DOCSLIB.ORG

MASTER THESIS in MATHEMATICS / APPLIED MATHEMATICS Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MASTER THESIS in MATHEMATICS / APPLIED MATHEMATICS Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative MASTER THESIS IN MATHEMATICS / APPLIED MATHEMATICS Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate Environment: A comparative analysis of Lognormal and Normal Model by Shadrack Lutembeka Masterarbete i matematik / tillämpad matematik DIVISION OF APPLIED MATHEMATICS MÄLARDALEN UNIVERSITY SE-721 23 VÄSTERÅS, SWEDEN Master thesis in mathematics / applied mathematics Date: 2017-01-19 Project name: Hedging Interest Rate Derivatives (Evidence from Swaptions) in a Negative Interest Rate En- vironment: A comparative analysis of Lognormal and Normal Model Author: Shadrack Lutembeka Supervisor(s): Jan Röman and Richard Bonner Reviewer: Anatoliy Malyarenko Examiner: Linus Carlsson Comprising: 30 ECTS credits Abstract This thesis is about hedging interest rate derivatives in a negative interest rate environment. The main focus is on doing a comparative analysis on how risk varies between Lognormal and Normal models. This because Lognormal models do not work in the negative interest rate since they do not allow negative values, hence there is a need of using Normal models. The use of different models will yield identical price but different hedges. In order to study this we looked at the case of Swaptions and Swaps as an example of interest rate derivatives. To study risk in these two models we employed the method of risk matrices to measure and report risk. We created various risk matrices for both Black model and Normal Black model which included the price matrices, Delta and Vega matrices to study how Swaptions and Swaps with different maturities are sensitive to changes in different parameters. We also plotted how Delta and Vega vary between the two models. Acknowledgements First and foremost I would like to thank God for always taking care of me. Secondly I would like to convey my special thanks to the Swedish Institute (SI) for awarding me a full scholar- ship to study the masters program Financial Engineering at Mäladalen University. Thirdly i would like to convey my sincere thanks to my supervisor Jan Röman who invested alot of his time and efforts to ensure that this thesis becomes a success, am truly grateful. I wouldn’t also forget my other supervisor, Richard Bonner for his continuous and prompt guidance during the writing of this thesis. To my dear parents who have been so supportive since day one that I set foot at the nursery school. To my siblings; Lilian, Meshack, Godfrey and Gladness for their continuous support throughout this journey. Last but not least are my dear friends and family, Polite Mpofu, Erick Momamnyi, James Okemwa, Oliver Grace, Kakta Mpofu and Mahalet Haile Selassie for always being there throughout this journey. 1 Contents 1 Introduction5 1.1 Negative Interest Rates Environment......................5 1.2 Motivation for Negative Interest Rate Policies.................6 1.3 Motivation and Problem Formulation......................7 1.3.1 Motivation...............................7 1.3.2 Problem Formulation..........................8 1.4 Understanding Interest Rates..........................9 1.5 Overview and Outline.............................. 12 2 Lognormal Model versus Normal Model 13 2.1 Pricing Models................................. 13 2.2 Log Normal Models............................... 14 2.2.1 Black’s model............................. 15 2.2.2 The Constant Elasticity of Variance (CEV) Model........... 17 2.2.3 The Stochastic Alpha Beta Rho (SABR) Model............ 18 2.3 Normal Models................................. 19 2.3.1 Bachelier’s Model........................... 20 2.3.2 Normal SABR Model......................... 20 3 Hedging Parameters 22 3.1 Greeks in Black-Scholes............................ 22 3.2 Greeks in Other Models............................. 26 3.2.1 Black Model.............................. 26 3.2.2 Normal Model............................. 27 2 3.2.3 SABR Model.............................. 28 3.3 Hedging Strategies............................... 29 3.3.1 Risk Matrices.............................. 31 4 Interest Rate Derivatives 32 4.1 Derivatives................................... 32 4.2 Interest Rate Derivatives............................ 33 4.2.1 Forward rate agreement (FRA)..................... 33 4.2.2 Caps.................................. 35 4.2.3 Floors.................................. 36 4.2.4 Bond Options.............................. 37 4.2.5 Interest Rate Swap........................... 37 4.2.6 Swaption................................ 41 5 Implementation 44 5.1 Bootstrapping a Swap Curve.......................... 44 5.2 Premium and Risk Measures Calculations................... 45 5.2.1 Plotting Delta and Vega in Black and Normal Black Model...... 46 5.3 Risk Matrices for Swaptions and Swaps at Different Maturities........ 48 5.3.1 Price Sensitivity............................ 48 5.3.2 Delta and Vega Sensitivity....................... 49 6 Conclusion 51 7 Notes on fulfillment of Thesis objectives 52 Bibliography 54 A More mathematics 56 A.1 Singular Pertubation Technique......................... 