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Bieta, Volker, Udo Broll, and Wilfried Siebe. "Strategic option pricing." Technische Universität Dresden, 2020. https://tud.qucosa.de/id/qucosa%3A71719.
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In this paper an extension of the well-known binomial approach to option pricing is presented. The classical question is: What is the price of an option on the risky asset? The traditional answer is obtained with the help of a replicating portfolio by ruling out arbitrage. Instead a two-person game from the Nash equilibrium of which the option price can be derived is formulated. Consequently both the underlying asset’s price at expiration and the price of the option on this asset are endogenously determined. The option price derived this way turns out, however, to be identical to the classical no-arbitrage option price of the binomial model if the expiration-date prices of the underlying asset and the corresponding risk-neutral probability are properly adjusted according to the Nash equilibrium data of the game.
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劉伯文 and Pak-man Lau. "Option pricing: a survey." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 1994. http://hub.hku.hk/bib/B31977911.
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Gu, Chenchen. "Option Pricing Using MATLAB." Digital WPI, 2011. https://digitalcommons.wpi.edu/etd-theses/382.
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This paper describes methods for pricing European and American options. Monte Carlo simulation and control variates methods are employed to price call options. The binomial model is employed to price American put options. Using daily stock data I am able to compare the model price and market price and speculate as to the cause of difference. Lastly, I build a portfolio in an Interactive Brokers paper trading [1] account using the prices I calculate. This project was done a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.
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Lau, Pak-man. "Option pricing : a survey /." [Hong Kong : University of Hong Kong], 1994. http://sunzi.lib.hku.hk/hkuto/record.jsp?B14386057.
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Matsumoto, Manabu. "Options on portfolios of options and multivariate option pricing and hedging." Thesis, Imperial College London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.324627.
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Neset, Yngvild. "Spectral Discretizations of Option Pricing Models for European Put Options." Thesis, Norges teknisk-naturvitenskapelige universitet, Institutt for matematiske fag, 2014. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-26546.
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The aim of this thesis is to solve option pricing models efficiently by using spectral methods. The option pricing models that will be solved are the Black-Scholes model and Heston's stochastic volatility model. We will restrict us to pricing European put options. We derive the partial differential equations governing the two models and their corresponding weak formulations. The models are then solved using both the spectral Galerkin method and a polynomial collocation method. The numerical solutions are compared to the exact solution. The exact solution is also used to study the numerical convergence. We compare the results from the two numerical methods, and look at the time consumptions of the different methods. Analysis of the methods are also given. This includes coercivity, continuity, stability and convergence estimates.For Black-Scholes equation, we study both the original equation and the log transformed equation, and we also compare the results to a solution obtained by using a finite element method solver.
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Compiani, Vera. "Particle methods in option pricing." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13896/.
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Lo scopo di questa tesi è la calibrazione del modello di volatilità locale-stocastico (SLV) usando il metodo delle particelle. Il modello SLV riproduce il prezzo di un asset finanziario descritto da un processo stocastico. Il coefficiente di diffusione o volatilità del processo è costituito da una parte stocastica, la varianza, e da una parte locale chiamata funzione di leva che dipende dal processo stesso e che dà origine ad un'equazione differenziale alle derivate parziali (PDE) non lineare. La funzione di leva deve essere calibrata alla tipica curva che appare nella volatilità implicita dei dati di mercato, il volatility-smile. Per fa ciò si utilizza un metodo computazionale preso dalla fisica: il metodo delle particelle. Esso consiste nell'approssimare la distribuzione di probabilità del processo con una distribuzione empirica costituita da N particelle. Le N particelle consistono in N variabili aleatorie indipendenti e identicamente distribuite che seguono ciascuna l'equazione differenziale stocastica del prezzo con N moti Browniani indipendenti. La funzione di leva dipenderà così da una misura di probabilità casuale e la PDE non-lineare si ridurrà ad una PDE lineare con N gradi di libertà. Il risultato finale è una funzione di leva determinata dall'interazione tra tutte le particelle. La simulazione al computer viene eseguita tramite la tecnica di implementazione in parallelo che accelera i calcoli sfruttando l'architettura grafica della GPU.
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Belova, Anna, and Tamara Shmidt. "Meshfree methods in option pricing." Thesis, Högskolan i Halmstad, Sektionen för Informationsvetenskap, Data– och Elektroteknik (IDE), 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-16383.
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A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.
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Pour, Abdollah Farshchi Elham. "Option Pricing with Extreme Events." Thesis, Uppsala universitet, Analys och tillämpad matematik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-161963.
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Wiklund, Erik. "Asian Option Pricing and Volatility." Thesis, KTH, Matematisk statistik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-93714.
