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Pricing of discretely sampled Asian options under Lévy processesXie, Jiayao January 2012 (has links)
We develop a new method for pricing options on discretely sampled arithmetic average in exponential Lévy models. The main idea is the reduction to a backward induction procedure for the difference Wn between the Asian option with averaging over n sampling periods and the price of the European option with maturity one period. This allows for an efficient truncation of the state space. At each step of backward induction, Wn is calculated accurately and fast using a piece-wise interpolation or splines, fast convolution and either flat iFT and (refined) iFFT or the parabolic iFT. Numerical results demonstrate the advantages of the method.
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KMV model v podmínkách českého kapitálového trhu / KMV model in the Czech capital marketJezbera, Lukáš January 2010 (has links)
The thesis is focused on the options of quantifying credit risk by using the concept of the KMV model. The introduction outlines the basic approaches to measuring credit risk. In the following chapters is specified the nature of KMV model with the focus on its application in the Czech capital market. Self-calibration of the KMV model is made in this part. The analytical part related to the quantification of credit risk using the KMV model is implemented on selected companies which are traded on the Prague Stock Exchange. The results obtained are consequently confronted with the official rating degrees of agency Moody's.
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Optimizing the Number of Time-steps Used in Option Pricing / Optimering av Antal Tidssteg inom OptionsprissättningLewenhaupt, Hugo January 2019 (has links)
Calculating the price of an option commonly uses numerical methods and can becomputationally heavy. In general, longer computations result in a more precisresult. As such, improving existing models or creating new models have been thefocus in the research field. More recently the focus has instead shifted towardcreating neural networks that can predict the price of a given option directly.This thesis instead studied how the number of time-steps parameter can beoptimized, with regard to precision of the resulting price, and then predict theoptimal number of time-steps for other options. The number of time-stepsparameter determines the computation time of one of the most common models inoption pricing, the Cox-Ross-Rubinstein model (CRR). Two different methodsfor determining the optimal number of time-steps were created and tested. Bothmodels use neural networks to learn the relationship between the input variablesand the output. The first method tried to predict the optimal number oftime-steps directly. The other method instead tried to predict the parameters ofan envelope around the oscillations of the option pricing method. It wasdiscovered that the second method improved the performance of the neuralnetworks tasked with predicting the optimal number of time-steps. It was furtherdiscovered that even though the best neural network that was found significantlyoutperformed the benchmark method, there was no significant difference incalculation times, most likely because the range of log moneyness and pricesthat were used. It was also noted that the neural network tended tounderestimate the parameter and that might not be a desirable property of asystem in charge of estimating a price in the financial sector.
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O comportamento do valor extrínseco das opções de compra de ações no mercado de capitais brasileiro / The intrinsic value performance of the stock call options on the Brazilian capital marketFranco, Matheus Silveira 12 December 2014 (has links)
O valor extrínseco é uma das variáveis que mais carregam informações sobre o valor de uma opção; características como a probabilidade de exercício e a trajetória até o vencimento dizem muito sobre como o mercado está influenciando pelo menos a ação-objeto. Como o valor extrínseco é uma variável bastante sensível, é importante que sua mensuração seja a mais detalhada o possível. Este trabalho buscou captar a geração instantânea de valor intrínseco avaliando as cotações \"tick by tick\" das ações e opções de compra da VALE5, PETR4, OGXP3, BVMF3 e ITUB4 durante os anos de 2011 e 2012. Foi possível demonstrar que mesmo uma mínima diferença de valor extrínseco captado possibilitou uma maior capacidade de explicação em diferentes modelos testados, comparando com a captação de final de dia (EOD). / The intrinsic value is one of the variables that most carry information on the value of a stock option. Characteristics such as exercise probability and path to maturity summarize how the market is working at least for the stock. As extrinsic value is very sensitive, it is important that its measurements happen in the most possible detailed way. This study focus on capture the instantaneous extrinsic value generation, analyzing \"tick by tick\" quotes of VALE5, PETR4, OGXP3, BVMF3 e ITUB4 stocks and related call options in the years of 2011 and 2012. It was possible to demonstrate that even a minimal difference of extrinsic value captured resulted in a better explanation capacity based on different tested models, in comparison with the extrinsic value captured by end of day data.
