A Lattice Framework for Option Pricing with Two State Variables
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Up to EUR 3,500,000.00 7% Fixed Rate Bonds Due 6 April 2026 ISIN
Up to EUR 3,500,000.00 7% Fixed Rate Bonds due 6 April 2026 ISIN IT0005440976 Terms and Conditions Executed by EPizza S.p.A. 4126-6190-7500.7 This Terms and Conditions are dated 6 April 2021. EPizza S.p.A., a company limited by shares incorporated in Italy as a società per azioni, whose registered office is at Piazza Castello n. 19, 20123 Milan, Italy, enrolled with the companies’ register of Milan-Monza-Brianza- Lodi under No. and fiscal code No. 08950850969, VAT No. 08950850969 (the “Issuer”). *** The issue of up to EUR 3,500,000.00 (three million and five hundred thousand /00) 7% (seven per cent.) fixed rate bonds due 6 April 2026 (the “Bonds”) was authorised by the Board of Directors of the Issuer, by exercising the powers conferred to it by the Articles (as defined below), through a resolution passed on 26 March 2021. The Bonds shall be issued and held subject to and with the benefit of the provisions of this Terms and Conditions. All such provisions shall be binding on the Issuer, the Bondholders (and their successors in title) and all Persons claiming through or under them and shall endure for the benefit of the Bondholders (and their successors in title). The Bondholders (and their successors in title) are deemed to have notice of all the provisions of this Terms and Conditions and the Articles. Copies of each of the Articles and this Terms and Conditions are available for inspection during normal business hours at the registered office for the time being of the Issuer being, as at the date of this Terms and Conditions, at Piazza Castello n. -
Futures and Options Workbook
EEXAMININGXAMINING FUTURES AND OPTIONS TABLE OF 130 Grain Exchange Building 400 South 4th Street Minneapolis, MN 55415 www.mgex.com [email protected] 800.827.4746 612.321.7101 Fax: 612.339.1155 Acknowledgements We express our appreciation to those who generously gave their time and effort in reviewing this publication. MGEX members and member firm personnel DePaul University Professor Jin Choi Southern Illinois University Associate Professor Dwight R. Sanders National Futures Association (Glossary of Terms) INTRODUCTION: THE POWER OF CHOICE 2 SECTION I: HISTORY History of MGEX 3 SECTION II: THE FUTURES MARKET Futures Contracts 4 The Participants 4 Exchange Services 5 TEST Sections I & II 6 Answers Sections I & II 7 SECTION III: HEDGING AND THE BASIS The Basis 8 Short Hedge Example 9 Long Hedge Example 9 TEST Section III 10 Answers Section III 12 SECTION IV: THE POWER OF OPTIONS Definitions 13 Options and Futures Comparison Diagram 14 Option Prices 15 Intrinsic Value 15 Time Value 15 Time Value Cap Diagram 15 Options Classifications 16 Options Exercise 16 F CONTENTS Deltas 16 Examples 16 TEST Section IV 18 Answers Section IV 20 SECTION V: OPTIONS STRATEGIES Option Use and Price 21 Hedging with Options 22 TEST Section V 23 Answers Section V 24 CONCLUSION 25 GLOSSARY 26 THE POWER OF CHOICE How do commercial buyers and sellers of volatile commodities protect themselves from the ever-changing and unpredictable nature of today’s business climate? They use a practice called hedging. This time-tested practice has become a stan- dard in many industries. Hedging can be defined as taking offsetting positions in related markets. -
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. -
Finance: a Quantitative Introduction Chapter 9 Real Options Analysis
Investment opportunities as options The option to defer More real options Some extensions Finance: A Quantitative Introduction Chapter 9 Real Options Analysis Nico van der Wijst 1 Finance: A Quantitative Introduction c Cambridge University Press Investment opportunities as options The option to defer More real options Some extensions 1 Investment opportunities as options 2 The option to defer 3 More real options 4 Some extensions 2 Finance: A Quantitative Introduction c Cambridge University Press Investment opportunities as options Option analogy The option to defer Sources of option value More real options Limitations of option analogy Some extensions The essential economic characteristic of options is: the flexibility to exercise or not possibility to choose best alternative walk away from bad outcomes Stocks and bonds are passively held, no flexibility Investments in real assets also have flexibility, projects can be: delayed or speeded up made bigger or smaller abandoned early or extended beyond original life-time, etc. 3 Finance: A Quantitative Introduction c Cambridge University Press Investment opportunities as options Option analogy The option to defer Sources of option value More real options Limitations of option analogy Some extensions Real Options Analysis Studies and values this flexibility Real options are options where underlying value is a real asset not a financial asset as stock, bond, currency Flexibility in real investments means: changing cash flows along the way: profiting from opportunities, cutting off losses Discounted -
