Option Pricing Theory and Models
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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. -
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. -
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. -
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. -
The Evaluation of American Compound Option Prices Under Stochastic Volatility and Stochastic Interest Rates
THE EVALUATION OF AMERICAN COMPOUND OPTION PRICES UNDER STOCHASTIC VOLATILITY AND STOCHASTIC INTEREST RATES CARL CHIARELLA♯ AND BODA KANG† Abstract. A compound option (the mother option) gives the holder the right, but not obligation to buy (long) or sell (short) the underlying option (the daughter option). In this paper, we consider the problem of pricing American-type compound options when the underlying dynamics follow Heston’s stochastic volatility and with stochastic interest rate driven by Cox-Ingersoll-Ross (CIR) processes. We use a partial differential equation (PDE) approach to obtain a numerical solution. The problem is formulated as the solution to a two-pass free boundary PDE problem which is solved via a sparse grid approach and is found to be accurate and efficient compared with the results from a benchmark solution based on a least-squares Monte Carlo simulation combined with the PSOR. Keywords: American compound option, stochastic volatility, stochastic interest rates, free boundary problem, sparse grid, combination technique, least squares Monte Carlo. JEL Classification: C61, D11. 1. Introduction The compound option goes back to the seminal paper of Black & Scholes (1973). As well as their famous pricing formulae for vanilla European call and put options, they also considered how to evaluate the equity of a company that has coupon bonds outstanding. They argued that the equity can be viewed as a “compound option” because the equity “is an option on an option on an option on the firm”. Geske (1979) developed · · · the first closed-form solution for the price of a vanilla European call on a European call. -
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. -
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 -
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. -
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. -
Sequential Compound Options and Investments Valuation
Sequential compound options and investments valuation Luigi Sereno Dottorato di Ricerca in Economia - XIX Ciclo - Alma Mater Studiorum - Università di Bologna Marzo 2007 Relatore: Prof. ssa Elettra Agliardi Coordinatore: Prof. Luca Lambertini Settore scienti…co-disciplinare: SECS-P/01 Economia Politica ii Contents I Sequential compound options and investments valua- tion 1 1 An overview 3 1.1 Introduction . 3 1.2 Literature review . 6 1.2.1 R&D as real options . 11 1.2.2 Exotic Options . 12 1.3 An example . 17 1.3.1 Value of expansion opportunities . 18 1.3.2 Value with abandonment option . 23 1.3.3 Value with temporary suspension . 26 1.4 Real option modelling with jump processes . 31 1.4.1 Introduction . 31 1.4.2 Merton’sapproach . 33 1.4.3 Further reading . 36 1.5 Real option and game theory . 41 1.5.1 Introduction . 41 1.5.2 Grenadier’smodel . 42 iii iv CONTENTS 1.5.3 Further reading . 45 1.6 Final remark . 48 II The valuation of new ventures 59 2 61 2.1 Introduction . 61 2.2 Literature Review . 63 2.2.1 Flexibility of Multiple Compound Real Options . 65 2.3 Model and Assumptions . 68 2.3.1 Value of the Option to Continuously Shut - Down . 69 2.4 An extension . 74 2.4.1 The mathematical problem and solution . 75 2.5 Implementation of the approach . 80 2.5.1 Numerical results . 82 2.6 Final remarks . 86 III Valuing R&D investments with a jump-di¤usion process 93 3 95 3.1 Introduction . -
FX Options and Structured Products
FX Options and Structured Products Uwe Wystup www.mathfinance.com 7 April 2006 www.mathfinance.de To Ansua Contents 0 Preface 9 0.1 Scope of this Book ................................ 9 0.2 The Readership ................................. 9 0.3 About the Author ................................ 10 0.4 Acknowledgments ................................ 11 1 Foreign Exchange Options 13 1.1 A Journey through the History Of Options ................... 13 1.2 Technical Issues for Vanilla Options ....................... 15 1.2.1 Value ................................... 16 1.2.2 A Note on the Forward ......................... 18 1.2.3 Greeks .................................. 18 1.2.4 Identities ................................. 20 1.2.5 Homogeneity based Relationships .................... 21 1.2.6 Quotation ................................ 22 1.2.7 Strike in Terms of Delta ......................... 26 1.2.8 Volatility in Terms of Delta ....................... 26 1.2.9 Volatility and Delta for a Given Strike .................. 26 1.2.10 Greeks in Terms of Deltas ........................ 27 1.3 Volatility ..................................... 30 1.3.1 Historic Volatility ............................ 31 1.3.2 Historic Correlation ........................... 34 1.3.3 Volatility Smile .............................. 35 1.3.4 At-The-Money Volatility Interpolation .................. 41 1.3.5 Volatility Smile Conventions ....................... 44 1.3.6 At-The-Money Definition ........................ 44 1.3.7 Interpolation of the Volatility on Maturity -
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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.