American options under stochastic volatility

Detalhes bibliográficos
Autor(a) principal: Marinhas, Marta Carvalho
Data de Publicação: 2016
Tipo de documento: Dissertação
Idioma: eng
Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
Texto Completo: http://hdl.handle.net/10451/25232
Resumo: Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2016
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spelling American options under stochastic volatilityMatemática financeiraTeses de mestrado - 2016Domínio/Área Científica::Ciências Naturais::MatemáticasTese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2016An option is a contract that gives the holder the right to buy, in the case of a call, or sell, in the case of a put, an underlying asset at a pre-determined strike price. A European option allows the holder to exercise the option only on a pre-determined expiration date, while with an American option the holder can exercise the option at any point in time until the maturity date. Options can incorporate dividends, which are a portion of a company's earning distributed to its shareholders, that can be issued as cash payments, as shares of stock or other property. Black and Scholes (1973) derived a closed form solution for the value of European options with constant volatility, while Heston (1993) provides a solution for European options with stochastic volatility. It was proved that assuming constant volatility leads to considerable mispricing. Bakshi, Cao and Chen (1997) did a series of tests comparing the Black and Scholes (1973) with three models which allow for stochastic volatility. They showed that incorporating stochastic volatility reduces the absolute pricing error by 20% to 70%. For example a call option with the price $1:68, under the Black and Scholes model has an error of $0:78, while with a model with stochastic volatility the error is reduced to $0:42. Hence, models that allow the volatility of the underlying asset to be stochastic better describe the market behavior. Unlike European options, American options do not have a closed form solution for its value with constant or stochastic volatility, due to the fact that the price depends on the optimal exercise policy. The models on American options under stochastic volatility can be separated in two approaches: the Partial Differential Equation, PDE, based and the non PDE based. There are various numerical methods to price American options. For example, Brennan and Schwartz (1977) introduced finite difference methods; the least squares Monte Carlo is a model developed by Longstaff and Schwartz (2001), where the model uses simulations of cash flows generated by the option and compare them to the value of immediate exercise to calculate the price. In Beliaeva and Nawalkha (2010) a bivariate tree is used where two independent trees are created for the stock price and for the variance. Broadie and Detemple (1996) developed a method for lower and upper bounds on the prices of American options based on regression coefficients. In the Clarke and Parrott (1999) model they use the Heston PDE, transformed into a non dimensional form, with a multigrid iteration method to solve the problem of option pricing. Detemple and Tian (2002) determine the exercise region by a single exercise boundary under general conditions on the interest rate and the dividend yield and derive a recursive integral equation for the exercise boundary. In this work, we will develop an implementation based on the Heston model with the explicit method. First, we will derive the Heston PDE, showing how it is used in the method described. Then we will test the accuracy of the results, randomly creating options and using the various methods to price them and calculate the errors of each method.Dias, José Carlos GonçalvesRepositório da Universidade de LisboaMarinhas, Marta Carvalho2016-12-13T16:24:34Z201620162016-01-01T00:00:00Zinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisapplication/pdfhttp://hdl.handle.net/10451/25232TID:201616157enginfo:eu-repo/semantics/openAccessreponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãoinstacron:RCAAP2023-11-08T16:14:42Zoai:repositorio.ul.pt:10451/25232Portal AgregadorONGhttps://www.rcaap.pt/oai/openaireopendoar:71602024-03-19T21:42:15.412949Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informaçãofalse
dc.title.none.fl_str_mv American options under stochastic volatility
title American options under stochastic volatility
spellingShingle American options under stochastic volatility
Marinhas, Marta Carvalho
Matemática financeira
Teses de mestrado - 2016
Domínio/Área Científica::Ciências Naturais::Matemáticas
title_short American options under stochastic volatility
title_full American options under stochastic volatility
title_fullStr American options under stochastic volatility
title_full_unstemmed American options under stochastic volatility
title_sort American options under stochastic volatility
author Marinhas, Marta Carvalho
author_facet Marinhas, Marta Carvalho
author_role author
dc.contributor.none.fl_str_mv Dias, José Carlos Gonçalves
Repositório da Universidade de Lisboa
dc.contributor.author.fl_str_mv Marinhas, Marta Carvalho
dc.subject.por.fl_str_mv Matemática financeira
Teses de mestrado - 2016
Domínio/Área Científica::Ciências Naturais::Matemáticas
topic Matemática financeira
Teses de mestrado - 2016
Domínio/Área Científica::Ciências Naturais::Matemáticas
description Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2016
publishDate 2016
dc.date.none.fl_str_mv 2016-12-13T16:24:34Z
2016
2016
2016-01-01T00:00:00Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
format masterThesis
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://hdl.handle.net/10451/25232
TID:201616157
url http://hdl.handle.net/10451/25232
identifier_str_mv TID:201616157
dc.language.iso.fl_str_mv eng
language eng
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
instname:Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron:RCAAP
instname_str Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
instacron_str RCAAP
institution RCAAP
reponame_str Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
collection Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
repository.name.fl_str_mv Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos) - Agência para a Sociedade do Conhecimento (UMIC) - FCT - Sociedade da Informação
repository.mail.fl_str_mv
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