Seeking Income: Cash Flow Distribution Analysis of S&P 500
RESEARCH Income CONTRIBUTORS Berlinda Liu Seeking Income: Cash Flow Director Global Research & Design Distribution Analysis of S&P [email protected] ® Ryan Poirier, FRM 500 Buy-Write Strategies Senior Analyst Global Research & Design EXECUTIVE SUMMARY [email protected] In recent years, income-seeking market participants have shown increased interest in buy-write strategies that exchange upside potential for upfront option premium. Our empirical study investigated popular buy-write benchmarks, as well as other alternative strategies with varied strike selection, option maturity, and underlying equity instruments, and made the following observations in terms of distribution capabilities. Although the CBOE S&P 500 BuyWrite Index (BXM), the leading buy-write benchmark, writes at-the-money (ATM) monthly options, a market participant may be better off selling out-of-the-money (OTM) options and allowing the equity portfolio to grow. Equity growth serves as another source of distribution if the option premium does not meet the distribution target, and it prevents the equity portfolio from being liquidated too quickly due to cash settlement of the expiring options. Given a predetermined distribution goal, a market participant may consider an option based on its premium rather than its moneyness. This alternative approach tends to generate a more steady income stream, thus reducing trading cost. However, just as with the traditional approach that chooses options by moneyness, a high target premium may suffocate equity growth and result in either less income or quick equity depletion. Compared with monthly standard options, selling quarterly options may reduce the loss from the cash settlement of expiring calls, while selling weekly options could incur more loss. -
The Recovery Theorem
The Recovery Theorem The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Ross, Steve. “The Recovery Theorem.” The Journal of Finance 70, no. 2 (April 2015): 615–48. As Published http://dx.doi.org/10.1111/jofi.12092 Publisher American Finance Association/Wiley Version Original manuscript Citable link http://hdl.handle.net/1721.1/99126 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ The Recovery Theorem Journal of Finance forthcoming Steve Ross Franco Modigliani Professor of Financial Economics Sloan School, MIT Abstract We can only estimate the distribution of stock returns but from option prices we observe the distribution of state prices. State prices are the product of risk aversion – the pricing kernel – and the natural probability distribution. The Recovery Theorem enables us to separate these so as to determine the market’s forecast of returns and the market’s risk aversion from state prices alone. Among other things, this allows us to recover the pricing kernel, the market risk premium, the probability of a catastrophe, and to construct model free tests of the efficient market hypothesis. I want to thank the participants in the UCLA Finance workshop for their insightful comments as well as Richard Roll, Hanno Lustig, Rick Antle, Andrew Jeffrey, Peter Carr, Kevin Atteson, Jessica Wachter, Ian Martin, Leonid Kogan, Torben Andersen, John Cochrane, Dimitris Papanikolaou, William Mullins, Jon Ingersoll, Jerry Hausman, Andy Lo, Steve Leroy, George Skiadopoulos, Xavier Gabaix, Patrick Dennis, Phil Dybvig, Will Mullins, Nicolas Caramp, Rodrigo Adao, the referee and the associate editor and editor. -
Implied Volatility Modeling
Implied Volatility Modeling Sarves Verma, Gunhan Mehmet Ertosun, Wei Wang, Benjamin Ambruster, Kay Giesecke I Introduction Although Black-Scholes formula is very popular among market practitioners, when applied to call and put options, it often reduces to a means of quoting options in terms of another parameter, the implied volatility. Further, the function σ BS TK ),(: ⎯⎯→ σ BS TK ),( t t ………………………………(1) is called the implied volatility surface. Two significant features of the surface is worth mentioning”: a) the non-flat profile of the surface which is often called the ‘smile’or the ‘skew’ suggests that the Black-Scholes formula is inefficient to price options b) the level of implied volatilities changes with time thus deforming it continuously. Since, the black- scholes model fails to model volatility, modeling implied volatility has become an active area of research. At present, volatility is modeled in primarily four