This thesis introduces a novel theoretical option pricing ensemble framework to improve the bias and variance of option pricing models, especially those based on Levy Processes. In particular, we present a completely new, yet very general theoretical framework to calibrate and combine several option pricing models using ensemble methods. This framework has four main steps: general option pricing tasks, ensemble generation, ensemble pruning and ensemble integration. The modularity allows for a exible implementation in terms of asset classes, base models, pricing techniques and ensemble architecture.

Mathematical and statistical modeling have been at the forefront of many significant advances in many disciplines in both the academic and industry sectors. From behavioral sciences to hard core quantum mechanics in physics, mathematical modeling has made a compelling argument for its usefulness and its necessity in advancing the current state of knowledge in the 21rst century. In Finance and Insurance in particular, stochastic modeling has proven to be an effective approach in accomplishing a vast array of tasks: risk management, leveraging of investments, prediction, hedging, pricing, insurance, and so on. However, the magnitude of the damage incurred in recent market crisis of 1929 (the great depression), 1937 (recession triggered by lingering fears emanating from the great depression), 1990 (one year recession following a decade of steady expansion) and 2007 (the great recession triggered by the sub-prime mortgage crisis) has suggested that there are certain aspects of financial markets not accounted for in existing modeling. Explanations have abounded as to why the market underwent such deep crisis and how to account for regime change risk. One such explanation brought forth was the existence of regimes in the financial markets. The basic idea of market regimes underscored the principle that the market was intrinsically subjected to many different states and can switch from one state to another under unknown and uncertain internal and external perturbations. Implementation of such a theory has been done in the simplifying case of Markov regimes. The mathematical simplicity of the Markovian regime model allows for semi-closed or closed form solutions in most financial applications while it also allows for economically interpretable parameters. However, there is a hefty price to be paid for such practical conveniences as many assumptions made on the market behavior are quite unreasonable and restrictive. One assumes for instance that each market regime has a constant propensity of switching to any other state irrespective of the age of the current state. One also assumes that there are no intermediate states as regime changes occur in a discrete manner from one of the finite states to another. There is therefore no telling how meaningful or reliable interpretation of parameters in Markov regime models are. In this thesis, we introduced a sound theoretical and analytic framework for Levy driven linear stochastic models under a semi Markov market regime switching process and derived It\'o formula for a general linear semi Markov switching model generated by a class of Levy It'o processes (1). It'o formula results in two important byproducts, namely semi closed form formulas for the characteristic function of log prices and a linear combination of duration times (2). Unlike Markov markets, the introduction of semi Markov markets allows a time varying propensity of regime change through the conditional intensity matrix. This is more in line with the notion that the market's chances of recovery (respectively, of crisis) are affected by the recession's age (respectively, recovery's age). Such a change is consistent with the notion that for instance, the longer the market is mired into a recession, the more improbable a fast recovery as the the market is more likely to either worsens or undergo a slow recovery. Another interesting consequence of the time dependence of the conditional intensity matrix is the interpretation of semi Markov regimes as a pseudo-infinite market regimes models. Although semi Markov regime assume a finite number of states, we note that while in any give regime, the market does not stay the same but goes through an infinite number of changes through its propensity of switching to other regimes. Each of those separate intermediate states endows the market with a structure of pseudo-infinite regimes which is an answer to the long standing problem of modeling market regime with infinitely many regimes. We developed a version of Girsanov theorem specific to semi Markov regime switching stochastic models, and this is a crucial contribution in relating the risk neutral parameters to the historical parameters (3). Given that Levy driven markets and regime switching markets are incomplete, there are more than one risk neutral measures that one can use for pricing derivative contracts. Although much work has been done about optimal choice of the pricing measure, two