56 A.2 Solution to Black Model............................ 57 B APPENDIX B 59 B.1 Extract of VBA Program Codes........................ 59 3 B.2 Risk Matrices for Swaptions and Swaps with Different Maturities...... 61 4 Chapter 1 Introduction 1.1 Negative Interest Rates Environment In a normal world, one would expect that a lender to receive from a borrower a rate on the amount borrowed. It is also expected that when one deposits money in the bank, he would expect to get back some form of interest on his deposit. However, when we have a situation where lenders have to pay borrowers for lending from them or when depositors are charged for keeping their money with the bank instead of receiving an interest income, we have what we call "negative interest rate". One could argue that interest rates were modelled to be positive to compensate a lender for undertaking the risk of borrowing. Most of economic theory fact that nominal interest rates should have a zero lower bound. In 1995, Black(1995) stated explicitly in his paper that it is possible to have negative real interest rate but we cannot have the negative nominal short rate. After almost twenty years we question if Black’s assumption was correct. In the current negative interest rate environment with around $ 13.5 trillion of negative-yielding bonds as reported by financial times in August 20161 central banks such as the European Central Bank have cut the deposit rate to below zero per cent. As a result, instead of paying interest to the banks or financial institutions that deposits their excess reserves to the central bank, the central bank taxes these deposits. As irrational as this concept may seem to be, the main idea behind it is to discourage the banks from parking the balances at the central bank, instead increase their lending or investments. However negative nominal interest rates is not a completely new concept. One could trace negative nominal interest rates back in the 19th century when "Gesell Tax" was introduced to overcome the zero-lower-bound on nominal interest rates Menner(2011). Similary in the 1970s the Swiss National Bank also experimented with negative rates to control capital inflows in a bid to prevent the Swiss Franc from appreciating. Looking with fresh eyes, in the past few years, we have witnessed the changes of interest rate environment. The global financial system 1This information was retrieved from Financial Times website: https://www.ft.com/content/973b6060-60ce- 11e6-ae3f-77baadeb1c93 on 12-14-2016 5 has been venturing further into the whole new world of negative interest rates. Between 2014 and 2016, five central banks namely, European Central Bank (ECB), Sveriges Riksbank (SR), Bank of Japan, Denmark National bank (DN) and the Swiss National Bank (SNB) decided to implement negative rate policy. As it can be seen from Figure 1.1 below 2. ECB was the first bank to decrease their interest rates to below zero in June 2014, since then the rate has been dividing deeper below zero. Figure 1.1: This figure shows the European Central bank’s interest rates from 2008 until early 2016. 1.2 Motivation for Negative Interest Rate Policies There are different motivations for implementing negative interest rate policy. Bech and Malk- hozov(2016) mention different reasons for the implementation of the negative interest rate policy in Europe. One of the major reason to implement negative rate policy has been to boost the economy and to raise inflation which is currently below zero. The other reason is to pre- vent high rising of the currency. By lowering negative interest rates, investors are discouraged by banks from buying the local currency hence preventing its value from rising up. Table 13 below summarizes the rationale behind implementation of negative rate policy by different banks in Europe. 2The figure is extracted from https://www.bloomberg.com/quicktake/negative-interest-rates 3The table is extracted from Jackson(2015). In the table; bp stands for basis points and DKK is the ISO code for Danish krone 6 1.3 Motivation and Problem Formulation In this section, we discuss the motivation behind carrying out this research and highlight why studying risk in the negative interest rate environment is important. 1.3.1 Motivation After having looked at the current negative interest environment in the world (especially in Europe), it is important to now look at how all this has an impact on the interest rate derivatives traded in the market. There is need to incorporate this new reality of negative interest rates in our models and in our volatility assumptions. 7 The need to change models Black’s model has been used as the standard model in the market to price interest rate de- rivatives. As will be discussed later, the key feature of this model is that it assumes that the forward rates are lognormally distributed. This assumption allows the Black model to work only with positive values, the Black model valuation formula is constructed in such a way that it rejects any negative values. It is then clear that in negative interest rate environment where we have negative values we cannot use the standard models like Black model to price and hedge interest rate derivatives.