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Abstract An Asian option is a path-depending exotic option, which means that either the settlement price or the strike of the option is formed by some aggregation of underlying asset prices during the option lifetime. This thesis will focus on European style Arithmetic Asian options where the settlement price at maturity is formed by the arithmetic average price of the last seven days of the underlying asset. For this type of option it does not exist any closed form analytical formula for calculating the theoretical option value. There exist closed form approximation formulas for valuing this kind of option. One such, used in this thesis, approximate the value of an Arithmetic Asian option by conditioning the valuation on the geometric mean price. To evaluate the accuracy in this approximation and to see if it is possible to use the well known Black-Scholes formula for valuing Asian options, this thesis examines the bias between Monte-Carlo simulation pricing and these closed form approximate pricings. The bias examination is done for several different volatility schemes. In general the Asian approximation formula works very well for valuing Asian options. For volatility scenarios where there is a drastic volatility shift and the period with higher volatility is before the average period of the option, the Asian approximation formula will underestimate the option value. These underestimates are very significant for OTM options, decreases for ATM options and are small, although significant, for ITM options. The Black-Scholes formula will in general overestimate the Asian option value. This is expected since the Black-Scholes formula applies to standard European options which only, implicitly, considers the underlying asset price at maturity of the option as settlement price. This price is in average higher than the Asian option settlement price when the underlying asset price has a positive drift. However, for some volatility scenarios where there is a drastic volatility shift and the period with higher volatility is before the average period of the option, even the Black-Scholes formula will underestimate the option value. As for the Asian approximation formula, these over-and underestimates are very large for OTM options and decreases for ATM and ITM options.Sammanfattning En Asiatisk option är en vägberoende exotisk option, vilket betyder att antingen settlement-priset eller strike-priset beräknas utifrån någon form av aggregering av underliggande tillgångens priser under optionens livstid. Denna uppsats fokuserar på Aritmetiska Asiatiska optioner av Europeisk karaktär där settlement-priset vid lösen bestäms av det aritmetiska medelvärdet av underliggande tillgångens priser de sista sju dagarna. För denna typ av option finns det inga slutna analytiska formler för att beräkna optionens teoretiska värde. Det finns dock slutna approximativa formler för värdering av denna typ av optioner. En sådan, som används i denna uppsats, approximerar värdet av en Aritmetisk Asiatisk option genom att betinga värderingen på det geometriska medelpriset. För att utvärdera noggrannheten i denna approximation och för att se om det är möjligt att använda den väl kända Black-Scholes-formeln för att värdera Asiatiska optioner, så analyseras differenserna mellan Monte-Carlo-simulering och dessa slutna formlers värderingar i denna uppsats. Differenserna analyseras utifrån ett flertal olika scenarion för volatiliteten. I allmänhet så fungerar Asiatapproximationsformeln bra för värdering av Asiatiska optioner. För volatilitetsscenarion som innebär en drastisk volatilitetsförändring och där den perioden med högre volatilitet ligger innan optionens medelvärdesperiod, så undervärderar Asiatapproximationen optionens värde. Dessa undervärderingar är mycket påtagliga för OTM-optioner, avtar för ATM-optioner och är små, om än signifikanta, för ITM-optioner. Black-Scholes formel övervärderar i allmänhet Asiatiska optioners värde. Detta är väntat då Black-Scholes formel är ämnad för standard Europeiska optioner, vilka endast beaktar underliggande tillgångens pris vid optionens slutdatum som settlement-pris. Detta pris är i snitt högre än Asiatisk optioners settlement-pris när underliggande tillgångens pris har en positiv drift. Men, för vissa volatilitetsscenarion som innebär en drastisk volatilitetsförändring och där den perioden med högra volatilitet ligger innan optionens medelvärdesperiod, så undervärderar även Black-Scholes formel optionens värde. Som för Asiatapproximationen så är dessa över- och undervärderingar mycket påtagliga för OTM-optioner och avtar för ATM och ITM-optioner.
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Nisol, Gilles. "Option pricing with transaction costs." Thesis, KTH, Matematik (Inst.), 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-102780.
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Portfolio optimization is an important field of research within financial engineering. The aim of the optimization is to fins what is the best strategy for an investor when choosing how to allocate their money between a bank account and a constant number of risky assets. In our problem, the investor must pay transaction costs, meaning that every time he transfers money, he loses a certain percentage of the money transferred. Thus, we have made the assumption of proportional transaction costs. In a frictionless market, Merton has proven that the optimal policy consists of a constant proportion of wealth in the risky asset. This means that one must constantly rehedge the portfolio to keep this ratio constant regardless of the evolution of the risky asset´s value. When transaction costs are imposed, repeated rehedging becomes too expensive and the optimal policy of investment is different. The so-called transaction cost region will appear; the investor should buy, sell or stay idle depending on whether his position at current time is above, below or within this region. One can show that we can transform the portfolio optimization problem into a double obstacle problems. Using this latter form of the problem, we have created and algorithm unveiling the different transaction cost regions. The algorithm and results of this algorithm will be presented.
