The thesis focuses on methods of option prices calculations using two different pricing models which are Heston and Black-Scholes models. The first part describes theory of these two models and conlcudes with a comparison of the risk-neutral measures of these two models. In the second part, the relations between input parameters and the option price generated by these models are clarified. This part ends up with an analysis of the market data and it answers the question which model predicts better.

Financial engineering problems are of great importance in the academic community and BlackScholes equation is a revolutionary concept in the modern financial theory. Financial instruments such as stocks and derivatives can be evaluated using this model. Option evaluation, is extremely important to trade in the stocks. The numerical solutions of the Black-Scholes equation are used to simulate these options. In this thesis, the explicit and the implicit Euler methods are used for the approximation of Black-scholes partial differential equation and a second order finite difference scheme is used for the spatial derivatives. These temporal and spatial discretizations are used to gain an insight about the stability properties of the explicit and the implicit methods in general. The numerical results show that the explicit methods have some constraints on the stability, whereas, the implicit Euler method is unconditionally stable. It is also demostrated that both the explicit and the implicit Euler methods are only first order convergent in time and this implies too small step-sizes to achieve a good accuracy.

This paper considers dynamic optimal portfolio strategies of utility maximizing investors in the presence of risk constraints. In particular, we investigate the optimization problem with an additional constraint modeling bounded shortfall risk measured by Value at Risk or Expected Loss. Using the Black-Scholes model of a complete financial market and applying martingale methods we give analytic expressions for the optimal terminal wealth and the optimal portfolio strategies and present some numerical results.

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.

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

Darbe nagrinėjami barjero pasirinkimo sandorių įkainojimo metodai. Barjero pasirinkimo sandorių išmokos sutampa su įprastinių pasirinkimo sandorių išmokomis, jei išpildoma papildoma barjero sąlyga, kurią reikia įvertinti. Įkainojimui naudojami diskretieji modeliai: binominis ir trinominis, tiriama jų konvergavimas į klasikinę Black-Scholes formulę. Dėl modelio diskretumo ir barjero sąlygos konvergavimas tam tikrais atvejais yra lėtas ir nemonotoniškas. Todėl siūloma pritaikyti adaptyviojo tinklelio algoritmą, smulkinant trinominio medžio tinklelį kritinėse srityse. Šiame darbe pateikiami rezultatai, gauti palyginus barjero pasirinkimo sandorio įkainojimo modelius. / In this paper we consider barrier options pricing models. Barrier options are standard call or put options except that they disappear or appear if the asset price crosses a predeterminant set of fixing dates. Barrier options are priced using continuous state Black-Scholes model and numerical approximation techniques, such as binomial and trinomial. Because of the the barrier condition and discreteness of these models the convergence to Black-Scholes model sometimes is slow. It is offered to apply adaptive mesh model grafting small sections of fine high-resolution lattice onto a tree in trinomial model. In this work we present the comparison of the models with some numerical results for barrier options.

Financial derivatives have been constituting one of the most dynamic fields in the mathematical finance. The main task is represented by the valuation or pricing of these instruments. This theses deals with standard models and their limits, tries to explore advanced methods of continuous martingale measures and on their bases proposes numerical methods applicable to derivatives valuation. Some procedures leading to elimination of certain simplifying assumptions are presented as well.

In the present work, we study the topic of stochastic differential equations, their numerical solution and solution of models for pricing of options which follow from stochastic differential equations using the Itô calculus. We present several numerical methods for solving stochastic differential equations. These methods are then implemented in MATLAB and we investigate their properties, especially their convergence characteristics. Furthermore, we formulate two models for pricing of European call options. We solve these models using a variant of the spectral collocation method, again in MATLAB.