Load more

Recommended publications
  • Differences Between Financial Options and Real Options
    Lecture Notes in Management Science (2012) Vol. 4: 169–178 4th International Conference on Applied Operational Research, Proceedings © Tadbir Operational Research Group Ltd. All rights reserved. www.tadbir.ca ISSN 2008-0050 (Print), ISSN 1927-0097 (Online) Differences between financial options and real options Tero Haahtela Aalto University, BIT Research Centre, Helsinki, Finland [email protected] Abstract. Real option valuation is often presented to be analogous with financial options valuation. The majority of research on real options, especially classic papers, are closely connected to financial option valuation. They share most of the same assumption about contingent claims analysis and apply close form solutions for partial difference equations. However, many real-world investments have several qualities that make use of the classical approach difficult. This paper presents many of the differences that exist between the financial and real options. Whereas some of the differences are theoretical and academic by nature, some are significant from a practical perspective. As a result of these differences, the present paper suggests that numerical methods and models based on calculus and simulation may be more intuitive and robust methods with looser assumptions for practical valuation. New methods and approaches are still required if the real option valuation is to gain popularity outside academia among practitioners and decision makers. Keywords: real options; financial options; investment under uncertainty, valuation Introduction Differences between real and financial options Real option valuation is often presented to be significantly analogous with financial options valuation. The analogy is often presented in a table format that links the Black and Scholes (1973) option valuation parameters to the real option valuation parameters.
    [Show full text]
  • Valuation of Stock Options
    VALUATION OF STOCK OPTIONS The right to buy or sell a given security at a specified time in the future is an option. Options are “derivatives’, i.e. they derive their value from the security that is to be traded in accordance with the option. Determining the value of publicly traded stock options is not a problem as the market sets the price. The value of a non-publicly traded option, however, depends on a number of factors: the price of the underlying security at the time of valuation, the time to exercise the option, the volatility of the underlying security, the risk free interest rate, the dividend rate, and the strike price. In addition to these financial variables, subjective variables also play a role. For example, in the case of employee stock options, the employee and the employer have different reasons for valuation, and, therefore, differing needs result in a different valuation for each party. This appendix discusses the reasons for the valuation of non-publicly traded options, explores and compares the two predominant valuation models, and attempts to identify the appropriate valuation methods for a variety of situations. We do not attempt to explain option-pricing theory in depth nor do we attempt mathematical proofs of the models. We focus on the practical aspects of option valuation, directing the reader to the location of computer calculators and sources of the data needed to enter into those calculators. Why Value? The need for valuation arises in a variety of circumstances. In the valuation of a business, for example, options owned by that business must be valued like any other investment asset.
    [Show full text]
  • Modeling the Steel Price for Valuation of Real Options and Scenario Simulation
    Norwegian School of Economics Bergen, December, 2015 Modeling the Steel Price for Valuation of Real Options and Scenario Simulation Master thesis in Finance Norwegian School of Economics Written by: Lasse Berggren Supervisor: Jørgen Haug This thesis was written as a part of the Master of Science in Economics and Business Administration at NHH. Please note that neither the institution nor the examiners are responsible | through the approval of this thesis | for the theories and methods used, or results and conclusions drawn in this work. 1 Summary Steel is widely used in construction. I tried to model the steel price such that valuations and scenario simulations could be done. To achieve a high level of precision this is done with a continuous-time continuous-state model. The model is more precise than a binomial tree, but not more economically interesting. I have treated the nearest futures price as the steel price. If one considers options expiring at the same time as the futures, it will be the same as if the spot were traded. If the maturity is short such that details like this matters, one should treat the futures as a spot providing a convenience yield equal to the interest rate earned on the delayed payment. This will in the model be the risk-free rate. Then I have considered how the drift can be modelled for real world scenario simu- lation. It involves discretion, as opposed to finding a convenient AR(1) representation, because the ADF-test could not reject non-stationarity (unit root). Given that the underlying is traded in a well functioning market such that prices reflect investors attitude towards risk, will the drift of the underlying disappear in the one-factor model applied to value a real-option.