of them jump out of the current literature: the minimal martingale measure and the minimum entropy martingale measure. We first presented a general version of Girsanov theorem explicitly accounting for semi Markov regime. Then we presented Siu and Yang pricing kernel. In addition, we developed the conditional and unconditional minimum entropy martingale measure which minimized the dissimilarity between the historical and risk neutral probability measures through a version of Kulbach Leibler distance (4). Estimation of a European option price in a semi Markov market has been attempted before in the restricted case of the Black Scholes model. The problems encountered then were twofold: First, the author employed a Markov chain Monte Carlo methods which relied much on the tractability of the likelihood function of the normal random sequences. This tractability is unavailable for most Levy processes, hence the necessity of alternative pricing methods is essential. Second, the accuracy of the parameter estimates required tens of thousands of simulations as it is often the case with Metropolis Hasting algorithms with considerable CPU time demand. Both above outlined issues are resolved by the development of a semi-closed form expression of the characteristic function of log asset prices, and it opened the door to a Fourier transform method which is derived on the heels of Carr and Madan algorithm and the Fourier time stepping algorithm (5). A round of simulations and calibrations is performed to better capture the performance of the semi Markov model as opposed to Markov regime models. We establish through simulations that semi Markov parameters and the backward recurrence time have a substantial effect on option prices ( 6). Differences between Markov and Semi Markov market calibrations are quantified and the CPU times are reported. More importantly, interpretation of risk neutral semi Markov parameters offer more insight into the dynamic of market regimes than Markov market regime models ( 7). This has been systematically exhibited in this work as calibration results obtained from a set of European vanilla call options led to estimates of the shape and scale parameters of the Weibull distribution considered, offering a deeper view of the current market state as they determine the in-regime dynamic crucial to determining where the market is headed. After introducing semi Markov models through linear Levy driven models, we consider semi Markov markets with nonlinear multidimensional coupled asset price processes (8). We establish that the tractability of linear semi Markov market models carries over to multidimensional nonlinear asset price models. Estimating equations and pricing formula are derived for historical parameters and risk neutral parameters respectively (9). The particular case of basket of commodities is explored and we provide calibration formula of the model parameters to observed historical commodity prices through the LLGMM method. We also study the case of Heston model in a semi Markov switching market where only one parameter is subjected to semi Markov regime changes. Heston model is one the most popular model in option pricing as it reproduces many more stylized facts than Black Scholes model while retaining tractability. However, in addition to having a faster deceasing smiles than observed, one of the most damning shortcomings of most diffusion models such as Heston model, is their inability to accurately reproduce short term options prices. An avenue for solving these issues consists in generalizing Heston to account for semi Markov market regimes. Such a solution is implemented and a semi analytic formula for options is obtained.

In this thesis, we study the estimation of the nonlinear conditionalrisk-neutral density function (RND) in discrete time. Specifically, weevaluate the extent to which the estimated nonlinear conditional RNDvaluable insights to answer relevant economic questions regarding to optionpricing, the measurement of invertors' preferences and portfolio choice.We make use of large dataset of options contracts written on the S&P 500index from 1996 to 2011, to estimate the parameters of the conditional RNDfunctions by minimizing the squared option pricing errors delivered by thenonlinear models studied in the thesis.In the first essay, we show that a semi-nonparametric option pricing modelwith GARCH variance outperforms several benchmarks models in-sample andout-of-sample. In the second essay, we show that a simple two-state regimeswitching model in volatility is not able to fully account for the pricingkernel and the risk aversion puzzle; however, it provides a reasonablecharacterisation of the time-series properties of the estimated riskaversion.In the third essay, we evaluate linear stochastic discount factormodels using an out-of-sample financial metric. We find that multifactormodels outperform the CAPM when this metric is used, and that modelsproducing the best fit in-sample are also those exhibiting the bestperformance out-of-sample.

