Appendix 1: Option Valuation
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V. Black-Scholes Model: Derivation and Solution
V. Black-Scholes model: Derivation and solution Beáta Stehlíková Financial derivatives, winter term 2014/2015 Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava V. Black-Scholes model: Derivation and solution – p.1/36 Content Black-Scholes model: • Suppose that stock price follows a geometric ◦ S Brownian motion dS = µSdt + σSdw + other assumptions (in a moment) We derive a partial differential equation for the price of ◦ a derivative Two ways of derivations: • due to Black and Scholes ◦ due to Merton ◦ Explicit solution for European call and put options • V. Black-Scholes model: Derivation and solution – p.2/36 Assumptions Further assumptions (besides GBP): • constant riskless interest rate ◦ r no transaction costs ◦ it is possible to buy/sell any (also fractional) number of ◦ stocks; similarly with the cash no restrictions on short selling ◦ option is of European type ◦ Firstly, let us consider the case of a non-dividend paying • stock V. Black-Scholes model: Derivation and solution – p.3/36 Derivation I. - due to Black and Scholes Notation: • S = stock price, t =time V = V (S, t)= option price Portfolio: 1 option, stocks • δ P = value of the portfolio: P = V + δS Change in the portfolio value: • dP = dV + δdS From the assumptions: From the It¯o • dS = µSdt + σSdw, ∂V ∂V 1 2 2 ∂2V ∂V lemma: dV = ∂t + µS ∂S + 2 σ S ∂S2 dt + σS ∂S dw Therefore: • ∂V ∂V 1 ∂2V dP = + µS + σ2S2 + δµS dt ∂t ∂S 2 ∂S2 ∂V + σS + δσS dw ∂S V. Black-Scholes model: Derivation and solution – p.4/36 Derivation I. -
Incentives for Central Clearing and the Evolution of Otc Derivatives
INCENTIVES FOR CENTRAL CLEARING AND THE EVOLUTION OF OTC DERIVATIVES – [email protected] A CCP12 5F No.55 Yuanmingyuan Rd. Huangpu District, Shanghai, China REPORT February 2019 TABLE OF CONTENTS TABLE OF CONTENTS................................................................................................... 2 EXECUTIVE SUMMARY ................................................................................................. 5 1. MARKET OVERVIEW ............................................................................................. 8 1.1 CENTRAL CLEARING RATES OF OUTSTANDING TRADES ..................... 8 1.2 MARKET STRUCTURE – COMPRESSION AND BACKLOADING ............... 9 1.3 CURRENT CLEARING RATES ................................................................... 11 1.4 INITIAL MARGIN HELD AT CCPS .............................................................. 16 1.5 UNCLEARED MARKETS ............................................................................ 17 1.5.1 FX OPTIONS ...................................................................................... 18 1.5.2 SWAPTIONS ...................................................................................... 19 1.5.3 EUROPE ............................................................................................ 21 2. TRADE PROCESSING ......................................................................................... 23 2.1 TRADE PROCESSING OF NON-CLEARED TRADES ............................... 23 2.1.1 CUSTODIAL ARRANGEMENTS ....................................................... -
Section 1256 and Foreign Currency Derivatives
Section 1256 and Foreign Currency Derivatives Viva Hammer1 Mark-to-market taxation was considered “a fundamental departure from the concept of income realization in the U.S. tax law”2 when it was introduced in 1981. Congress was only game to propose the concept because of rampant “straddle” shelters that were undermining the U.S. tax system and commodities derivatives markets. Early in tax history, the Supreme Court articulated the realization principle as a Constitutional limitation on Congress’ taxing power. But in 1981, lawmakers makers felt confident imposing mark-to-market on exchange traded futures contracts because of the exchanges’ system of variation margin. However, when in 1982 non-exchange foreign currency traders asked to come within the ambit of mark-to-market taxation, Congress acceded to their demands even though this market had no equivalent to variation margin. This opportunistic rather than policy-driven history has spawned a great debate amongst tax practitioners as to the scope of the mark-to-market rule governing foreign currency contracts. Several recent cases have added fuel to the debate. The Straddle Shelters of the 1970s Straddle shelters were developed to exploit several structural flaws in the U.S. tax system: (1) the vast gulf between ordinary income tax rate (maximum 70%) and long term capital gain rate (28%), (2) the arbitrary distinction between capital gain and ordinary income, making it relatively easy to convert one to the other, and (3) the non- economic tax treatment of derivative contracts. Straddle shelters were so pervasive that in 1978 it was estimated that more than 75% of the open interest in silver futures were entered into to accommodate tax straddles and demand for U.S. -