    [Show full text]
  • The Promise and Peril of Real Options
    1 The Promise and Peril of Real Options Aswath Damodaran Stern School of Business 44 West Fourth Street New York, NY 10012 [email protected] 2 Abstract In recent years, practitioners and academics have made the argument that traditional discounted cash flow models do a poor job of capturing the value of the options embedded in many corporate actions. They have noted that these options need to be not only considered explicitly and valued, but also that the value of these options can be substantial. In fact, many investments and acquisitions that would not be justifiable otherwise will be value enhancing, if the options embedded in them are considered. In this paper, we examine the merits of this argument. While it is certainly true that there are options embedded in many actions, we consider the conditions that have to be met for these options to have value. We also develop a series of applied examples, where we attempt to value these options and consider the effect on investment, financing and valuation decisions. 3 In finance, the discounted cash flow model operates as the basic framework for most analysis. In investment analysis, for instance, the conventional view is that the net present value of a project is the measure of the value that it will add to the firm taking it. Thus, investing in a positive (negative) net present value project will increase (decrease) value. In capital structure decisions, a financing mix that minimizes the cost of capital, without impairing operating cash flows, increases firm value and is therefore viewed as the optimal mix.
    [Show full text]
  • Negative Rates in Derivatives Pricing. Theory and Practice Agustín Pineda
    Negative Rates in Derivatives Pricing. Theory and Practice Agustín Pineda García Trabajo de investigación 021/017 Master en Banca y Finanzas Cuantitativas Tutores: Dr. Manuel Moreno Fuentes Gregorio Vargas Martínez Universidad Complutense de Madrid Universidad del País Vasco Universidad de Valencia Universidad de Castilla-La Mancha www.finanzascuantitativas.com Negative rates in derivatives pricing. Theory and Practice A MSc Thesis submitted to the Complutense University of Madrid in partial fulfilment of the requirements for the degree Master’s degree in Banking and Quantitative Finance July, 2017 Agust´ın Pineda Garcia† Academic supervisor: Manuel Moreno Fuentes†† Industry supervisor: Gregorio Vargas Mart´ınez‡‡ †[email protected] ††Associate Professor at University of Castilla-La Mancha ‡‡Market Risk Manager at EY Acknowledgments To my supervisors, Gregorio and Manuel, for their constant guidance and support. Your selfless help has been one of the main pillars of this project. To Alvaro,´ Alberto and Samu, who needed between 5 to 7 minutes to find and down- load those non-standard volatilities I had previously been searching for too many days. You are stars. To my friends, for su↵ering my bad mood when things just would not work. Wherever I am, you are always with me. Alba, Tamara, Fran, Miguel, Rober, Rub´en,Carlos, Joan, Ferran:Ideservenoneofyou,sothankyouall. To Eli, for an unforgettable journey full of laughter and companionship. Late-night discussions about quantile regression, friends, Archimedean copulae, life, martingale rep- resentation theorem and less important topics made my day. Everyday. To Dani, for assuming the role of being the best friend one could only imagine. You played it nicely, as every time you are on the stage.
    [Show full text]
  • Numerical Valuation of European and American Options Under Kou's
    Numerical Valuation of European and American Options under Kou’s Jump-Diffusion Model∗ Jari Toivanen† Abstract Numerical methods are developed for pricing European and American op- tions under Kou’s jump-diffusion model which assumes the price of the un- derlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a Eu- ropean option is given by a partial integro-differential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differ- ences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are de- scribed: an operator slitting method and a penalty method. Numerical experi- ments confirm that the developed methods are very efficient as fairly accurate option prices can be computed in a few milliseconds on a PC. Keywords: option pricing, jump-diffusion model, partial integro-differential equa- tion, linear complementarity problem, finite difference method, operator splitting method, penalty method 1 Introduction The amount of financial option trading has grown to enormous scale since the pio- neering work by Black and Scholes
    [Show full text]
  • The Valuation of Volatility Options
    99s-43 The Valuation of Volatility Options Jérôme Detemple, Carlton Osakwe Série Scientifique Scientific Series Montréal Décembre 1999 CIRANO Le CIRANO est un organisme sans but lucratif constitué en vertu de la Loi des compagnies du Québec. Le financement de son infrastructure et de ses activités de recherche provient des cotisations de ses organisations- membres, d=une subvention d=infrastructure du ministère de la Recherche, de la Science et de la Technologie, de même que des subventions et mandats obtenus par ses équipes de recherche. CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère de la Recherche, de la Science et de la Technologie, and grants and research mandates obtained by its research teams. Les organisations-partenaires / The Partner Organizations $École des Hautes Études Commerciales $École Polytechnique $Université Concordia $Université de Montréal $Université du Québec à Montréal $Université Laval $Université McGill $MEQ $MRST $Alcan Aluminium Ltée $Banque Nationale du Canada $Banque Royale du Canada $Bell Québec $Développement des ressources humaines Canada (DRHC) $Egis $Fédération des caisses populaires Desjardins de Montréal et de l=Ouest-du-Québec $Hydro-Québec $Imasco $Industrie Canada $Microcell Labs inc. $Raymond Chabot Grant Thornton $Téléglobe Canada $Ville de Montréal © 1999 Jérôme Detemple et Carlton Osakwe. Tous droits réservés. All rights reserved. Reproduction partielle permise avec citation du document source, incluant la notice ©. Short sections may be quoted without explicit permission, provided that full credit, including © notice, is given to the source. Ce document est publié dans l=intention de rendre accessibles les résultats préliminaires de la recherche effectuée au CIRANO, afin de susciter des échanges et des suggestions.