[EN] In the stock markets, the process of estimating a fair price for a stock, option or commodity is consider the corner stone for this trade. There are several attempts to obtain a suitable mathematical model in order to enhance the estimation process for evaluating the options for short or long periods. The Black-Scholes partial differential equation (PDE) and its analytical solution, 1973, are considered a breakthrough in the mathematical modeling for the stock markets. Because of the ideal assumptions of Black-Scholes several alternatives have been developed to adequate the models to the real markets. Two strategies have been done to capture these behaviors; the first modification is to add jumps into the asset following Lévy processes, leading to a partial integro-differential equation (PIDE); the second is to allow the volatility to evolve stochastically leading to a PDE with two spatial variables. Here in this work, we solve numerically PIDEs for a wide class of Lévy processes using finite difference schemes for European options and also, the associated linear complementarity problem (LCP) for American option. Moreover, the models for options under stochastic volatility incorporated with jump-diffusion are considered. Numerical analysis for the proposed schemes is studied since it is the efficient and practical way to guarantee the convergence and accuracy of numerical solutions. In fact, without numerical analysis, careless computations may waste good mathematical models. This thesis consists of four chapters; the first chapter is an introduction containing historically review for stochastic processes, Black-Scholes equation and preliminaries on numerical analysis. Chapter two is devoted to solve the PIDE for European option under CGMY process. The PIDE for this model is solved numerically using two distinct discretization approximations; the first approximation guarantees unconditionally consistency while the second approximation provides unconditional positivity and stability. In the first approximation, the differential part is approximated using the explicit scheme and the integral part is approximated using the trapezoidal rule. In the second approximation, the differential part is approximated using the Patankar-scheme and the integral part is approximated using the four-point open type formula. Chapter three provides a unified treatment for European and American options under a wide class of Lévy processes as CGMY, Meixner and Generalized Hyperbolic. First, the reaction and convection terms of the differential part of the PIDE are removed using appropriate mathematical transformation. The differential part for European case is explicitly discretized , while the integral part is approximated using Laguerre-Gauss quadrature formula. Numerical properties such as positivity, stability and consistency for this scheme are studied. For the American case, the differential part of the LCP is discretized using a three-time level approximation with the same integration technique. Next, the Projected successive over relaxation and multigrid techniques have been implemented to obtain the numerical solution. Several numerical examples are given including discussion of the errors and computational cost. Finally in Chapter four, the PIDE for European option under Bates model is considered. Bates model combines both stochastic volatility and jump diffusion approaches resulting in a PIDE with a mixed derivative term. Since the presence of cross derivative terms involves the existence of negative coefficient terms in the numerical scheme deteriorating the quality of the numerical solution, the mixed derivative is eliminated using suitable mathematical transformation. The new PIDE is solved numerically and the numerical analysis is provided. Moreover, the LCP for American option under Bates model is studied. / [ES] El proceso de estimación del precio de una acción, opción u otro derivado en los mercados de valores es objeto clave de estudio de las matemáticas financieras. Se pueden encontrar diversas técnicas para obtener un modelo matemático adecuado con el fin de mejorar el proceso de valoración de las opciones para periodos cortos o largos. Históricamente, la ecuación de Black-Scholes (1973) fue un gran avance en la elaboración de modelos matemáticos para los mercados de valores. Es un modelo práctico para estimar el valor razonable de una opción. Sobre unos supuestos determinados, F. Black y M. Scholes obtuvieron una ecuación diferencial parcial lineal y su solución analítica. Desde entonces se han desarrollado modelos más complejos para adecuarse a la realidad de los mercados. Un tipo son los modelos con volatilidad estocástica que vienen descritos por una ecuación en derivadas parciales con dos variables espaciales. Otro enfoque consiste en añadir saltos en el precio del subyacente por medio de modelos de Lévy lo que lleva a resolver una ecuación integro-diferencial