Introduction to Options
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options|the so-called plain vanilla options. We consider the payoffs to these options for holders and writers and describe both the time vale and intrinsic value of an option. We explain why options are traded and examine some of the properties of put and call option prices. We shall show that longer dated options typically have higher prices and that call prices are higher when the strike price is lower and that put prices are higher when the strike price is higher. Keywords: Put, call, strike price, maturity date, in-the-money, time value, intrinsic value, parity value, American option, European option, early exer- cise, bull spread, bear spread, butterfly spread. 1 Options Options are derivative assets. That is their payoffs are derived from the pay- off on some other underlying asset. The underlying asset may be an equity, an index, a bond or indeed any other type of asset. Options themselves are classified by their type, class and their series. The most important distinc- tion of types is between put options and call options. A put option gives the owner the right to sell the underlying asset at specified dates at a specified price. A call option gives the owner the right to buy at specified dates at a specified price. Options are different from forward/futures contracts as a put (call) option gives the owner right to sell (buy) the underlying asset but not an obligation. The right to buy or sell need not be exercised. -
The Synthetic Collateralised Debt Obligation: Analysing the Super-Senior Swap Element
The Synthetic Collateralised Debt Obligation: analysing the Super-Senior Swap element Nicoletta Baldini * July 2003 Basic Facts In a typical cash flow securitization a SPV (Special Purpose Vehicle) transfers interest income and principal repayments from a portfolio of risky assets, the so called asset pool, to a prioritized set of tranches. The level of credit exposure of every single tranche depends upon its level of subordination: so, the junior tranche will be the first to bear the effect of a credit deterioration of the asset pool, and senior tranches the last. The asset pool can be made up by either any type of debt instrument, mainly bonds or bank loans, or Credit Default Swaps (CDS) in which the SPV sells protection1. When the asset pool is made up solely of CDS contracts we talk of ‘synthetic’ Collateralized Debt Obligations (CDOs); in the so called ‘semi-synthetic’ CDOs, instead, the asset pool is made up by both debt instruments and CDS contracts. The tranches backed by the asset pool can be funded or not, depending upon the fact that the final investor purchases a true debt instrument (note) or a mere synthetic credit exposure. Generally, when the asset pool is constituted by debt instruments, the SPV issues notes (usually divided in more tranches) which are sold to the final investor; in synthetic CDOs, instead, tranches are represented by basket CDSs with which the final investor sells protection to the SPV. In any case all the tranches can be interpreted as percentile basket credit derivatives and their degree of subordination determines the percentiles of the asset pool loss distribution concerning them It is not unusual to find both funded and unfunded tranches within the same securitisation: this is the case for synthetic CDOs (but the same could occur with semi-synthetic CDOs) in which notes are issued and the raised cash is invested in risk free bonds that serve as collateral. -
Understanding the Z-Spread Moorad Choudhry*
Learning Curve September 2005 Understanding the Z-Spread Moorad Choudhry* © YieldCurve.com 2005 A key measure of relative value of a corporate bond is its swap spread. This is the basis point spread over the interest-rate swap curve, and is a measure of the credit risk of the bond. In its simplest form, the swap spread can be measured as the difference between the yield-to-maturity of the bond and the interest rate given by a straight-line interpolation of the swap curve. In practice traders use the asset-swap spread and the Z- spread as the main measures of relative value. The government bond spread is also considered. We consider the two main spread measures in this paper. Asset-swap spread An asset swap is a package that combines an interest-rate swap with a cash bond, the effect of the combined package being to transform the interest-rate basis of the bond. Typically, a fixed-rate bond will be combined with an interest-rate swap in which the bond holder pays fixed coupon and received floating coupon. The floating-coupon will be a spread over Libor (see Choudhry et al 2001). This spread is the asset-swap spread and is a function of the credit risk of the bond over and above interbank credit risk.1 Asset swaps may be transacted at par or at the bond’s market price, usually par. This means that the asset swap value is made up of the difference between the bond’s market price and par, as well as the difference between the bond coupon and the swap fixed rate. -