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Ross, David James. "Topics in American option pricing." Thesis, Imperial College London, 2004. http://hdl.handle.net/10044/1/8337.
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Panas, Vassilios Gerassimos. "Option pricing with transaction costs." Thesis, Imperial College London, 1993. http://hdl.handle.net/10044/1/7362.
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Chen, Miao. "Option pricing in incomplete markets." Thesis, University of Warwick, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.491472.
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The seminal paper of Black and Scholes (1973) led to the explosive growth of option pricing and hedging theory. However, the assumptions of the Black-Scholes model contradict reality. In the past three decades, a large volume ofresearch has been conducted on the problem of pricing and hedging contingent claims under more realistic assumptions. In particular, two streams of the . literature are directly related to this thesis. One is the development of stochastic volatility jump diffusion models and their option pricing formulas. The other is optimal hedging under market frictions. This thesis consists of four essays. The first two essays propose an affine stochastic volatility jump diffusion model for equity index. This is a rich model motivated by other empirical work. It includes two stochastic volatility factors, jumps in volatility process, and leverage effects. An option pricing formula is obtained by using the integral transform approach. Empirically, the model is nicely calibrated to the FTSEIOO index options data. Once the structural parameters are obtained, we examine the performance of several different calibration schemes as well as the dynamics ofthe state variables. The third and the fourth essays study the problem of optimal hedging of contingent claims in the presence of transactions costs. In the third essay, the market is described by pure diffusion. We introduce a local time analysis approach to this class of problem. This approach is new to the literature. It provides solutions that are consistent to the literature. More importantly, it is capable of providing deeper insights. The approach should stimulate further research. The fourth essay studies the optimal hedging problem with a jump diffusion market in the presence of transactions costs. The local time analysis is no longer appropriate because of the discontinuity in the stock price process, so we use a dynamic programming approach instead. The numerical results in particular extend our knowledge beyond the scope of the current literature. The essay is focused particularly on the impact ofjumps on the optimal hedging policy. Keywords: Option pricing, incomplete markets, stochastic volatility, jump diffusion, transactions costs, local time.
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Jönsson, Ola. "Option pricing and Bayesian learning /." Lund: Univ., Dep. of Economics, 2007. http://www.gbv.de/dms/zbw/541563130.pdf.
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Guichard, Regis Stephane Hubert. "Two topics in option pricing." Thesis, Imperial College London, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.312324.
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Whalley, A. E. "Option pricing with transaction costs." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298265.
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Chen, Kan. "Approximate methods for option pricing." Thesis, University of Strathclyde, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.501648.
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As the important achievement of Mathematical Finance and the fundamental result of pricing theory, the Black & Scholes model and analysis have profound influence and have been studied by many people. However, some biases of BS fl formula can be observed in real financial markets. People attribute this behavior to the fact that some simple assumptions of the BS model such as constant volatility are violated in practice and do many works to find some ways to improve and generalize it. This thesis will focus on discussing three key factors of pricing theory to modify BS model: pricing models, pricing rules, and approximation methods.
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Fei, Bingxin. "Computational Methods for Option Pricing." Digital WPI, 2011. https://digitalcommons.wpi.edu/etd-theses/381.
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This paper aims to practice the use of Monte Carlo methods to simulate stock prices in order to price European call options using control variates. American put options are priced using the binomial model separately. Finally, we use the information to form a portfolio position using an Interactive Brokers paper trading account.
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Marques, Catarina Neto. "Option pricing under variable volatility." Master's thesis, Instituto Superior de Economia e Gestão, 2017. http://hdl.handle.net/10400.5/14171.
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Mestrado em Mathematical FinanceA teoria de valorização de opções que conhecemos hoje em dia deu o seu maior passo quando Fischer Black e Myron Scholes escreveram um artigo com uma fórmula fechada que permitia calcular os preços de opções Europeias de compra e venda cujo subjacente é uma acção ou um índice. No entanto, evidências mostram que a fórmula anterior não funciona bem num grande número de situações reais: os preços estimados desviam-se significativamente dos preços de mercado. Isto deve-se às hipóteses, muito restritivas, em que o modelo está assente. Esta dissertação alarga o modelo de Black-Scholes a situações em que a volatilidade do preço do ativo é variável. As implicações desta extensão são estudadas tanto de um ponto de vista teórico como prático. Existem muitos modelos propostos para o caso em estudo e este trabalho foca-se nos modelos de volatilidade local porque mantêm as características mais importantes do modelo original. Foram selecionadas quatro funções diferentes para descrever a volatilidade do preço do ativo e um método de diferenças finitas foi implementado para obter estimações e previsões do preço de opções. Os resultados obtidos realmente indicam que os modelos de volatilidade local estimam melhor os preços das opções do que o modelo de Black-Scholes original.
