Khaliq of high order was applied to nonlinear Black-Scholes equation in case of American options. Here, we use this method oh fourth order in time and space to raise American option price accuracy.

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Angeli, Andrea, and Cornelius Bonz. "Changes in the creditability of the Black-Scholes option pricing model due to financial turbulences." Thesis, Umeå University, Umeå School of Business, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-34873.

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This study examines whether the performance of the Black-Scholes model to price stock index options is influenced by the general conditions of the financial markets. For this purpose we calculated the theoretical values of 5814 options (3366 put option price observations and 2448 call option price observations) under the Black-Scholes assumptions. We compared these theoretical values with the real market prices in order to put the degree of deviations in two different time windows built around the bankruptcy of Lehman Brothers (September 15th 2008) to the test. We find clear evidences to state that the Black-Scholes model performed differently in the period after Lehman Brothers than in the period before; therefore we are able to blame this event for our findings.

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Dissertations / Theses on the topic 'Black Scholes'

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