In the commodity and energy markets, there are two kinds of risk that traders and analysts are concerned a lot about: multiple underlying risk and average price risk. Spread options, swaps and swaptions are widely used to hedge multiple underlying risks and Asian (average price) options can deal with average price risk. But when those two risks are combined together, then we need to consider Asian spread options and Asian-European spread options for hedging purposes. For an Asian or Asian-European spread call option, its payoff depends on the difference of two underlyings' average price or of one average price and one final (at expiration) price. Asian and Asian-European spread option pricing is challenging work. Even under the basic assumption that each underlying price follows a log-normal distribution, the average price does not have a distribution with a simple form. In this dissertation, for the first time, a systematic analysis of Asian spread option and Asian-European spread option pricing is proposed, several original approaches for the Black-Scholes-Merton model and a special stochastic volatility model are developed and some numerical computation tests are conducted as well.

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In this dissertation, we introduce novel exponential jump-diffusion models for pricing options. Firstly, the normal convolution gamma mixture jump-diffusion model is presented. This model generalizes Merton's jump-diffusion and Kou's double exponential jump-diffusion. We show that the normal convolution gamma mixture jump-diffusion model captures some economically important features of the asset price, and that it exhibits heavier tails than both Merton jump-diffusion and double exponential jump-diffusion models. Secondly, the normal convolution double gamma jump-diffusion model for pricing options is presented. We show that under certain configurations of both the normal convolution gamma mixture and the normal convolution double gamma jump-diffusion models, the latter exhibits a heavier left or right tail than the former. For both models, the maximum likelihood procedure for estimating the model parameters under the physical measure is fairly straightforward; moreover, the likelihood function is given in closed form thereby eliminating the need to embed a probability density function recovery procedure such as the fast Fourier transform or the Fourier-cosine expansion methods in the parameter estimation procedure. In addition, both models can reproduce the implied volatility surface observed in the options data and provide a good fit to the market-quoted European option prices.

Jump-diffusion models can under certain assumptions be expressed as partial integro-differential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using finite differences and the second order backward difference formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only $O(n\log n)$ operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model first order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized. / Master of Science

[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. / [CA] 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]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/53917

Hedge funds use a variety of different financial instruments in order to try to achieve over-average returns without taking on excessive risk - options being one of the most common of these instruments. Basket options is a type of option that is written on several underlying assets that can be used to hedge risky positions. This project has been working together with the hedge fund Proxy P to develop software to construct basket options and to analyze their use as a hedging strategy. Construction of basket options can be performed through the use of several different mathematical models. These models range from complex continuous models, such as Monte Carlo simulations, to simple discrete models, such as the binomial option pricing model. In this project, the binomial option pricing model was chosen as the main tool to determine some quantities of basket options. It can conveniently handle both European and American options, independently of whether these are put or call options. The quantities calculated, the option price and option Delta, are dependent on the volatility and the initial price of the underlying. When evaluating the basket option there are two key assumptions that need to be studied. These key assumptions are if the weights and the initial price of the underlying change with each time step, or if they are held constant. It was found that both the weights and the price of the underlying should change dynamically with each time step. Furthermore, in order to evaluate the performance of the basket options used as a hedge, the project used historical data and measured how the options neutralized negative movements in the underlying. This was done through the use of the option Delta and the hedge ratio. What could be concluded was that the put basket option can serve as a relatively inexpensive hedge and minimize the risk on the downside in a sufficient matter. / Hedgefonder använder en rad olika finansiella instrument, där optioner är ett av de mest förekommande av dessa, för att generera överavkastning utan att ta överdriven risk. Korgoptioner, eller basket options som de kallas på engelska, är en typ av option som är skriven på flertalet underliggande tillgångar som kan användas för att gardera finansiella institutioner mot risk. Det här projektet har samarbetat med den svenska hedgefonden Proxy P för att utveckla programvara för att konstruera korgoptioner och evaluera hur de kan användas som hedgingstrategi. Konstrueringen av dessa korgoptioner kan göras med hjälp av flertalet matematiska mo-deller. Allt ifrån komplexa kontinuerliga modeller, som Monte Carlo simulering, till mer simpla diskreta modeller, som binomialprissättningsmodellen, kan användas. I detta projekt kommer binomialprissättningsmodellen användas för att beräkna relevanta kvantiteter gällande korgoptioner. Modellen kan hantera både optioner av den amerikanska och euro-peiska varianten, samt sälj- och köpoptioner. Relevanta kvantiteterna som benämnts gäller optionspriset samt optionens Delta, där dessa beror på marknadsvolatiliteten och startpriset på den underliggande tillgången. Vid utvärdering av korgoptionen behöver två antaganden tas i beaktande: att vikterna och initiala priset på underliggande ändras vid varje tidssteg eller om de hålls konstanta. Slutsatsen kunde dras att både vikterna och den underliggande tillgångens pris skulle vara dynamiska och därmed ändras vid varje tidssteg. För att kunna utvärdera hur väl korgoptioner fungerade som en hedge använde projektet historisk data för att utvärdera hur optionen neutraliserade negativa rörelser i den under-liggande tillgången. Denna utvärdering gjordes med avseende på Deltat hos optionen och hedgekvoten. Slutsatsen som kunde dras var att korgoptioner är ett relativt billigt sätt att hedga och minimera nedsidans risk.