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Numerical methods for foreign exchange option pricing under hybrid stochastic and local volatility modelsCozma, Andrei January 2017 (has links)
In this thesis, we study the FX option pricing problem and put forward a 4-factor hybrid stochastic-local volatility model. The model, which describes the dynamics of an exchange rate, its volatility and the domestic and foreign short rates, allows for a perfect calibration to European options and has a good hedging performance. Due to the high-dimensionality of the problem, we propose a Monte Carlo simulation scheme that combines the full truncation Euler scheme for the stochastic volatility component and the stochastic short rates with the log-Euler scheme for the exchange rate. We analyze exponential integrability properties of Euler discretizations for the square-root process driving the stochastic volatility and the short rates, properties which play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a large class of stochastic differential equations in finance, including the ones studied in this thesis. Hence, we prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. Then, we calibrate the model to market data and discuss its fitness for pricing FX options. Next, due to the relatively slow convergence of the Monte Carlo method in the number of simulations, we examine a variance reduction technique obtained by mixing Monte Carlo and finite difference methods via conditioning. We consider a purely stochastic version of the model and price vanilla and exotic options by simulating the paths of the volatility and the short rates, and then evaluating the "inner" Black-Scholes-type expectation by means of a partial differential equation. We prove the convergence of numerical approximations and carry out a theoretical variance reduction analysis. Finally, we illustrate the efficiency of the method through a detailed quantitative assessment.
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A precificação de opções para processos de mistura de brownianos / Option pricing using mixture of Brownian motion processesKimura, Herbert 14 September 1998 (has links)
O estudo apresenta um modelo de precificação de derivativos financeiros baseado em processos de mistura de movimentos brownianos. A partir de uma modelagem probabilística, são apresentados ajustes ao modelo tradicional de Black-Scholes-Merton para contemplar situações em que o retorno do ativo-objeto não segue uma distribuição normal. O trabalho discute ainda um mecanismo de estimação de parâmetros da mistura de normais. O resultado da pesquisa possibilita a análise de preço de opções em situações mais gerais. / The study presents a model for pricing financial derivatives based on a mixture of Brownian motion processes. From a probabilistic modeling, the research focuses on adjustments to the traditional Black- Scholes- Merton model to address situations where the return of the underlying asset does not follow a normal distribution. The paper also discusses a mechanism to estimate parameters of a mixture of normal distributions. The result of the study allows an analysis of option price in more general situations.
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[en] GARCH OPTION PRICING MODEL VIA FILTERED HISTORICAL SIMULATION: AN APPLICATION ON THE BRAZILIAN MARKET / [pt] MODELO GARCH DE APREÇAMENTO DE OPÇÕES VIA SIMULAÇÃO HISTÓRICA FILTRADA: UMA APLICAÇÃO PARA O MERCADO BRASILEIRONAYARA LOPES GOMES 09 October 2012 (has links)
[pt] O modelo implementado neste trabalho, proposto em Barone-Adesi,
Engle e Mancini (2008), utiliza o método da Simulação Histórica Filtrada em
conjunto com a simulação de Monte Carlo para calibração de parâmetros de um
modelo GARCH a partir do qual opções do mercado brasileiro são apreçadas.
