Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: In modern theory of finance, the so-called First and Second Fundamental Theorems of Asset Pricing play an important role in pricing options with no-arbitrage. These theorems gives a necessary and sufficient conditions for a market to have no-arbitrage and for a market to be complete. An early version of the First Fundamental Theorem of Asset Pricing was proven by Harrison and Kreps [30] in the case of a finite probability space. A more general version was proven by Harrison and Pliska [31] in the case of a finite probability space and discrete time. In the case of continuous time, Delbaen and Schachermayer [19] introduced a more general concept of no-arbitrage called "No-Free Lunch With Vanishing Risk" (NFLVR), and showed that for a locally-bounded semimartingale price process NFLVR is essentially equivalent to the existence of an equivalent local martingale measure. The goal of this thesis is to review the theory of arbitrage pricing and the extension of this theory to include liquidity risk. At the current time, liquidity risk is a key challenge faced by investors. Consequently there is a need to develop more realistic pricing models that include liquidity risk. We present an approach to liquidity risk by Çetin, Jarrow and Protter [10]. In to this approach the liquidity risk is embedded into the classical theory of arbitrage pricing by having investors act as price takers, and assuming the existence of a supply curve where prices depend on trade size. This framework assumes that the quantity impact on the price transacted is momentary. Using trading strategies that are both continuous and of finite variation allows one to avoid liquidity costs. Therefore, the First and Second Fundamental Theorems of Asset Pricing and the Black-Scholes model can be extended. / AFRIKAANSE OPSOMMING: In moderne finansiële teorie speel die sogenaamde Eerste en Tweede Fundamentele Stellings van Bateprysbepaling ’n belangrike rol in die prysbepaling van opsies in arbitrage-vrye markte. Hierdie stellings gee nodig en voldoende voorwaardes vir ’n mark om vry van arbitrage te wees, en om volledig te wees. ’n Vroeë weergawe van die Eerste Fundamentele Stelling was deur Harrison en Kreps [30] bewys in die geval van ’n eindige waarskynlikheidsruimte. ’n Meer algemene weergawe was daarna gepubliseer deur Harrison en Pliska [31] in die geval van ’n eindige waarskynlikheidsruimte en diskrete tyd. In die geval van kontinue tyd het Delbaen en Schachermayer [19] ’n meer algemene konsep van arbitragevryheid ingelei, naamlik “No–Free–Lunch–With–Vanishing–Risk" (NFLVR), en aangetoon dat vir lokaalbegrensde semimartingaalprysprosesse NFLVR min of meer ekwivalent is aan die bestaan van ’n lokaal martingaalmaat. Die doel van hierdie tesis is om ’n oorsig te gee van beide klassieke arbitrageprysteorie, en ’n uitbreiding daarvan wat likideit in ag neem. Hedendaags is likiditeitsrisiko ’n vooraanstaande uitdaging wat beleggers die hoof moet bied. Gevolglik is dit noodsaaklik om meer realistiese modelle van prysbepaling wat ook likiditeitsrisiko insluit te ontwikkel. Ons bespreek die benadering van Çetin, Jarrow en Protter [10], waar likiditeitsrisiko in die klassieke arbitrageprysteorie ingesluit word deur die bestaan van ’n aanbodkromme aan te neem, waar pryse afhanklik is van handelsgrootte. In hierdie raamwerk word aangeneem dat die impak op die transaksieprys slegs tydelik is. Deur gebruik te maak van handelingsstrategië wat beide kontinu en van eindige variasie is, is dit dan moontlik om likiditeitskoste te vermy. Die Eerste en Tweede Fundamentele Stellings van Bateprysbepaling en die Black–Scholes model kan dus uitgebrei word om likiditeitsrisiko in te sluit.

本論文的研究目的，主要希望藉由對於保本型及非保本型商品的實證研究分析，使得投資人更加了解投資匯率衍生性金融商品所會面臨的報酬與風險，另外藉由一連串的範例探討設計原理，俾能更加了解金融商品設計之關鍵所在。 如何將基本的金融商品相結合以創造出更具競爭力的新金融商品，如何將金融商品評價以了解報酬與風險之所在，係學習財務工程者的目標。本論文之研究成果可分為下列幾項： 一、在匯率衍生性金融商品評價模型方面，本論文引用Black-Scholes模型及Martingale Pricing為推導模型，找出保本及非保本商品之封閉解。 二、進一步運用Delta、Gamma、Vega及Theta求出相關匯率衍生性金融商品的敏感度分析，以了解風險範疇。 三、將數學及matlab程式軟體應用於論文中，在求算避險參數時，以簡化的表格及圖形表達複雜的微分及數學運算結果。 四、引述實務界商品，分析其基本設計架構，冀能合併並引發新的金融商品設計理念並創造獲利。

Title: Black-Scholes Models of Option Pricing Author: Martin Cekal Department: Department of Probability and Mathematical Statistics Supervisor: prof. RNDr. Bohdan Maslowski, DrSc., Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics. Abstract: In the present master thesis we study a generalization of Black-Scholes model using fractional Brownian motion and jump processes. The main goal is a derivation of the price of call option in a fractional jump market model. The first chapter introduces long memory and its modelling by discrete and continuous time models. In the second chapter fractional Brownian motion is defined, appropriate stochastic analysis is developed and we generalize the notion of Lévy and jump processes. The third chapter introduces fractional Black-Scholes model. In the fourth chapter, tools developed in the second chapter are used for the construction of jump fractional Black-Scholes model and derivation of explicit formula for the price of european call option. In the fifth chapter, we analyze long memory contained in simulated and empirical time series. Keywords: Black-Scholes model, fractional Brownian motion, fractional jump process, long- memory, options pricing.

