Modeling, Pricing and Hedging of Assets and Derivatives
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University of Illinois at Urbana-Champaign Department of Mathematics
University of Illinois at Urbana-Champaign Department of Mathematics ASRM 410 Investments and Financial Markets Fall 2018 Chapter 5 Option Greeks and Risk Management by Alfred Chong Learning Objectives: 5.1 Option Greeks: Black-Scholes valuation formula at time t, observed values, model inputs, option characteristics, option Greeks, partial derivatives, sensitivity, change infinitesimally, Delta, Gamma, theta, Vega, rho, Psi, option Greek of portfolio, absolute change, option elasticity, percentage change, option elasticity of portfolio. 5.2 Applications of Option Greeks: risk management via hedging, immunization, adverse market changes, delta-neutrality, delta-hedging, local, re-balance, from- time-to-time, holding profit, delta-hedging, first order, prudent, delta-gamma- hedging, gamma-neutrality, option value approximation, observed values change inevitably, Taylor series expansion, second order approximation, delta-gamma- theta approximation, delta approximation, delta-gamma approximation. Further Exercises: SOA Advanced Derivatives Samples Question 9. 1 5.1 Option Greeks 5.1.1 Consider a continuous-time setting t ≥ 0. Assume that the Black-Scholes framework holds as in 4.2.2{4.2.4. 5.1.2 The time-0 Black-Scholes European call and put option valuation formula in 4.2.9 and 4.2.10 can be generalized to those at time t: −δ(T −t) −r(T −t) C (t; S; δ; r; σ; K; T ) = Se N (d1) − Ke N (d2) P = Ft;T N (d1) − PVt;T (K) N (d2); −r(T −t) −δ(T −t) P (t; S; δ; r; σ; K; T ) = Ke N (−d2) − Se N (−d1) P = PVt;T (K) N (−d2) − Ft;T N (−d1) ; where d1 and d2 are given by S 1 2 ln K + r − δ + 2 σ (T − t) d1 = p ; σ T − t S 1 2 p ln K + r − δ − 2 σ (T − t) d2 = p = d1 − σ T − t: σ T − t 5.1.3 The option value V is a function of observed values, t and S, model inputs, δ, r, and σ, as well as option characteristics, K and T . -
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. -
Rational Multi-Curve Models with Counterparty-Risk Valuation Adjustments
Post-Crisis Interest Rate Markets and Models Rational Multi-Curve Models Rational Bilateral Counterparty Risk Model Conclusion Rational Multi-Curve Models with Counterparty-Risk Valuation Adjustments Stéphane Crépey Université d'Evry Val-d'Essonne, Laboratoire de Mathématiques et Modélisation d'Evry (LaMME) Joint work with A. Macrina, T. M. Nguyen and D. Skovmand 7th General AMaMeF and Swissquote Conference EPFL, 7-10 September 2015 The research presented in these slides beneted from the support of the Chair Markets in Transition under the aegis of Louis Bachelier laboratory, a joint initiative of École polytechnique, Université d'Évry Val d'Essonne and Fédération Bancaire Française 1 / 49 Post-Crisis Interest Rate Markets and Models Rational Multi-Curve Models Rational Bilateral Counterparty Risk Model Conclusion Outline 1 Post-Crisis Interest Rate Markets and Models 2 Rational Multi-Curve Models 3 Rational Bilateral Counterparty Risk Model 4 Conclusion 2 / 49 Post-Crisis Interest Rate Markets and Models Rational Multi-Curve Models Rational Bilateral Counterparty Risk Model Conclusion Libor Most interest-rate derivatives have Libor-indexed cash-ows (Libor xings) What is Libor? Libor stands for London InterBank Oered Rate. It is produced for 10 currencies with 15 maturities quoted for each, ranging from overnight to 12 Months producing 150 rates each business day. Libor is computed as a trimmed average of the interbank borrowing rates assembled from the Libor contributing banks. More precisely, every contributing bank has to submit an -
Ice Crude Oil
ICE CRUDE OIL Intercontinental Exchange® (ICE®) became a center for global petroleum risk management and trading with its acquisition of the International Petroleum Exchange® (IPE®) in June 2001, which is today known as ICE Futures Europe®. IPE was established in 1980 in response to the immense volatility that resulted from the oil price shocks of the 1970s. As IPE’s short-term physical markets evolved and the need to hedge emerged, the exchange offered its first contract, Gas Oil futures. In June 1988, the exchange successfully launched the Brent Crude futures contract. Today, ICE’s FSA-regulated energy futures exchange conducts nearly half the world’s trade in crude oil futures. Along with the benchmark Brent crude oil, West Texas Intermediate (WTI) crude oil and gasoil futures contracts, ICE Futures Europe also offers a full range of futures and options contracts on emissions, U.K. natural gas, U.K power and coal. THE BRENT CRUDE MARKET Brent has served as a leading global benchmark for Atlantic Oseberg-Ekofisk family of North Sea crude oils, each of which Basin crude oils in general, and low-sulfur (“sweet”) crude has a separate delivery point. Many of the crude oils traded oils in particular, since the commercialization of the U.K. and as a basis to Brent actually are traded as a basis to Dated Norwegian sectors of the North Sea in the 1970s. These crude Brent, a cargo loading within the next 10-21 days (23 days on oils include most grades produced from Nigeria and Angola, a Friday). In a circular turn, the active cash swap market for as well as U.S. -
Yuichi Katsura
P&TcrvvCJ Efficient ValtmVmm-and Hedging of Structured Credit Products Yuichi Katsura A thesis submitted for the degree of Doctor of Philosophy of the University of London May 2005 Centre for Quantitative Finance Imperial College London nit~ýl. ABSTRACT Structured credit derivatives such as synthetic collateralized debt obligations (CDO) have been the principal growth engine in the fixed income market over the last few years. Val- uation and risk management of structured credit products by meads of Monte Carlo sim- ulations tend to suffer from numerical instability and heavy computational time. After a brief description of single name credit derivatives that underlie structured credit deriva- tives, the factor model is introduced as the portfolio credit derivatives pricing tool. This thesis then describes the rapid pricing and hedging of portfolio credit derivatives using the analytical approximation model and semi-analytical models under several specifications of factor models. When a full correlation structure is assumed or the pricing of exotic structure credit products with non-linear pay-offs is involved, the semi-analytical tractability is lost and we have to resort to the time-consuming Monte Carlo method. As a Monte Carlo variance reduction technique we extend the importance sampling method introduced to credit risk modelling by Joshi [2004] to the general n-factor problem. The large homogeneouspool model is also applied to the method of control variates as a new variance reduction tech- niques for pricing structured credit products. The combination of both methods is examined and it turns out to be the optimal choice. We also show how sensitivities of a CDO tranche can be computed efficiently by semi-analytical and Monte Carlo framework. -
