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.

Intrinsic connectivity networks (ICNs) refer to brain functional networks that are consistently found under various conditions, during tasks or at rest. Some studies demonstrated that while some stimuli do not impact intrinsic connectivity, other stimuli actually activate intrinsic connectivity through suppression, excitation, moderation or modi cation. Most analyses of functional magnetic resonance imaging (fMRI) data use ad-hoc methods to estimate the latent structure of ICNs. Modeling the effects on ICNs has also not been fully investigated. Bayesian Intrinsic Connectivity Network (BICNet) captures the ICN structure with We propose a BICNet model, an extended Bayesian dynamic sparse latent factor model, to identify the ICNs and quantify task-related effects on the ICNs. BICNet has the following advantages: (1) It simultaneously identifies the individual and group-level ICNs; (2) It robustly identifies ICNs by jointly modeling resting-state fMRI (rfMRI) and task-related fMRI (tfMRI); (3) Compared to independent component analysis (ICA)-based methods, it can quantify the difference of ICNs amplitudes across different states; (4) The sparsity of ICNs automatically performs feature selection, instead of ad-hoc thresholding. We apply BICNet to the rfMRI and language tfMRI data from the Human Connectome Project (HCP) and identify several ICNs related to distinct language processing functions.

Merton’s Optimal Investment-Consumption Problem is a classic optimization problem in finance. It aims to find the optimal controls for a portfolio with both risky and risk-less assets, inorder to maximize an investor’s utility function. One of the controls is the optimal allocationof wealth invested in a risky asset and the other control is the consumption rate. The problemis solved by using Dynamic Programming and the related Hamilton-Jacobi-Bellman equation.One of the disadvantages of the original problem is the consideration of constant volatility. Inthis thesis, we extend Merton’s problem considering the Grzelak-Oosterlee-Van Veeren modelthat describes the dynamics of a risky asset with stochastic volatility and stochastic interestrate. We derive the related Hamilton-Jacobi-Bellman for Merton’s problem considering theGrzelak-Oosterlee-Van Veeren model. We simulate the controls from Merton’s problem intwo different cases, one case where the volatility and interest rate are stochastic, following theGOVV-model. In the other case, the volatility and interest rate are assumed to be constant, asin Merton’s problem. The results obtained from simulations show that the case with stochasticvolatility and interest gave the same results as the case where the volatility and the interest ratewere assumed to be constant.

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.

This thesis studies realized volatility (RV), implied volatility (IV) and their applications in stochastic volatility models. The first essay uses both daytime and overnight high-frequency price data for equity index futures to estimate the RV of the S\&P500 and NASDAQ 100 indexes. Empirical results reveal strong inter-correlation between the regular-trading-time and after-hour RVs, as well as a significant predictive power of overnight RV on daytime RV and vice versa. We propose a new day-night realized stochastic volatility (DN-SV-RV) model, where the daytime and overnight returns are jointly modeled with their RVs, and their latent volatilities are correlated. The newly proposed DN-SV-RV model has the best out-of-sample return distribution forecasts among the models considered. The second essay extends the realized stochastic volatility model by jointly estimating return, RV and IV. We examine how RV and IV enhance the estimation of the latent volatility process for both the S\&P500 index and individual stocks. The third essay re-examines asymmetric stochastic volatility (ASV) models with different return-volatility correlation structures given RV and IV. We show by simulation that estimating the ASV models with return series alone may infer erroneous estimations of the correlation coefficients. The incorporation of volatility measures helps identify the true return-volatility correlation within the ASV framework. Empirical evidence on global equity market indices verifies that ASV models with additional volatility measures not only obtain significantly different estimations of the correlations compared to the benchmark ASV models, but also improve out-of-sample return forecasts. / Thesis / Doctor of Philosophy (PhD)