Os retornos da simulação são gerados a partir das inovações empíricas obtidas
no modelo GARCH assimétrico ajustado aos retornos diários das ações. Os
resultados obtidos apontam para ajustes satisfatórios dentro da amostra, quando
comparado ao modelo de Black E Scholes. No entanto, fora da amostra,
resultados similares foram verificados para ambos os modelos de apreçamento. / [en] The model implemented in this work, proposed by Barone-Adesi, Engle,
and Mancini (2008), applies the Filtered Historical Simulation method based on
Monte Carlo simulation to calibrate the parameters of a GARCH model in which
options from Brazilian market are priced. The simulated returns are generated
from empirical innovations obtained by an asymmetric GARCH model adjusted
for daily stock returns. The results suggest a satisfactory in-sample fit when
compared to the Black E Scholes model. However, similar results were observed
out-of-sample for both pricing models.
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[en] OPTION PRICING VIA NONPARAMETRIC ESSCHER TRANSFORM / [pt] APREÇAMENTO DE OPÇÕES VIA TRANSFORMADA DE ESSCHER NÃO PARAMÉTRICAMANOEL FRANCISCO DE SOUZA PEREIRA 01 March 2012 (has links)
[pt] O apreçamento de opções é um dos temas mais importantes da economia
financeira. Este estudo introduz uma versão não paramétrica da Transformada de
Esscher para o apreçamento neutro ao risco de opções financeiras. Os tradicionais
métodos paramétricos exigem a formulação de um modelo neutro ao risco
explícito e são operacionalmente apenas para poucas funções densidade de
probabilidade. Em nossa proposta, com simples suposições, evitamos a
necessidade da formulação de um modelo neutro ao risco para os retornos.
Primeiro, simulamos uma amostra de trajetórias de retornos sob a distribuição
original P. Então, baseado na Transformada de Esscher, a amostra é reponderada,
dando origem a uma amostra com risco neutralizado. Em seguida, os preços dos
derivativos são obtidos através de uma simples média dos payoffs de cada
trajetória da opção. Comparamos nossa proposta com alguns métodos de
apreçamento tradicionais, aplicando quatro exercícios em situações diferentes,
para destacar as diferenças e as semelhanças entre os métodos. Sob as mesmas
condições e em situações similares, o método proposto reproduz os resultados dos
métodos de apreçamento estabelecidos na literatura, o modelo de Black e Scholes
(1973) e o método de Duan (1995). Quando as condições são diferentes, o método
proposto indica que há mais risco do que outros métodos podem capturar. / [en] Option valuation is one of the most important topics in financial
economics. This study introduces a nonparametric version of the Esscher
transform for risk neutral option pricing. Traditional parametric methods require
the formulation of an explicit risk-neutral model and are operational only for a
few probability density functions. In our proposal, we make only mild
assumptions on the price kernel and there is no need for the formulation of the
risk-neutral model for the returns. First, we simulate sample paths for the returns
under the historical distribution P. Then, based on the Esscher transform, the
sample is reweighted, giving rise to a risk-neutralized sample from which
derivative prices can be obtained by a simple average of the pay-offs of the option
to each path. We compare our proposal with some traditional pricing methods,
applying four exercises under different situations, which seek to highlight the
differences and similarities between the methods. Under the same conditions and
in similar situations, the option pricing method proposed reproduces the results of
pricing methods fully established in the literature, the Black and Scholes [3]
model and the Duan [13] method. When the conditions are different, the results
show that the method proposed indicates that there is more risk than the other
methods can capture.
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Stochastic Volatility Models in Option PricingKalavrezos, Michail, Wennermo, Michael January 2008 (has links)
<p>In this thesis we have created a computer program in Java language which calculates European call- and put options with four different models based on the article The Pricing of Options on Assets with Stochastic Volatilities by John Hull and Alan White. Two of the models use stochastic volatility as an input. The paper describes the foundations of stochastic volatility option pricing and compares the output of the models. The model which better estimates the real option price is dependent on further research of the model parameters involved.</p>
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Essays on Gaussian Probability Laws with Stochastic Means and Variances : With Applications to Financial EconomicsEriksson, Anders January 2005 (has links)
<p>This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. </p><p>In the fourth paper a Lévy process with second order stochastic volatility is presented, option pricing under such a process is also considered.</p>
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