The modern theory of option pricing gave its biggest step when Fischer Black and Myron Scholes wrote a paper with a closed form solution for the prices of European call and put options on a single stock or index. However, evidence shows that the former formula no longer holds in many real cases: the estimated prices deviate significantly from the market ones. This is due to the very restrictive assumptions on which the model is based. This dissertation extends the Black-Scholes model by making the volatility of the asset price variable. The implications of this extension are studied from both theoretical and practical points of view. Several models have been proposed and this work focuses on the local volatility models because they maintain the most important features of the classical model. Four different functions were selected to describe the volatility of the asset price and a finite difference method was implemented in order to obtain the estimations and predictions of the option prices. The results suggest that indeed local volatility models have a better performance than the classical Black-Scholes model in estimating option prices.
info:eu-repo/semantics/publishedVersion
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Timsina, Tirtha Prasad. "Sensitivities in Option Pricing Models." Diss., Virginia Tech, 2007. http://hdl.handle.net/10919/28904.
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The inverse problem in finance consists of determining the unknown parameters of
the pricing equation from the values quoted from the market. We formulate the
inverse problem as a minimization problem for an appropriate cost function to minimize
the difference between the solution of the model and the market observations.
Efficient gradient based optimization requires accurate gradient estimation of the
cost function. In this thesis we highlight the adjoint method for computing gradients
of the cost function in the context of gradient based optimization and show its
importance. We derive the continuous adjoint equations with appropriate boundary
conditions for three main option pricing models: the Black-Scholes model, the Hestonâ s
model and the jump diffusion model, for European type options. These adjoint
equations can be used to compute the gradient of the cost function accurately for
parameter estimation problems.
The adjoint method allows efficient evaluation of the gradient of a cost function F(¾)
with respect to parameters ¾ where F depends on ¾ indirectly, via an intermediate
variable. Compared to the finite difference method and the sensitivity equation
method, the adjoint equation method is very efficient in computing the gradient
of the cost function. The sensitivity equations method requires solving a PDE
corresponding to each parameter in the model to estimate the gradient of the cost
function. The adjoint method requires solving a single adjoint equation once. Hence,
for a large number of parameters in the model, the adjoint equation method is very
efficient.
Due to its nature, the adjoint equation has to be solved backward in time. The
adjoint equation derived from the jump diffusion model is harder to solve due to its
non local integral term. But algorithms that can be used to solve the Partial Integro-
Differential Equation (PIDE) derived from jump diffusion model can be modified to
solve the adjoint equation derived from the PIDE.Ph. D.
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Christoforidou, Amalia. "Regime-switching option pricing models." Thesis, University of Glasgow, 2015. http://theses.gla.ac.uk/6684/.
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Part I: This chapter develops a lattice method for option evaluation aiming to investigate whether the option prices reflect the shifts in the distributions of the underlying asset returns and the risk-free interest rate. More precisely we try to investigate whether the option prices reflect the switches in the correlation between the underlying and risk-free bond returns that characterise different states of the economy. For this reason we develop and test two models. In the first model we allow all the parameters to follow a regime-switching process while in the second model, in order to isolate the regime-switching correlation effect on the option prices, we allow only the correlation to follow a regime-switching process. The models developed use pentanomial lattices to represent the evolution of the regime-switching underlying assets. Our findings suggest that the option prices reflect the regime-switches and that a model which considers these switches could produce more accurate results than a single-regime model. Part II: This part develops a class of closed-form models for options on commodities evaluation under the assumptions of mean-reversion in the commodity prices and factors’ values and regime-switching in the volatilities and correlations. At first we develop novel closed-form solutions of the 1-, 2- and 3-factors models and later in the paper these three models are transformed into regime switching models. The six models (three with and three without regime-switching) are then tested and compared on real market data. Our findings suggest that the by increasing the stochastic factors and assuming regime-switching in the models their flexibility and thus their accuracy increases.
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Stafford, D. (Daniel). "Machine learning in option pricing." Master's thesis, University of Oulu, 2019. http://urn.fi/URN:NBN:fi:oulu-201901091016.
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This paper gives an overview of the research that has been conducted regarding neural networks in option pricing. The paper also analyzes whether a deep neural network model has advantages over the Black-Scholes option pricing model in terms of pricing and hedging European-style call options on the S&P 500 stock index with data ranging from 1996 to 2014.