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

In this thesis we consider the calibration of models based on Lévy processes to option prices observed in some market. This means that we choose the parameters of the option pricing models such that the prices calculated using the models correspond as closely as possible to these option prices. We demonstrate the ability of relatively simple Lévy option pricing models to nearly perfectly replicate option prices observed in nancial markets. We speci cally consider calibrating option pricing models to barrier option prices and we demonstrate that the option prices obtained under one model can be very accurately replicated using another. Various types of calibration are considered in the thesis. We calibrate a wide range of Lévy option pricing models to option price data. We con- sider exponential Lévy models under which the log-return process of the stock is assumed to follow a Lévy process. We also consider linear Lévy models; under these models the stock price itself follows a Lévy process. Further, we consider time changed models. Under these models time does not pass at a constant rate, but follows some non-decreasing Lévy process. We model the passage of time using the lognormal, Pareto and gamma processes. In the context of time changed models we consider linear as well as exponential models. The normal inverse Gaussian (N IG) model plays an important role in the thesis. The numerical problems associated with the N IG distribution are explored and we propose ways of circumventing these problems. Parameter estimation for this distribution is discussed in detail. Changes of measure play a central role in option pricing. We discuss two well-known changes of measure; the Esscher transform and the mean correcting martingale measure. We also propose a generalisation of the latter and we consider the use of the resulting measure in the calculation of arbitrage free option prices under exponential Lévy models. / PhD (Risk Analysis), North-West University, Potchefstroom Campus, 2015