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: This work employs perturbation techniques to price and hedge financial derivatives in a stochastic volatility framework. Fouque et al. [44] model volatility as a function of two processes operating on different time-scales. One process is responsible for the fast-fluctuating feature of volatility and corresponds to the slow time-scale and the second is for slowfluctuations or fast time-scale. The former is an Ergodic Markov process and the latter is a strong solution to a Lipschitz stochastic differential equation. This work mainly involves modelling, analysis and estimation techniques, exploiting the concept of mean reversion of volatility. The approach used is robust in the sense that it does not assume a specific volatility model. Using singular and regular perturbation techniques on the resulting PDE a first-order price correction to Black-Scholes option pricing model is derived. Vital groupings of market parameters are identified and their estimation from market data is extremely efficient and stable. The implied volatility is expressed as a linear (affine) function of log-moneyness-tomaturity ratio, and can be easily calibrated by estimating the grouped market parameters from the observed implied volatility surface. Importantly, the same grouped parameters can be used to price other complex derivatives beyond the European and American options, which include Barrier, Asian, Basket and Forward options. However, this semi-analytic perturbative approach is effective for longer maturities and unstable when pricing is done close to maturity. As a result a more accurate technique, the decomposition pricing approach that gives explicit analytic first- and second-order pricing and implied volatility formulae is discussed as one of the current alternatives. Here, the method is only employed for European options but an extension to other options could be an idea for further research. The only requirements for this method are integrability and regularity of the stochastic volatility process. Corrections to [3] remarkable work are discussed here. / AFRIKAANSE OPSOMMING: Hierdie werk gebruik steuringstegnieke om finansiële afgeleide instrumente in ’n stogastiese wisselvalligheid raamwerk te prys en te verskans. Fouque et al. [44] gemodelleer wisselvalligheid as ’n funksie van twee prosesse wat op verskillende tyd-skale werk. Een proses is verantwoordelik vir die vinnig-wisselende eienskap van die wisselvalligheid en stem ooreen met die stadiger tyd-skaal en die tweede is vir stadig-wisselende fluktuasies of ’n vinniger tyd-skaal. Die voormalige is ’n Ergodiese-Markov-proses en die laasgenoemde is ’n sterk oplossing vir ’n Lipschitz stogastiese differensiaalvergelyking. Hierdie werk behels hoofsaaklik modellering, analise en skattingstegnieke, wat die konsep van terugkeer to die gemiddelde van die wisseling gebruik. Die benadering wat gebruik word is rubuust in die sin dat dit nie ’n aanname van ’n spesifieke wisselvalligheid model maak nie. Deur singulêre en reëlmatige steuringstegnieke te gebruik op die PDV kan ’n eerste-orde pryskorreksie aan die Black-Scholes opsie-waardasiemodel afgelei word. Belangrike groeperings van mark parameters is geïdentifiseer en hul geskatte waardes van mark data is uiters doeltreffend en stabiel. Die geïmpliseerde onbestendigheid word uitgedruk as ’n lineêre (affiene) funksie van die log-geldkarakter-tot-verval verhouding, en kan maklik gekalibreer word deur gegroepeerde mark parameters te beraam van die waargenome geïmpliseerde wisselvalligheids vlak. Wat belangrik is, is dat dieselfde gegroepeerde parameters gebruik kan word om ander komplekse afgeleide instrumente buite die Europese en Amerikaanse opsies te prys, dié sluit in Barrier, Asiatiese, Basket en Stuur opsies. Hierdie semi-analitiese steurings benadering is effektief vir langer termyne en onstabiel wanneer pryse naby aan die vervaldatum beraam word. As gevolg hiervan is ’n meer akkurate tegniek, die ontbinding prys benadering wat eksplisiete analitiese eerste- en tweede-orde pryse en geïmpliseerde wisselvalligheid formules gee as een van die huidige alternatiewe bespreek. Hier word slegs die metode vir Europese opsies gebruik, maar ’n uitbreiding na ander opsies kan’n idee vir verdere navorsing wees. Die enigste vereistes vir hierdie metode is integreerbaarheid en reëlmatigheid van die stogastiese wisselvalligheid proses. Korreksies tot [3] se noemenswaardige werk word ook hier bespreek.

Understanding financial asset return correlation is a key facet in asset allocation and investor’s portfolio optimization strategy. For the last decades, several studies have investigated this relationship between stock and bond returns. But, fewer studies have dealt with multi-asset return dynamics. While initial literature attempted to understand the fundamental pattern of comovements, later studies model the economic state variables influencing such time-varying comovements of primarily stock and bond returns. Research widely acknowledges that return distributions of financial assets are non-normal. When the joint distributions of the asset returns follow a non-elliptical structure, linear correlation fails to provide sufficient information of their dependence structure. In particular two issues arise from this existing empirical evidence. The first is to propose a more reliable alternative density specification for a higher-dimensional case. The second is to formulate a measure of the variables’ dependence structure which is more instructive than linear correlation. In this work I use a time-varying conditional multivariate elliptical and non-elliptical copula to examine the return comovements of three different asset classes: financial assets, commodities and real estate in the US market. I establish the following stylized facts about asset return comovements. First, the static measures of asset return comovements overestimate the asset return comovements in the economic expansion phase, while underestimating it in the periods of economic contraction. Second, Student t-copulas outperform both elliptical and non-elliptical copula models, thus confirming the ii dominance of Student t-distribution. Third, findings show a significant increase in asset return comovements post August 2007 subprime crisis ... [cont.].