While the previous literature has focused on shallow MLP-styled neural networks, this paper applies a deeper network structure of convolutional neural networks to the problem of pricing and hedging options. Convolutional neural networks are previously known for their success in image classification.
The first chapters of this paper focus on both introducing neural networks for a reader, who is not familiar with the topic a priori, as well as giving an overview of the relevant previous literature regarding neural networks in option pricing. The latter chapters present the empirical methodology and the empirical results.
The empirical study of this thesis focuses on comparing an option pricing model learned by a convolutional neural network to the Black-Scholes option pricing model. The comparison of the two pricing models is two-fold: the first part of the comparison focuses on pricing performance. Both models will be tested under a test set of data, computing error measures between the price predictions of each model against the true price of an option contract. The second part of the comparison focuses on hedging performance: both models will be used in a dynamic delta-hedging strategy to hedge an option position using the data that is available in the test set. The models are compared to each other using discounted mean absolute tracking error as a measure-of-fit. Bootstrapped confidence intervals are provided for all relevant performance measures.
The empirical results are in line with the previous literature in terms of pricing performance and show that a convolutional neural network is superior to the Black-Scholes option pricing model in all error measures. The pricing results also show that a convolutional neural network is better than neural networks in previous studies with superior performance in pricing accuracy also when the data is partitioned by moneyness and maturity. The empirical results are not in line with the previous literature in terms of hedging results and show that a convolutional neural network is inferior to the Black-Scholes option pricing model in terms of discounted mean absolute tracking error.
The main findings show that combining a neural network with a traditional parametric pricing formula gives the best possible outcome in terms of pricing and hedging options.
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Dokuchaev, Mikhail. "Numerical Methods for Option Pricing." Thesis, Curtin University, 2021. http://hdl.handle.net/20.500.11937/86211.
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The thesis studies numerical method for solving partial differential equations arising in financial modelling. More precisely, the thesis is focused on methods of solutions of parabolic equations with state dependent coefficients describing the fair price for European options and American options with parameters that depend on the state price.
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Preo, Alice <1990>. "Option Pricing with Genetic Programming." Master's Degree Thesis, Università Ca' Foscari Venezia, 2015. http://hdl.handle.net/10579/5981.
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In the last decades, an increasing attention has been directed to the application of Genetic Programming. This approach has been adopted to a wide number of fields, reporting successful results also in finance domain as automated computational programming tool. Starting from the theoretical research in Genetic Programming, the aim of this work is to focus on the actual implementation of this methodology to the financial field, especially on the prediction of derivative securities behavior. The attention will be centered on the option pricing and empirical tests will be carried out on market data. Proceeding from the analysis of already developed and qualified studies in the existing literature, this work examines developed models and the reached conclusions.
Further, the research will be amplified including the examination of the effect of different variables on the option price behavior, particularly after the variation of different parameters.
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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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Duan, Fangjing. "Option pricing models and volatility surfaces." St. Gallen, 2005. http://www.biblio.unisg.ch/org/biblio/edoc.nsf/wwwDisplayIdentifier/03607991001/$FILE/03607991001.pdf.
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D'Elia, Riccardo Giuseppe. "Deep Learning for American Option Pricing." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2019. http://amslaurea.unibo.it/19526/.
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American Option pricing efficiency is still on interesting topic in Computational Mathematichs and Finance. These types of derivatives offer great flexibility in all financial and trading markets due to the possibility of an early exercise. However, there is not any closed form analytical valuation of American option because of the optimal exercise problem created by the early exercise. This is the reason why the
problem is faced by a numerical approximation. Per contra, traditional numerical methods suffer from the curse of dimensionality. The classical approaches yield good results for up to 3 dimensional problem. To solve problems in higher dimensional space, Longstaff and Schwartz in [1] have proposed a regression based Monte Carlo method which approximates conditional expectation by projections on a finite set of basis function. The main problem of this method is that the number of paths required for convergence should grow exponentially with the number of basis functions. Consequently, the algorithm of Longstaff and Schwartz failed. During the last years, this problem has been treated through Artificial Intelligence tools. In [2], S. Becker, P. Cheridito and A. Jentzen have developed a Deep Learning method for optimal stopping problems that learns optimal stopping rule from Monte Carlo samples. By the virtue of this algorithm, they were able to achieve extraordinary results with a reasonable computational cost.
Therefore, in the first chapter of this thesis we introduce the problem and its formulation from a theoretical point of view. Then we examine in depth the least square regression method of Longstaff and Schwartz. Finally we present the "Deep Optimal Stopping" method by S. Becker, P. Cheridito and A. Jentzen.
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Kalavrezos, Michail, and Michael Wennermo. "Stochastic Volatility Models in Option Pricing." Thesis, Mälardalen University, Department of Mathematics and Physics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-538.
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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.