parcial (EIDP). En esta memoria se aborda la resolución numérica de una amplia clase de modelos con procesos de Lévy. Se desarrollan esquemas en diferencias finitas para opciones europeas y también para opciones americanas con su problema de complementariedad lineal (PCL) asociado. Además se tratan modelos con volatilidad estocástica incorporando difusión con saltos. Se plantea el análisis numérico ya que es el camino eficiente y práctico para garantizar la convergencia y precisión de las soluciones numéricas. De hecho, la ausencia de análisis numérico debilita un buen modelo matemático. Esta memoria está organizada en cuatro capítulos. El primero es una introducción con un breve repaso de los procesos estocásticos, el modelo de Black-Scholes así como nociones preliminares de análisis numérico. En el segundo capítulo se trata la EIDP para las opciones europeas según el modelo CGMY. Se proponen dos esquemas en diferencias finitas; el primero garantiza consistencia incondicional de la solución mientras que el segundo proporciona estabilidad y positividad incondicionales. Con el primer enfoque, la parte diferencial se discretiza por medio de un esquema explícito y para la parte integral se usa la regla del trapecio. En la segunda aproximación, para la parte diferencial se usa un esquema tipo Patankar y la parte integral se aproxima por medio de la fórmula de tipo abierto con cuatro puntos. En el capítulo tercero se propone un tratamiento unificado para una amplia clase de modelos de opciones en procesos de Lévy como CGMY, Meixner e hiperbólico generalizado. Se eliminan los términos de reacción y convección por medio de un apropiado cambio de variables. Después la parte diferencial se aproxima por un esquema explícito mientras que para la parte integral se usa la fórmula de cuadratura de Laguerre-Gauss. Se analizan positividad, estabilidad y consistencia. Para las opciones americanas, la parte diferencial del LCP se discretiza con tres niveles temporales mediante cuadratura de Laguerre-Gauss para la integración numérica. Finalmente se implementan métodos iterativos de proyección y relajación sucesiva y la técnica de multimalla. Se muestran varios ejemplos incluyendo estudio de errores y coste computacional. El capítulo 4 está dedicado al modelo de Bates que combina los enfoques de volatilidad estocástica y de difusión con saltos derivando en una EIDP con un término con derivadas cruzadas. Ya que la discretización de una derivada cruzada comporta la existencia de coeficientes negativos en el esquema que deterioran la calidad de la solución numérica, se propone un cambio de variables que elimina dicha derivada cruzada. La EIDP transformada se resuelve numéricamente y se muestra el análisis numérico. Por otra parte se estudia el LCP para opciones americanas con el modelo de Bates. / [CAT] El procés d'estimació del preu d'una acció, opció o un altre derivat en els mercats de valors és objecte clau d'estudi de les matemàtiques financeres . Es poden trobar diverses tècniques per a obtindre un model matemàtic adequat a fi de millorar el procés de valoració de les opcions per a períodes curts o llargs. Històricament, l'equació Black-Scholes (1973) va ser un gran avanç en l'elaboració de models matemàtics per als mercats de valors. És un model matemàtic pràctic per a estimar un valor raonable per a una opció. Sobre uns suposats F. Black i M. Scholes van obtindre una equació diferencial parcial lineal amb solució analítica. Des de llavors s'han desenrotllat models més complexos per a adequar-se a la realitat dels mercats. Un tipus és els models amb volatilitat estocástica que ve descrits per una equació en derivades parcials amb dos variables espacials. Un altre enfocament consistix a afegir bots en el preu del subjacent per mitjà de models de Lévy el que porta a resoldre una equació integre-diferencial parcial (EIDP) . En esta memòria s'aborda la resolució numèrica d'una àmplia classe de models baix processos de Lévy. Es desenrotllen esquemes en diferències finites per a opcions europees i també per a opcions americanes amb el seu problema de complementarietat lineal (PCL) associat. A més es tracten models amb volatilitat estocástica incorporant difusió amb bots. Es planteja l'anàlisi numèrica ja que és el camí eficient i pràctic per a garantir la convergència i precisió de les solucions numèriques. De fet, l'absència d'anàlisi numèrica debilita un bon model matemàtic. Esta memòria està organitzada en quatre capítols. El primer és una introducció amb un breu repàs dels processos estocásticos, el model de Black-Scholes així com nocions preliminars d'anàlisi numèrica. En el segon capítol es tracta l'EIDP per a les opcions europees segons el model CGMY. Es proposen dos esquemes en diferències finites; el primer garantix consistència incondicional de la solució mentres que el segon proporciona estabilitat i positivitat incondicionals. Amb el primer enfocament, la part