Computation of Binomial Option Pricing Model with Parallel
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by KHALSA PUBLICATIONS I S S N 2 2 7 8 - 5 6 1 2 Volume 11 Number 5 International Journal of Management and Information T e c h n o l o g y Computation Of Binomial Option Pricing Model With Parallel Processing On A Linux Cluster Harya Widiputra Faculty of Information Technology, Perbanas Institute Jakarta, Indonesia [email protected] ABSTRACT Binary Option Pricing Model (BOPM) is one approach that can be utilized to calculate the value of either call or put option. BOPM generally works by building a binomial tree diagram, also known as lattice diagram to explore all possible option values that occur based on the intrinsic price of the underlying asset in a range of specific time period. Due to its basic characteristics BOPM is renowned for its longer lead times in calculating option values while also acknowledged as a technique that can provide an assessment of the most fair option value. Therefore, this study aims to shorten the BOPM completion time of calculating the value of an option by building a computer cluster and presently implement the BOPM algorithm into a form that can be run in parallel processing. As an outcome, a Linux-based computer cluster has been built and conversion of BOPM algorithm so that it can be executed in parallel has been performed and also implemented using Python in this research. Conclusively, results of conducted experiments confirm that the implementation of BOPM algorithm in a form that can be executed in parallel and its application on a computer cluster was able to limit the growth of the completion time whilst at the same time maintain quality of calculated option values. -
Chapter 2 Roche Holding: the Company, Its
Risk Takers Uses and Abuses of Financial Derivatives John E. Marthinsen Babson College PEARSON Addison Wesley Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal Preface xiii Parti 1 Chapter 1 Employee Stock Options: What Every MBA Should Know 3 Introduction 3 Employee Stock Options: A Major Pillar of Executive Compensation 6 Why Do Companies Use Employee Stock Options 6 Aligning Incentives 7 Hiring and Retention 7 Postponing the Timing of Expenses 8 Adjusting Compensation to Employee Risk Tolerance Levels 8 Tax Advantages for Employees 9 Tax Advantages for Companies 9 Cash Flow Advantages for Companies 10 Option Valuation Differences and Human Resource Management 12 Problems with Employee Stock Options 26 Employee Motivation 26 The Share Price of "Good" Companies 27 Motivation of Undesired Behavior 28 vi Contents Performance Improvement 28 Absolute Versus Relative Performance 29 Some Innovative Solutions to Employee Stock Option Problems 29 Premium-Priced Stock Options 30 Index Options 30 Restricted Shares 34 Omnibus Plans 34 Conclusion 35 Review Questions 36 Bibliography 38 Chapter 2 Roche Holding: The Company, Its Financial Strategy, and Bull Spread Warrants 41 Introduction 41 Roche Holding AG: Transition from a Lender to a Borrower 43 Loss of the Valium Patent in the United States 43 Rapid Increase in R&D Costs, Market Growth, and Industry Consolidation 43 Roche's Unique Capital Structure 44 Roche Brings in a New Leader and Replaces Its Management Committee -
The Forward Smile in Stochastic Local Volatility Models
The forward smile in stochastic local volatility models Andrea Mazzon∗ Andrea Pascucciy Abstract We introduce an approximation of forward start options in a multi-factor local-stochastic volatility model. We derive explicit expansion formulas for the so-called forward implied volatility which can be useful to price complex path-dependent options, as cliquets. The expansion involves only polynomials and can be computed without the need for numerical procedures or special functions. Recent results on the exploding behaviour of the forward smile in the Heston model are confirmed and generalized to a wider class of local-stochastic volatility models. We illustrate the effectiveness of the technique through some numerical tests. Keywords: forward