    [Show full text]
  • Chapter 21 . Options
    Chapter 21 : Options-1 CHAPTER 21 . OPTIONS Contents I. INTRODUCTION · BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes Option Pricing Model (1973, BSOPM) E. Put-Call Parity III. EXERCISE FOR PAYOFF DIAGRAM IV. ASSIGNED PROBLEMS FROM THE TEXTBOOK V. SELF-EXERCISE PROBLEMS I. INTRODUCTION · BASIC TERMS 1. Option : right to buy (or sell) an asset at a fixed price on or before a given date Right ® buyer of option has no obligation, seller of option is obligated Call ® right to buy Put ® right to sell Note: Option may be written on any type of asset => most common is stock 2. Exercising the option - buying or selling asset by using option 3. Strike (or exercise) price - price at which asset may be bought or sold 4. Expiration date - last date on which option may be exercised 5. European option - may be exercised only at expiration date 6. American option - may be exercised on or before expiration date 7. In-the-money - positive cash flow if exercised ® call [put] =? 8. Out-of-the-money - negative cash flow if exercised ® call [put] = ? Chapter 21 : Options-2 9. At-the-money - zero cash flow if exercised ® call [put] = ? Chapter 21 : Options-3 II. VALUATION OF OPTIONS A. Minimum Values of Options 1. Minimum Value of Call A call option is an instrument with limited liability. If the call holder sees that it is advantageous to exercise it, the call will be exercised. If exercising it will decrease the call holder's wealth, the holder will not exercise it.
    [Show full text]
  • Download Download
    CBU INTERNATIONAL CONFERENCE ON INNOVATION, TECHNOLOGY TRANSFER AND EDUCATION MARCH 25-27, 2015, PRAGUE, CZECH REPUBLIC WWW.CBUNI.CZ, OJS.JOURNALS.CZ MODIFICATION OF DELTA FOR CHOOSER OPTIONS Marek Ďurica1 Abstract: Correctly used financial derivatives can help investors increase their expected returns and minimize their exposure to risk. To ensure the specific needs of investors, a large number of different types of non- standard exotic options is used. Chooser option is one of them. It is an option that gives its holder the right to choose at some predetermined future time whether the option will be a standard call or put with predetermined strike price and maturity time. Although the chooser options are more expensive than standard European-style options, in many cases they are a more suitable instrument for investors in hedging their portfolio value. For an effective use of the chooser option as a hedging instrument, it is necessary to check the values of the Greek parameters delta and gamma for the options. Especially, if the value of the parameter gamma is too large, hedging of the portfolio value using only parameter delta is insufficient and brings high transaction costs because the portfolio has to be reviewed relatively often. Therefore, in this article, a modification of delta-hedging as well as using the value of parameter gamma is suggested. Error of the delta modification is analyzed and compared with the error of widely used parameter delta. Typical patterns for the modified hedging parameter variation with various time to choose time for chooser options are also presented in this article.