diferencial es discretiza per mitjà d'un esquema explícit i per a la part integral s'empra la regla del trapezi. En la segona aproximació, per a la part diferencial s'usa l'esquema tipus Patankar i la part integral s'aproxima per mitjà de la fórmula de tipus obert amb quatre punts. En el capítol tercer es proposa un tractament unificat per a una àmplia classe de models d'opcions en processos de Lévy com ara CGMY, Meixner i hiperbòlic generalitzat. S'eliminen els termes de reacció i convecció per mitjà d'un apropiat canvi de variables. Després la part diferencial s'aproxima per un esquema explícit mentres que per a la part integral s'usa la fórmula de quadratura de Laguerre-Gauss. S'analitzen positivitat, estabilitat i consistència. Per a les opcions americanes, la part diferencial del LCP es discretiza amb tres nivells temporals amb quadratura de Laguerre-Gauss per a la integració numèrica. Finalment s'implementen mètodes iteratius de projecció i relaxació successiva i la tècnica de multimalla. Es mostren diversos exemples incloent estudi d'errors i cost computacional. El capítol 4 està dedicat al model de Bates que combina els enfocaments de volatilitat estocástica i de difusió amb bots derivant en una EIDP amb un terme amb derivades croades. Ja que la discretización d'una derivada croada comporta l'existència de coeficients negatius en l'esquema que deterioren la qualitat de la solució numèrica, es proposa un canvi de variables que elimina dita derivada croada. La EIDP transformada es resol numèricament i es mostra l'anàlisi numèrica. D'altra banda s'estudia el LCP per a opcions americanes en el model de Bates. / El-Fakharany, MMR. (2015). Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917 / TESIS

[pt] Esta dissertação visa analisar como o modelo de apreçamento de opções, utilizando o conceito de árvore trinomial implícita, pode ser aplicado no mercado acionário brasileiro, com resultados mais consistentes, se comparado ao modelo de Black-Scholes. Esse modelo incorpora o conceito de volatilidade implícita, sendo consideradas as expectativas futuras em relação ao preço de um ativo. A volatilidade implícita apresenta diferentes valores para diferentes preços de exercício ao longo do tempo. A denominação sorriso de volatilidade deve-se ao formato da curva da volatilidade implícita em função do preço de exercício. O formato do sorriso varia de acordo com o ativo-objeto da opção. Assim, a volatilidade varia ao longo tempo no cálculo da árvore, pois leva em considerando as oscilações do mercado, o que, conseqüentemente, impacta no preço do ativo e sua opção. / [en] This Paper aims to analyze how the option pricing model, using the concept of Implied Trinomial Trees can be applied to the Brazilian stock market, achieving more accurate results, if compared to the Black-Scholes model. This model includes the Implied Volatility concept, which means that future expectations are considered to price an asset. It presents different values for different Strike Prices through time. The volatility smile is named this way because of the shape of the Implied Volatility x Strike Price curve, which reminds a smile. Its shape changes according to the asset to be priced. Thus, as volatility varies with time, the option pricing using Implied Trinomial Trees is affected by the market’s oscillations, whose consequences can be observed in the asset’s price and its option price, consequently.

[pt] O objetivo deste trabalho é realizar uma comparação entre os prêmios de referência da BMEFBovespa e os modelos de Garman Kohlhagen, Corrado-Su Modificado, Difusão com Saltos de Merton, Black e o modelo de Black adaptado para assimetria e curtose para o apreçamento de opções de dólar e sobre futuro de Ibovespa. Para isso, foram definidos cenários de análise e comparados os resultados com os prêmios de referência calculados pela BMEFBovespa no período janeiro de 2006 a setembro de 2011. Os resultados obtidos mostram que, em grande parte dos casos, os prêmios de referência calculados pela Bolsa são superestimados, além de revelar que os valores calculados pelos três modelos para as opções de compra e de venda de dólar e de futuro de Ibovespa encontram-se muito próximos. / [en] This paper proposes a comparison between option reference premiums supplied by BMEF Bovespa and those obtained by the following models: Garman Kohlhagen, modified Corrado-Su, Merton s jump diffusion model, Black and an alternative version of the model, adapted for asymmetry and kurtosis. The underlying assets are futures contracts for Reais/Dolars exchange rate and Ibovespa futures contracts. Base scenarios were created and the results were compared between the models for the January 2006 – September 2011 period. The results show that the majority of the premiums calculated by BMEF Bovespa are overestimated when compared to the proposed models. Furthermore, the results obtained by this models are very similar to one another.