implied volatility, cliquet option, local volatility, stochastic volatility, analytical ap- proximation Key messages • approximation for the forward implied volatility • local stochastic volatility models • explosion of the out-of-the-money forward smile 1 Introduction In an arbitrage-free market, we consider the risk-neutral dynamics described by the d-dimensional Markov diffusion dXt = µ(t; Xt)dt + σ(t; Xt)dWt; (1.1) where W is a m-dimensional Brownian motion. The first component X1 represents the log-price of an asset, while the other components of X represent a number of things, e.g., stochastic volatilities, economic indicators or functions of these quantities. We are interested in the forward start payoff + X1 −X1 k e t+τ t − e (1.2) ∗Gran Sasso Science Institute, viale Francesco Crispi 7, 67100 L'Aquila, Italy ([email protected]) yDipartimento di Matematica, Universit`a di Bologna, Piazza di Porta S. -
Dixit-Pindyck and Arrow-Fisher-Hanemann-Henry Option Concepts in a Finance Framework
Dixit-Pindyck and Arrow-Fisher-Hanemann-Henry Option Concepts in a Finance Framework January 12, 2015 Abstract We look at the debate on the equivalence of the Dixit-Pindyck (DP) and Arrow-Fisher- Hanemann-Henry (AFHH) option values. Casting the problem into a financial framework allows to disentangle the discussion without unnecessarily introducing new definitions. Instead, the option values can be easily translated to meaningful terms of financial option pricing. We find that the DP option value can easily be described as time-value of an American plain-vanilla option, while the AFHH option value is an exotic chooser option. Although the option values can be numerically equal, they differ for interesting, i.e. non-trivial investment decisions and benefit-cost analyses. We find that for applied work, compared to the Dixit-Pindyck value, the AFHH concept has only limited use. Keywords: Benefit cost analysis, irreversibility, option, quasi-option value, real option, uncertainty. Abbreviations: i.e.: id est; e.g.: exemplum gratia. 1 Introduction In the recent decades, the real options1 concept gained a large foothold in the strat- egy and investment literature, even more so with Dixit & Pindyck (1994)'s (DP) seminal volume on investment under uncertainty. In environmental economics, the importance of irreversibility has already been known since the contributions of Arrow & Fisher (1974), Henry (1974), and Hanemann (1989) (AFHH) on quasi-options. As Fisher (2000) notes, a unification of the two option concepts would have the benefit of applying the vast results of real option pricing in environmental issues. In his 1The term real option has been coined by Myers (1977). -
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. -
New Frontiers in Practical Risk Management
New Frontiers in Practical Risk Management English edition Issue n.6-S pring 2015 Iason ltd. and Energisk.org are the editors of Argo newsletter. Iason is the publisher. No one is al- lowed to reproduce or transmit any part of this document in any form or by any means, electronic or mechanical, including photocopying and recording, for any purpose without the express written permission of Iason ltd. Neither editor is responsible for any consequence directly or indirectly stem- ming from the use of any kind of adoption of the methods, models, and ideas appearing in the con- tributions contained in Argo newsletter, nor they assume any responsibility related to the appropri- ateness and/or truth of numbers, figures, and statements expressed by authors of those contributions. New Frontiers in Practical Risk Management Year 2 - Issue Number 6 - Spring 2015 Published in June 2015 First published in October 2013 Last published issues are available online: www.iasonltd.com www.energisk.org Spring 2015 NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT Editors: Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl) Andrea RONCORONI (ESSEC Business School, Paris) Executive Editor: Luca OLIVO (Iason ltd) Scientific Editorial Board: Fred Espen BENTH (University of Oslo) Alvaro CARTEA (University College London) Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl) Mark CUMMINS (Dublin City University Business School) Gianluca FUSAI (Cass Business School, London) Sebastian JAIMUNGAL (University of Toronto) Fabio MERCURIO (Bloomberg LP) Andrea RONCORONI (ESSEC Business School, Paris) Rafal WERON (Wroclaw University of Technology) Iason ltd Registered Address: 6 O’Curry Street Limerick 4 Ireland Italian Address: Piazza 4 Novembre, 6 20124 Milano Italy Contact Information: [email protected] www.iasonltd.com Energisk.org Contact Information: [email protected] www.energisk.org Iason ltd and Energisk.org are registered trademark.