RT Options Scanner Guide
RT Options Scanner Introduction Our RT Options Scanner allows you to search in real-time through all publicly traded US equities and indexes options (more than 170,000 options contracts total) for trading opportunities involving strategies like: - Naked or Covered Write Protective Purchase, etc. - Any complex multi-leg option strategy – to find the “missing leg” - Calendar Spreads Search is performed against our Options Database that is updated in real-time and driven by HyperFeed’s Ticker-Plant and our Implied Volatility Calculation Engine. We offer both 20-minute delayed and real-time modes. Unlike other search engines using real-time quotes data only our system takes advantage of individual contracts Implied Volatilities calculated in real-time using our high-performance Volatility Engine. The Scanner is universal, that is, it is not confined to any special type of option strategy. Rather, it allows determining the option you need in the following terms: – Cost – Moneyness – Expiry – Liquidity – Risk The Scanner offers Basic and Advanced Search Interfaces. Basic interface can be used to find options contracts satisfying, for example, such requirements: “I need expensive Calls for Covered Call Write Strategy. They should expire soon and be slightly out of the money. Do not care much as far as liquidity, but do not like to bear high risk.” This can be done in five seconds - set the search filters to: Cost -> Dear Moneyness -> OTM Expiry -> Short term Liquidity -> Any Risk -> Moderate and press the “Search” button. It is that easy. In fact, if your request is exactly as above, you can just press the “Search” button - these values are the default filtering criteria. -
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. -
Economic Information Transmissions and Liquidity Between Shipping Markets: New Evidence from Freight Derivatives ⇑ G
Transportation Research Part E 98 (2017) 82–104 Contents lists available at ScienceDirect Transportation Research Part E journal homepage: www.elsevier.com/locate/tre Economic information transmissions and liquidity between shipping markets: New evidence from freight derivatives ⇑ G. Alexandridis a, S. Sahoo a, I. Visvikis b, a ICMA Centre, Henley Business School, University of Reading, Whiteknights, Reading RG6 6BA, UK b World Maritime University, Fiskehamnsgatan 1, SE-211 18 Malmö, Sweden article info abstract Article history: Economic return and volatility spillovers of derivatives markets on a number of assets have Received 6 July 2016 been extensively examined in the general economics literature. However, there are only a Received in revised form 24 November 2016 limited number of studies that investigate such interactions between freight rates and the Accepted 5 December 2016 freight futures, and no studies that also consider potential linkages with freight options. This study fills this gap by investigating the economic spillovers between time-charter rates, freight futures and freight options prices in the dry-bulk sector of the international JEL Classification: shipping industry. Empirical results indicate the existence of significant information trans- C32 mission in both returns and volatilities between the three related markets, which we attri- G13 G14 bute to varying trading activity and market liquidity. The results also point out that, consistent with theory, the freight futures market informationally leads the freight rate Keywords: market, though surprisingly, freight options lag behind both futures and physical freight Freight derivatives rates. The documented three-way economic interactions between the related markets Options contracts can be used to enhance budget planning and risk management strategies, potentially Price discovery attract more investors, and thus, improve the liquidity of the freight derivatives market. -
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. -
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. -
Option Greeks Demystified
Webinar Presentation Option Greeks Demystified Presented by Trading Strategy Desk 1 Fidelity Brokerage Services, Member NYSE, SIPC, 900 Salem Street, Smithfield, RI 02917. © 2016 FMR LLC. All rights reserved. 764207.1.0 Disclosures Options trading entails significant risk and is not appropriate for all investors. Certain complex options strategies carry additional risk. Before trading options, please read Characteristics and Risks of Standardized Options, and call 800-544- 5115 to be approved for options trading. Supporting documentation for any claims, if applicable, will be furnished upon request. Examples in this presentation do not include transaction costs (commissions, margin interest, fees) or tax implications, but they should be considered prior to entering into any transactions. The information in this presentation, including examples using actual securities and price data, is strictly for illustrative and educational purposes only and is not to be construed as an endorsement, recommendation. Active Trader Pro PlatformsSM is available to customers trading 36 times or more in a rolling 12-month period; customers who trade 120 times or more have access to Recognia anticipated events and Elliott Wave analysis. Greeks are mathematical calculations used to determine the effect of various factors on options. Goal of this webinar Demystify what Options Greeks are and explain how they are used in plain English. What we will cover: What the Greeks are What the Greeks tell us How the Greeks can help with planning option trades How the Greeks can help with managing option trades The Greeks What the Greeks are: • Delta • Gamma • Vega • Theta • Rho But what do they mean? Greeks What do they tell me? In simplest terms, Greeks give traders a theoretical way to judge their exposure to various options pricing inputs.