This diploma thesis describes problem of change-points in volatility of the time-series and their impact on price of nancial assets. Those change-points are estimated by using statistical methods and tests. Change-point estimation was tested on simulated datas and real world driven datas. Simulation helped to discover signi cant characteristics of change-point test, those data were simulated with using stochastic calculus. Google share prices and prices of call options were chosen to analyse impact of volatility change on those prices. Also implied volatility and its impact to call option price was analysed.

In the present thesis we deal with several possible approaches to financial de- rivatives pricing. In the first part, we introduce the basic types of derivatives and the methods of trading. Furthermore, we present several models for the valuati- on of specific financial derivative, i.e. options. Firstly we describe Black-Scholes model in detail, which considers that the development of the underlying asset price is governed by Wiener process. Following are the jumps diffusion models that are extension of the Black-Scholes model with jumps. Then we get to jump models, which are based on Lévy processes. Finally, we will deal with the model, which considers that the development of the underlying asset price is governed by fractional Brownian motion with Hurst's coefficient greater than 1/2. All models are suplemented with sample examples. 1

Derivative pricing, and in particular the pricing of options, is an important area of current research in financial mathematics. Experts debate on the best method of pricing and the most appropriate model of a price process to use. In this thesis, a ``Switching Black-Scholes'' model of a price process is proposed. This model is based on the standard geometric Brownian motion (or Black-Scholes) model of a price process. However, the drift and volatility parameters are permitted to vary between a finite number of possible values at known times, according to the state of a hidden Markov chain. This type of model has been found to replicate the Black-Scholes implied volatility smiles observed in the market, and produce option prices which are closer to market values than those obtained from the traditional Black-Scholes formula. As the Markov chain incorporates a second source of uncertainty into the Black-Scholes model, the Switching Black-Scholes market is incomplete, and no unique option pricing methodology exists. In this thesis, we apply the methods of mean-variance hedging, Esscher transforms and minimum entropy in order to price options on assets which evolve according to the Switching Black-Scholes model. C programs to compute these prices are given, and some particular numerical examples are examined. Finally, filtering techniques and reference probability methods are applied to find estimates of the model parameters and state of the hidden Markov chain. / Thesis (Ph.D.)--Applied Mathematics, 2003.

Cette thèse consiste en une série d'applications du calcul stochastique aux mathématiques financières. Elle est composée de quatre chapitres. Dans le premier on étudie le rapport entre la complétude du marché et l'extrémalité des mesures martingales equivalentes dans le cas d'une infinité d'actifs. Dans le deuxième on trouve des conditions équivalentes à l'existence et unicité d'une mesure martingale equivalente sous la quelle le processus des prix suit des lois n-dimensionnelles données à n fixe. Dans le troisième on étend à un marché admettant une infinité dénombrable d'actifs une charactérisation de la stratégie de couverture optimale (pour le critère moyenne-variance) basé sur une technique de changement de numéraire et extension artificielle. Enfin, dans le quatrième on s'occupe du problème de couverture d'un actif contingent dans un marché avec information asymetrique.

<p>Syftet med denna uppsats är att med tre GARCH-modeller skatta volatiliteten för fjorton aktier med t- och normalfördelade slumptermer. Dessa volatiliteter implementeras sedan i Black-Scholes modell samt i Monte-Carlo simuleringar och utfallen av dessa två värderingsmetoder jämförs.</p><p>Författarna har kommit fram till att GARCH-modeller behövs för att skatta volatiliteten för de aktier som ingår i arbetet då modellerna tar hänsyn till den föreliggande heteroskedasticiteten.</p><p>De skillnader som uppstår mellan Monte-Carlo simuleringar och Black-Scholes modell beror främst på skillnader mellan normal- och t-fördelningen samt att volatiliteten ger större effekt i Monte-Carlo simuleringarna. Författarna kan inte uttala sig om huruvida Monte-Carlo skattningarna ger bättre resultat än den vedertagna Black-Scholes modell, däremot är Monte-Carlo mer teoretiskt korrekt.</p>

<p>The present mini-thesis seeks to explore and investigate the mathematical theory and concepts that underpins the valuation of derivative securities, particularly European plainvanilla options. The main argument that we emphasise is that novel models of option pricing, as is suggested by Hull and White (1987) [1] and others, must account for the discrepancy observed on the implied volatility &ldquo / smile&rdquo / curve. To achieve this we also propose that market volatility be modeled as random or stochastic as opposed to certain standard option pricing models such as Black-Scholes, in which volatility is assumed to be constant.</p>

Options are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.