    [Show full text]
  • Annual Report 2014
    CDP - Bilancio2014_Cover-eng 8:Layout 1 25/06/15 16:37 Pagina 1 Cassa ANNUAL REPORT depositi Rome Milan Brussels e prestiti Via Goito, 4 Palazzo Busca Square de Meeûs, 37 00185 Rome Corso Magenta, 71 (7th floor) Italy 20123 Milan 1000 Bruxelles Tel +39 06 4221.1 Italy Belgium Tel +39 02 4674.4322 Tel +32 2 2131950 ANNUAL REPORT www.cdp.it 2014 (Translation from the Italian original) Contents 5 Introduction Role and mission of the CDP Group 6 Company Officers 12 Letter from the Chairman 16 Letter from the CEO 18 21 Report on operations of the Group 1. Overview of 2014 22 2. Macroeconomic scenario and the market 27 3. Composition of the CDP Group 36 4. Financial position and performance 48 5. Operating performance 66 6. Outlook 119 7. Corporate Governance 120 8. Relations of the Parent Company with the MEF 141 9. Proposed allocation of net income for the year 144 145 Allocation of net income for the year 149 Separate Financial Statements 327 Annexes 337 Report of the Board of Auditors 343 Report of the independent auditors 347 Certification of the separate financial statements pursuant to Article 154-bis of Legislative Decree 58/1998 351 Consolidated Financial Statements 621 Annexes 631 Report of the independent auditors 635 Certification of the consolidated financial statements pursuant to Article 154-bis of Legislative Decree 58/1998 Role and mission of the CDP Group PRESENTATION OF THE CDP GROUP The CDP Group (the “Group”) works to support growth in Italy. It employs its resources – mainly fund- ed through its management of postal savings (postal savings bonds and postal passbook savings ac- counts) – in accordance with its institutional mission, in its capacity as a: • leader financier of investments by the public administration; • catalyst for infrastructure development; • key player in supporting the Italian economy and business system.
    [Show full text]
  • Option Prices and Pricing Theory: Combining Financial Mathematics with Statistical Modeling
    Option Prices and Pricing Theory: Combining Financial Mathematics with Statistical Modeling Ling Chen Securities Division, Goldman Sachs Tze Leung Lai Department of Statistics, Stanford University Tiong Wee Lim Department of Statistics and Applied Probability, National University of Singapore Keywords Option pricing, Substantive models, Nonparametric regression, Semiparametric regression, Time series modeling Abstract After an overview of important developments of option pricing theory, this article describes statistical ap- proaches to modeling the difference between the theoretical and actual prices. An empirical study is given to compare various approaches. A cornerstone of Financial Mathematics is option pricing theory, which Ross1 has described as “the most successful theory not only in finance, but in all of economics.” A call (put) option gives the holder the right to buy (sell) the underlying asset by a certain expiration date T at a certain price K, which is called the strike price. If the option can be exercised only at T , it is termed European or European-style, whereas if early exercise is also allowed at any time before T , the option is termed American or American-style. In the seminal work of Black and Scholes,2 a dynamic hedging argument was used in conjunction with a no- arbitrage condition to derive closed-form pricing formulas for European options written on an asset whose price St at time t is modeled by a geometric Brownian motion 2 St = S0 expf(µ − σ =2)t + σBtg; (1) where µ and σ are the mean and standard deviation of the asset’s return dSt=St and fBt; t ≥ 0g is a standard Brownian motion; σ is commonly known as the asset’s volatility.
    [Show full text]
  • Numerical Methods for the Valuation of Financial Derivatives
    NUMERICAL METHODS FOR THE VALUATION OF FINANCIAL DERIVATIVES DAVIS BUNDI NTWIGA A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science, in the Department of Mathematics and Applied Mathematics, University of Western Cape, South Africa. Supervisor: Prof. Jean Claude Ndogmo November 2005 NUMERICAL METHODS FOR THE VALUATION OF FINANCIAL DERIVATIVES. DAVIS BUNDI NTWIGA KEYWORDS Black Scholes. Financial Derivatives. Exotic Options. Futures. American and European Options. Stochastic Differential Equations. Multi Period Binomial Model. Finite Difference Method. Monte Carlo Simulation. Differential Equations. i ABSTRACT NUMERICAL METHODS FOR THE VALUATION OF FINANCIAL DERIVATIVES. DAVIS BUNDI NTWIGA MSc thesis, Department of Mathematics and Applied Mathematics, University of Western Cape. Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps.
    [Show full text]