This thesis concerns with a comparison of two advanced option-pricing techniques applied on European-style DAX index options. Specifically, the study examines the performance of both the stochastic volatility model based on asymmetric nonlinear GARCH, which was proposed by Heston and Nandi (2000), and the artificial neural network, where the conventional Black-Scholes-Merton model serves as a benchmark. These option-pricing models are tested with the use of the dataset covering the period 3rd July 2006 - 30th October 2009 as well as of its two subsets labelled as "before crisis" and "in crisis" data where the breakthrough day is the 17th March 2008. Finding the most appropriate option-pricing method for the whole periods as well as for both the "before crisis" and the "in crisis" datasets is the main focus of this work. The first two chapters introduce core issues involved in option pricing, while the subsequent third section provides a theoretical background related to all of above-mentioned pricing methods. At the same time, the reader is provided with an overview of the theoretical frameworks of various nonlinear optimization techniques, i.e. descent gradient, quassi-Newton method, Backpropagation and Levenberg-Marquardt algorithm. The empirical part of the thesis then shows that none of the...

The thesis compares GARCH volatility models and Stochastic Volatility (SV) models with Student's t distributed errors and its empirical forecasting per- formance of Value at Risk on five stock price indices: S&P, NASDAQ Com- posite, CAC, DAX and FTSE. It introduces in details the problem of SV models Maximum Likelihood examinations and suggests the newly devel- oped approach of Efficient Importance Sampling (EIS). EIS is a procedure that provides an accurate Monte Carlo evaluation of likelihood function which depends upon high-dimensional numerical integrals. Comparison analysis is divided into in-sample and out-of-sample forecast- ing performance and evaluated using standard statistical probability back- testig methods as conditional and unconditional coverage. Based on empirical analysis thesis shows that SV models can perform at least as good as GARCH models if not superior in forecasting volatility and parametric VaR. 1

自1980年代後期Hull & White、Wiggins、Johnson & Shanno等人相繼發表關於隨機波動度模型的文獻後，就有諸多的文獻對於在選擇權定價中考慮隨機波動度作更深入的分析與模型探討，然而關於隨機波動度的研究，在早期大多採用蒙地卡羅模擬法來分析選擇權的價格行為，但蒙地卡羅模擬法受限於運算效率不高與缺乏彈性，故在評價新奇選擇權，如美式選擇權、障礙選擇權時，並無法應用。故本文以Leisen（2000）的二元樹狀模型出發，探討在不同相關係數及參數設定下之各類選擇權的定價、避險參數及隱含波動度曲面模擬計算等主題。 最後我們得到下面幾點結論： 1.在收斂速度與運算效率方面，我們可以發現二元樹狀模型在分割期數n大於20時，計算價格與收斂價格的差距就非常微小，而若我們計算不同切割期數的最大價格差異也會發現其實都不到百分之一，因此整體而言，收斂速度是令人非常滿意的。 2.當期初波動度提高時，會縮小價外選擇權與B-S價格之間的價格誤差。當到期期限增加時，隱含波動度曲線會有整體提高的趨勢。 3.若提高波動係數σ為2.5時，則不論相關係數的正負情形，價內外的程度，皆會大幅提高選擇權的隱含波動度。而在相關係數為-0.5的時候，可以發現實證中常觀察到的隱含波動度微笑曲線，這可能代表著市場上的波動係數比我們預期中的都還來的高。 4.在進行不同相關係數及不同價內外程度下二元樹狀與單元樹狀模型的美式選擇權價格比較時，我們可以發現，若以二元樹狀模型為正確價格，當相關係數為負的時候，在價外的時候，單元樹狀模型有價格低估的現象，在價內的時候，則有價格高估的現象，而在相關係數為正的時候，則反之。 5.Leisen二元樹狀與封閉解的歐式向上出局賣權價格比較，在特定的參數設定之下，Leisen二元樹狀模型在評價歐式向上出局賣權的時候，當相關係數為負的時候，在價外的時候，模型價格會高於封閉解，在價內的時候，模型價格則會低於封閉解，而在相關係數為正的時候，則反之。