Thesis (MComm (Statistics and Actuarial Science)--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Life insurance and pension funds offer a wide range of products that are invested in a mix of assets. These portfolios (II), underlying the products, are rebalanced back to predetermined fixed proportions on a regular basis. This is done by selling the better performing assets and buying the worse performing assets. Life insurance or pension fund contracts can offer the client a minimum payout guarantee on the contract by charging them an extra premium (a). This problem can be changed to that of the pricing of a put option with underlying . It forms a liability for the insurance firm, and therefore needs to be managed in terms of risks as well. This can be done by studying the option’s sensitivities. In this thesis the premium and sensitivities of this put option are calculated, using different Monte Carlo methods, in order to find the most efficient method. Using general Monte Carlo methods, a simplistic pricing method is found which is refined by applying mathematical techniques so that the computational time is reduced significantly. After considering Antithetic Variables, Control Variates and Latin Hypercube Sampling as variance reduction techniques, option prices as Control Variates prove to reduce the error of the refined method most efficiently. This is improved by considering different Quasi-Monte Carlo techniques, namely Halton, Faure, normal Sobol’ and other randomised Sobol’ sequences. Owen and Faure-Tezuke type randomised Sobol’ sequences improved the convergence of the estimator the most efficiently. Furthermore, the best methods between Pathwise Derivatives Estimates and Finite Difference Approximations for estimating sensitivities of this option are found. Therefore by using the refined pricing method with option prices as Control Variates together with Owen and Faure-Tezuke type randomised Sobol’ sequences as a Quasi-Monte Carlo method, more efficient methods to price this option (compared to simplistic Monte Carlo methods) are obtained. In addition, more efficient sensitivity estimators are obtained to help manage risks. / AFRIKAANSE OPSOMMING: Lewensversekering en pensioenfondse bied die mark ’n wye reeks produkte wat belê word in ’n mengsel van bates. Hierdie portefeuljes (II), onderliggend aan die produkte, word op ’n gereelde basis terug herbalanseer volgens voorafbepaalde vaste proporsies. Dit word gedoen deur bates wat beter opbrengste gehad het te verkoop, en bates met swakker opbrengste aan te koop. Lewensversekeringof pensioenfondskontrakte kan ’n kliënt ’n verdere minimum uitbetaling aan die einde van die kontrak waarborg deur ’n ekstra premie (a) op die kontrak te vra. Die probleem kan verander word na die prysing van ’n verkoopopsie met onderliggende bate . Hierdie vorm deel van die versekeringsmaatskappy se laste en moet dus ook bestuur word in terme van sy risiko’s. Dit kan gedoen word deur die opsie se sensitiwiteite te bestudeer. In hierdie tesis word die premie en sensitiwiteite van die verkoopopsie met behulp van verskillende Monte Carlo metodes bereken, om sodoende die effektiefste metode te vind. Deur die gebruik van algemene Monte Carlo metodes word ’n simplistiese prysingsmetode, wat verfyn is met behulp van wiskundige tegnieke wat die berekeningstyd wesenlik verminder, gevind. Nadat Antitetiese Veranderlikes, Kontrole Variate en Latynse Hiperkubus Steekproefneming as variansiereduksietegnieke oorweeg is, word gevind dat die verfynde metode se fout die effektiefste verminder met behulp van opsiepryse as Kontrole Variate. Dit word verbeter deur verskillende Quasi-Monte Carlo tegnieke, naamlik Halton, Faure, normale Sobol’ en ander verewekansigde Sobol’ reekse, te vergelyk. Die Owen en Faure-Tezuke tipe verewekansigde Sobol’ reeks verbeter die konvergensie van die beramer die effektiefste. Verder is die beste metode tussen Baanafhanklike Afgeleide Beramers en Eindige Differensie Benaderings om die sensitiwiteit vir die opsie te bepaal, ook gevind. Deur dus die verfynde prysingsmetode met opsiepryse as Kontrole Variate, saam met Owen en Faure-Tezuke tipe verewekansigde Sobol’ reekse as ’n Quasi-Monte Carlo metode te gebruik, word meer effektiewe metodes om die opsie te prys, gevind (in vergelyking met simplistiese Monte Carlo metodes). Verder is meer effektiewe sensitiwiteitsberamers as voorheen gevind wat gebruik kan word om risiko’s te help bestuur.

Energy markets feature a wide range of unusual price behaviour along with a complicated dependence structure between electricity, natural gas, coal and carbon, as well as other variables. We approach this broad modelling challenge by firstly developing a structural framework to modelling spot electricity prices, through an analysis of the underlying supply and demand factors which drive power prices, and the relationship between them. We propose a stochastic model for fuel prices, power demand and generation capacity availability, as well as a parametric form for the bid stack function which maps these price drivers to the spot electricity price. Based on the intuition of cost-related bids from generators, the model describes mathematically how different fuel prices drive different portions of the bid stack (i.e., the merit order) and hence influence power prices at varying levels of demand. Using actual bid data, we find high correlations between the movements of bids and the corresponding fuel prices (coal and gas). We fit the model to the PJM and New England markets in the US, and assess the performance of the model, in terms of capturing key properties of simulated price trajectories, as well as comparing the model’s forward prices with observed data. We then discuss various mathematical techniques (explicit solutions, approximations, simulations and other numerical techniques) for calibrating to observed fuel and electricity forward curves, as well as for pricing of various single and multi-commodity options. The model reveals that natural gas prices are historically the primary driver of power prices over long horizons in both markets, with shorter term dynamics driven also by fluctuations in demand and reserve margin. However, the framework developed in this thesis is very flexible and able to adapt to different markets or changing conditions, as well as capturing automatically the possibility of changes in the merit order of fuels. In particular, it allows us to begin to understand price movements in the recently-formed carbon emissions markets, which add a new level of complexity to energy price modelling. Thus, the bid stack model can be viewed as more than just an original and elegant new approach to spot electricity prices, but also a convenient and intuitive tool for understanding risks and pricing contracts in the global energy markets, an important, rapidly-growing and fascinating area of research.