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Sherwani, Yasir. "Binomial approximation methods for option pricing." Thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120639.
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Lei, Ngai Heng. "Martingale method in option pricing theory." Thesis, University of Macau, 2003. http://umaclib3.umac.mo/record=b1447303.
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Inkaya, Alper. "Option Pricing With Fractional Brownian Motion." Master's thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613736/index.pdf.
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Traditional financial modeling is based on semimartingale processes with stationary and independent
increments. However, empirical investigations on financial data does not always
support these assumptions. This contradiction showed that there is a need for new stochastic
models. Fractional Brownian motion (fBm) was proposed as one of these models by Benoit
Mandelbrot. FBm is the only continuous Gaussian process with dependent increments. Correlation
between increments of a fBm changes according to its self-similarity parameter H. This
property of fBm helps to capture the correlation dynamics of the data and consequently obtain
better forecast results. But for values of H different than 1/2, fBm is not a semimartingale and
classical Ito formula does not exist in that case. This gives rise to need for using the white noise
theory to construct integrals with respect to fBm and obtain fractional Ito formulas. In this
thesis, the representation of fBm and its fundamental properties are examined. Construction of
Wick-Ito-Skorohod (WIS) and fractional WIS integrals are investigated. An Ito type formula
and Girsanov type theorems are stated. The financial applications of fBm are mentioned and
the Black&Scholes price of a European call option on an asset which is assumed to follow a geometric fBm is derived. The statistical aspects of fBm are investigated. Estimators for the self-similarity parameter H and simulation methods of fBm are summarized. Using the R/S methodology of Hurst, the estimations of the parameter H are obtained and these values are used to evaluate the fractional Black&
Scholes prices of a European call option with different maturities. Afterwards, these values are compared to Black&
Scholes price of the same option to demonstrate the effect of long-range dependence on the option prices. Also, estimations of H at different time scales are obtained to investigate the multiscaling in financial data. An outlook of the future work is given.
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Sheng, Gong. "Filtered historical simulation and option pricing." Thesis, Uppsala universitet, Analys och tillämpad matematik, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-154743.
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Demin, Mikhail. "Finite Volume Methods for Option Pricing." Thesis, Högskolan i Halmstad, Tillämpad matematik och fysik (MPE-lab), 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:hh:diva-16397.
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Kazantzaki, Savina. "Aspects of exotic option pricing theory." Thesis, Imperial College London, 2006. http://hdl.handle.net/10044/1/11787.
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Jahandideh, Mohammad Taghi. "Option pricing for infinite variance data." Thesis, University of Ottawa (Canada), 2004. http://hdl.handle.net/10393/26665.
Full textAbstract:
Infinite variance distributions are among the competing models used to explain the non-normality of stock price changes (Mandelbrot, 1963; Fama, 1965; Mandelbrot and Taylor, 1967; Rachev and Samorodnitsky, 1993). We investigate the asymptotic option price formula in infinite variance setting for both independent and correlated data using point processes. As we shall see the application of point process models can also lead us to investigate a more general option price formula. We also apply a recursion technique to quantify various characteristics of the resulting formulas. It shows that such formulas, and even their approximations, may be difficult to apply in practice. A nonparametric bootstrap method is proposed as one alternative approach and its asymptotic consistency is established under a resampling scheme of m = o(n). Some empirical evidence is provided showing the method works in principle, although large sample sizes appear to be needed for accuracy. This method is also illustrated using publicly available financial data.
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Cartea, A. I. G. "Option pricing with Levy-Stable processes." Thesis, University of Oxford, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.270341.
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Nordström, Walter. "Adaptive tree techniques in option pricing." Thesis, KTH, Numerisk analys, NA, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-167979.
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When pricing american option with discrete cash dividends standard tree techniques are insufficient. J. W. Nieuwenhuis and M. H. Vellekoop have presented a new tree technique involving interpolation to solve the problem. At ORC it has been observed that when using an adaptive mesh to increase the resolution of the tree around the dividends the speed of convergence is improved n this paper we isolate the sources of errors in the tree model and explain why the adaptive mesh has a good effect. Using that knowledge we further improve the algorithm. We found that we could both improve the accuracy and reduce execution time for the algorithm.När man prissätter amerikanska optioner där underliggande har diskreta utdelningar så kan vanliga trädmodeller vara otillräckliga. J. W. Nieuwenhuis och M. H. Vellekoop har hittat en ny trädmetod där interpolation används för att lösa de problem som uppstår. På ORC har man upptäckt att genom att använda ett adaptivt träd för att öka punkttätheten i trädet kring utdelningarna så kan man få snabbare konvergens av optionspriset. I detta arbete undersöker vi det adaptiva trädets effekter på prissättningsalgoritmen och isolerar olika felkällor. Vi använder den kunskapen till att effektivisera algoritmen för optimal noggrannhet och prestanda.