The 2008/9 financial crisis intensified the search for realistic return models, that capture real market movements. The assumed underlying statistical distribution of financial returns plays a crucial role in the evaluation of risk measures, and pricing of financial instruments. In this dissertation, we discuss an empirical study on the evaluation of the traditional portfolio risk measures, and option pricing under the hybrid Brownian motion model, developed by Shaw and Schofield. Under this model, we derive probability density functions that have a fat-tailed property, such that “25-sigma” or worse events are more probable. We then estimate Value-at-Risk (VaR) and Expected Shortfall (ES) using four equity stocks listed on the Johannesburg Stock Exchange, including the FTSE/JSE Top 40 index. We apply the historical method and Variance-Covariance method (VC) in the valuation of VaR. Under the VC method, we adopt the GARCH(1,1) model to deal with the volatility clustering phenomenon. We backtest the VaR results and discuss our findings for each probability density function. Furthermore, we apply the hybrid model to price European style options. We compare the pricing performance of the hybrid model to the classical Black-Scholes model. / Dissertation (MSc)--University of Pretoria, 2017. / National Research Fund (NRF), University of Pretoria Postgraduate bursary and the General Studentship bursary / Mathematics and Applied Mathematics / MSc / Unrestricted

The objective of the present study is to investigate option pricing and forecasting problems in finance. This is achieved by developing stochastic models in the framework of classical modeling approach. In this study, by utilizing the stock price data, we examine the correctness of the existing Geometric Brownian Motion (GBM) model under standard statistical tests. By recognizing the problems, we attempted to demonstrate the development of modified linear models under different data partitioning processes with or without jumps. Empirical comparisons between the constructed and GBM models are outlined. By analyzing the residual errors, we observed the nonlinearity in the data set. In order to incorporate this nonlinearity, we further employed the classical model building approach to develop nonlinear stochastic models. Based on the nature of the problems and the knowledge of existing nonlinear models, three different nonlinear stochastic models are proposed. Furthermore, under different data partitioning processes with equal and unequal intervals, a few modified nonlinear models are developed. Again, empirical comparisons between the constructed nonlinear stochastic and GBM models in the context of three data sets are outlined. Stochastic dynamic models are also used to predict the future dynamic state of processes. This is achieved by modifying the nonlinear stochastic models from constant to time varying coefficients, and then time series models are constructed. Using these constructed time series models, the prediction and comparison problems with the existing time series models are analyzed in the context of three data sets. The study shows that the nonlinear stochastic model 2 with time varying coefficients is robust with respect different data sets. We derive the option pricing formula in the context of three nonlinear stochastic models with time varying coefficients. The option pricing formula in the frame work of hybrid systems, namely, Hybrid GBM (HGBM) and hybrid nonlinear stochastic models are also initiated. Finally, based on our initial investigation about the significance of presented nonlinear stochastic models in forecasting and option pricing problems, we propose to continue and further explore our study in the context of nonlinear stochastic hybrid modeling approach.

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.

An exact solution for the valuation of the options of the European style can be obtained using the Black-Scholes model. However, some of the limitations of the Black-Scholes model are said to be inconsistent such as the constant volatility of the stock price which is not the case in real life. In this thesis, the Black-Scholes model is extended to a model where the volatility is fully stochastic and changing over time, modelled by Markov chain with three states - high, medium and low. Under this model, we price options of both types, European and American, using Monte Carlo simulation.