different ways which are : a) The stochastic volatility model which assumes a stochastic nature of volatility [1]. The problem with this approach often lies in finding the market price of volatility risk which can’t be observed in the market. b) The deterministic volatility function (DVF) which assumes that volatility is a function of time alone and is completely deterministic [2,3]. This fails because as mentioned before the implied volatility surface changes with time continuously and is unpredictable at a given point of time. Ergo, the lattice model [2] & the Dupire approach [3] often fail[4] c) a factor based approach which assumes that implied volatility can be constructed by forming basis vectors. Further, one can use implied volatility as a mean reverting Ornstein-Ulhenbeck process for estimating implied volatility[5]. -
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. -
A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model
risks Article A Discrete-Time Approach to Evaluate Path-Dependent Derivatives in a Regime-Switching Risk Model Emilio Russo Department of Economics, Statistics and Finance, University of Calabria, Ponte Bucci cubo 1C, 87036 Rende (CS), Italy; [email protected] Received: 29 November 2019; Accepted: 25 January 2020 ; Published: 29 January 2020 Abstract: This paper provides a discrete-time approach for evaluating financial and actuarial products characterized by path-dependent features in a regime-switching risk model. In each regime, a binomial discretization of the asset value is obtained by modifying the parameters used to generate the lattice in the highest-volatility regime, thus allowing a simultaneous asset description in all the regimes. The path-dependent feature is treated by computing representative values of the path-dependent function on a fixed number of effective trajectories reaching each lattice node. The prices of the analyzed products are calculated as the expected values of their payoffs registered over the lattice branches, invoking a quadratic interpolation technique if the regime changes, and capturing the switches among regimes by using a transition probability matrix. Some numerical applications are provided to support the model, which is also useful to accurately capture the market risk concerning path-dependent financial and actuarial instruments. Keywords: regime-switching risk; market risk; path-dependent derivatives; insurance policies; binomial lattices; discrete-time models 1. Introduction With the aim of providing an accurate evaluation of the risks affecting financial markets, a wide range of empirical research evidences that asset returns show stochastic volatility patterns and fatter tails with respect to the standard normal model. -
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 Me the Money: Option Moneyness Concentration and Future Stock Returns Kelley Bergsma Assistant Professor of Finance Ohio Un
Show Me the Money: Option Moneyness Concentration and Future Stock Returns Kelley Bergsma Assistant Professor of Finance Ohio University Vivien Csapi Assistant Professor of Finance University of Pecs Dean Diavatopoulos* Assistant Professor of Finance Seattle University Andy Fodor Professor of Finance Ohio University Keywords: option moneyness, implied volatility, open interest, stock returns JEL Classifications: G11, G12, G13 *Communications Author Address: Albers School of Business and Economics Department of Finance 901 12th Avenue Seattle, WA 98122 Phone: 206-265-1929 Email: [email protected] Show Me the Money: Option Moneyness Concentration and Future Stock Returns Abstract Informed traders often use options that are not in-the-money because these options offer higher potential gains for a smaller upfront cost. Since leverage is monotonically related to option moneyness (K/S), it follows that a higher concentration of trading in options of certain moneyness levels indicates more informed trading. Using a measure of stock-level dollar volume weighted average moneyness (AveMoney), we find that stock returns increase with AveMoney, suggesting more trading activity in options with higher leverage is a signal for future stock returns. The economic impact of AveMoney is strongest among stocks with high implied volatility, which reflects greater investor uncertainty and thus higher potential rewards for informed option traders. AveMoney also has greater predictive power as open interest increases. Our results hold at the portfolio level as well as cross-sectionally after controlling for liquidity and risk. When AveMoney is calculated with calls, a portfolio long high AveMoney stocks and short low AveMoney stocks yields a Fama-French five-factor alpha of 12% per year for all stocks and 33% per year using stocks with high implied volatility.