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Shen, Liya. "Option pricing with the wavelet method." Thesis, University of Essex, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.437680.
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PEREIRA, MANOEL FRANCISCO DE SOUZA. "OPTION PRICING VIA NONPARAMETRIC ESSCHER TRANSFORM." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2011. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=19219@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIORO 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.
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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Hao, Wenyan. "Quantum mechanics approach to option pricing." Thesis, University of Leicester, 2018. http://hdl.handle.net/2381/43020.
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Options are financial derivatives on an underlying security. The Schrodinger and Heisenberg approach to the quantum mechanics together with the Dirac matrix approaches are applied to derive the Black-Scholes formula and the quantum Cox- Rubinstein formula. The quantum mechanics approach to option pricing is based on the interpretation of the option price as the Schrodinger wave function of a certain quantum mechanics model determined by Hamiltonian H. We apply this approach to continuous time market models generated by Levy processes. In the discrete time formulization, we construct both self-adjoint and non selfadjoint quantum market. Moreover, we apply the discrete time formulization and analyse the quantum version of the Cox-Ross-Rubinstein Binomial Model. We find the limit of the N-period bond market, which convergences to planar Brownian motion and then we made an application to option pricing in planar Brownian motion compared with Levy models by Fourier techniques and Monte Carlo method. Furthermore, we analyse the quantum conditional option price and compare for the conditional option pricing in the quantum formulization. Additionally, we establish the limit of the spectral measures proving the convergence to the geometric Brownian motion model. Finally, we found Binomial Model formula and Path integral formulization gave are close to the Black-Scholes formula.
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Wang, Junxiong. "Option Pricing Using Monte Carlo Methods." Digital WPI, 2011. https://digitalcommons.wpi.edu/etd-theses/331.
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This project is devoted primarily to the use of Monte Carlo methods to simulate stock prices in order to price European call options using control variates, and to the use of the binominal model to price American put options. At the end, we can use the information to form a portfolio position using an Interactive Brokers paper trading account. This project was done as a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.
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Lu, Mengliu. "Option Pricing Using Monte Carlo Methods." Digital WPI, 2011. https://digitalcommons.wpi.edu/etd-theses/380.
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This paper aims to use Monte Carlo methods to price American call options on equities using the variance reduction technique of control variates and to price American put options using the binomial model. We use this information to form option positions. This project was done a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.
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LUO, SHAN-MING, and 羅善明. "The Comparison of BS Option Pricing Model and GARCH Option Pricing Model in Index Options." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/58041960664220628803.
Full textAbstract:
碩士國立臺北大學
統計學系
98
Options have been playing an important role in real financial market since the first option had traded. However, it is a major subject that how the rational price of options had been made. Black & Scholes (1973) set up the landmark of option pricing after they proposed the famous Black-Scholes option pricing model. In practice, one of the assumptions made in Black-Scholes (BS) option pricing model, namely volatility is a fixed constant, isn’t in accordance with the practices in real world. Scholars afterward try to release the assumption made by Black-Scholes and proposed so called stochastic volatility model which can categorized in discrete- and continuous-time model. Among discrete-time models, Duan (1995) introduced the model in quantitative economics, Generalized Autoregressive Conditional Heteroskedasticity (GARCH), to amend the assumption that volatility in Black-Scholes option pricing model is a fixed constant. Like most other option pricing models, the closed-form solutions do not exist. Heston & Nandi (2000) proposed a GARCH option pricing model with specific closed-form solution that can be directly derived by using numerical integration techniques. This research attempted to check out how the Heston & Nandi (2000) GARCH option pricing model perform in Taiwan Stock Exchange Capitalization Weighted Stock Index options (TXO) based on Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and compared with BS option pricing model in option mispricing. The results show that it is outperformed by BS option pricing model both in in-sample and out-of-sample valuation though Heston & Nandi (2000) released the assumption mentioned above. The significant mispricing could be caused by different data definitions. On the other hand, like Heston & Nandi (2000), the significant mispricing may also caused by the poor estimates of the parameters in the model.
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Yun, Wen Chen, and 陳韻文. "Pricing of Barrier Option ─ Using GARCH Option Pricing Model." Thesis, 2004. http://ndltd.ncl.edu.tw/handle/05072525095887761139.
Full textAbstract:
碩士世新大學
財務金融學系
92
Warrant is a kind of option. The call warrant have been sold in Taiwan in 1997. We also trade stock option after January 1st.2003. Because of trading option, we can expect that exotic option will be an important product in the market of Taiwan. Generally speaking, the way to pricing option are Black-Scholes model and numerical analysis. There are many different kinds of numerical analysis. For example, there are Binomial model and Monte Carlo Simulation. This thesis will pricing up-barrier warrant by using the trinomial GARCH model which creating by Ritchken & Trevor（1999）. Besides, this thesis will also compare the result which be calculated by Black-Scholes model, Binomial model, and Trinomial model.
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"Option pricing theory." Chinese University of Hong Kong, 1993. http://library.cuhk.edu.hk/record=b5887791.
Full textAbstract:
by Ka-kit Chan.Thesis (M.Phil.)--Chinese University of Hong Kong, 1993.
Includes bibliographical references (leaves 71-73).
Chapter I. --- Introduction to Stochastic Calculus --- p.1
Stochastic Processes --- p.2
Stochastic Integration --- p.6
Quadratic Variation Processes and Mutual Variation Process --- p.11
The Ito Formula --- p.13
Girsanov's Theorem --- p.16
Stochastic Differential Equations --- p.18
Chapter II. --- Pricing American Equity Options --- p.21
A Representation Formula for European Put Option --- p.22
The Free Boundary Formulation of American Put Option --- p.24
A Representation Formula for American Put Option --- p.27
An Alternative Representation Formula for American Put Option --- p.35
The Optimal Exercise Boundary --- p.37
Numerical Valuations of the Representation Formulae --- p.39
Chapter III. --- The Effects of Margin Requirements on Option Prices --- p.42
Pricing European Options --- p.44
Pricing American Options --- p.46
Chapter IV. --- General Pricing Theory --- p.49
Transformations of Price Processes --- p.50
No Arbitrage Condition and Completeness of Market --- p.52
More on Market Completeness --- p.58
Term Structure of Interest Rate and Interest Rate Options --- p.61
Pricing Equity Options --- p.67
Bibliography --- p.71
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Cheng, Tsun-Hung, and 鄭圳宏. "Deep Learning for Option Pricing Using TAIEX Options." Thesis, 2019. http://ndltd.ncl.edu.tw/handle/ff87uk.
Full textAbstract:
碩士國立臺北商業大學
財務金融系研究所
107
This paper explores the pricing of TAIEX put options with deep learning. We first choose the contracts which volume over 1000 lots to avoid the problem of liquidity, then convert call premiums with high strike price to those of put by using the put-call parity. Moreover, we use TAIEX future, historical volatility, and 30-days money market rate as underlying, volatility, and risk-free rate respectively. The inputs of deep learning include the underlying, strike price, volatility, time to maturity, and risk-free rate, as those in the Black-Scholes-Merton Framework. The results show that the pricing error of deep learning is small than that of Black-Scholes-Merton model for the at-the-money and out-of-the money contracts, based on the criteria of mean square error and mean absolute error. However, the result is opposite for the in-the-money contract. The summary is that the performance of deep learning is slightly better than that of Black-Scholes-Merton model for all contracts.
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Huang, Teng-Ching, and 黃騰進. "The Pricing Performance of Markov Chain Option Pricing Algorithm on Barrier Options." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/c4e6v4.
Full textAbstract:
碩士銘傳大學
財務金融學系碩士班
93
In this study we adapt the numerical method of Markov chain option pricing algorithm to price barrier warrants in Taiwan and compare it with the method of Rubinstein and Reiner（1991）. In theory Markov chain is more accurate and flexible , and the empirical results show that Markov chain option pricing algorithm has lower pricing errors . Finally , by this study we also find that issuers will set a volatility markup when they issue warrants , and the volatility markup is around 15％~25％.
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Sun, Ya-ling, and 孫雅玲. "Pricing double barrier option." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/57516550309939540038.
Full textAbstract:
碩士國立成功大學
數學系應用數學碩博士班
95
This study, a boundary element method is designed to deal with double touch barrier options with rebate. That is, a dobule touch barrier call is knocked out when the price reaches the upper or lower barreir before expiration day, but the seller have to pay rebate to buyer. In this thesis, first the pricing of option、barrier option and double barrier option are reviewed. The black-scholes model is converted to a heat equation boundary value problem. Then use a boundary element method to deal with the bounday value problem and solve the price of continuous double barrier options with rebate. Finally,some sensitive analysis are present.
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Yu, Shang-En, and 余尚恩. "Fuzzy Option Pricing Model." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/13974811463039935724.
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碩士朝陽科技大學
財務金融系碩士班
90
Option is a tool that investors often use to arbitrage or hedge. However, either Black-Scholes model or CRR model can only provide a theoretical reference value. This paper applies fuzzy set to the CRR model. It is expected that the fuzzy volatility, instead of the crisp one as in conventional CRR model, can provide reasonable ranges and corresponding memberships of option prices. As a result, investors with various risk preferences can interpret the optimal differently.