Exotic Options and Hybrids
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Variance Reduction with Control Variate for Pricing Asian Options in a Geometric Lévy Model
IAENG International Journal of Applied Mathematics, 41:4, IJAM_41_4_05 ______________________________________________________________________________________ Variance Reduction with Control Variate for Pricing Asian Options in a Geometric Levy´ Model Wissem Boughamoura and Faouzi Trabelsi Abstract—This paper is concerned with Monte Carlo simu- use of the path integral formulation. [19] studied a cer- lation of generalized Asian option prices where the underlying tain one-dimensional, degenerate parabolic partial differential asset is modeled by a geometric (finite-activity) Levy´ process. equation with a boundary condition which arises in pricing A control variate technique is proposed to improve standard Monte Carlo method. Some convergence results for plain Monte of Asian options. [25] considered the approximation of Carlo simulation of generalized Asian options are proven, and a the optimal stopping problem associated with ultradiffusion detailed numerical study is provided showing the performance processes in valuating Asian options. The value function is of the variance reduction compared to straightforward Monte characterized as the solution of an ultraparabolic variational Carlo method. inequality. Index Terms—Levy´ process, Asian options, Minimal-entropy martingale measure, Monte Carlo method, Variance Reduction, For discontinuous markets, [8] generalized Laplace trans- Control variate. form for continuously sampled Asian option where under- lying asset is driven by a Levy´ Process, [18] presented a binomial tree method for Asian options under a particular I. INTRODUCTION jump-diffusion model. But such Market may be incomplete. He pricing of Asian options is a delicate and interesting Therefore, it will be a big class of equivalent martingale T topic in quantitative finance, and has been a topic of measures (EMMs). For the choice of suitable one, many attention for many years now. -
Interest Rate Options
Interest Rate Options Saurav Sen April 2001 Contents 1. Caps and Floors 2 1.1. Defintions . 2 1.2. Plain Vanilla Caps . 2 1.2.1. Caplets . 3 1.2.2. Caps . 4 1.2.3. Bootstrapping the Forward Volatility Curve . 4 1.2.4. Caplet as a Put Option on a Zero-Coupon Bond . 5 1.2.5. Hedging Caps . 6 1.3. Floors . 7 1.3.1. Pricing and Hedging . 7 1.3.2. Put-Call Parity . 7 1.3.3. At-the-money (ATM) Caps and Floors . 7 1.4. Digital Caps . 8 1.4.1. Pricing . 8 1.4.2. Hedging . 8 1.5. Other Exotic Caps and Floors . 9 1.5.1. Knock-In Caps . 9 1.5.2. LIBOR Reset Caps . 9 1.5.3. Auto Caps . 9 1.5.4. Chooser Caps . 9 1.5.5. CMS Caps and Floors . 9 2. Swap Options 10 2.1. Swaps: A Brief Review of Essentials . 10 2.2. Swaptions . 11 2.2.1. Definitions . 11 2.2.2. Payoff Structure . 11 2.2.3. Pricing . 12 2.2.4. Put-Call Parity and Moneyness for Swaptions . 13 2.2.5. Hedging . 13 2.3. Constant Maturity Swaps . 13 2.3.1. Definition . 13 2.3.2. Pricing . 14 1 2.3.3. Approximate CMS Convexity Correction . 14 2.3.4. Pricing (continued) . 15 2.3.5. CMS Summary . 15 2.4. Other Swap Options . 16 2.4.1. LIBOR in Arrears Swaps . 16 2.4.2. Bermudan Swaptions . 16 2.4.3. Hybrid Structures . 17 Appendix: The Black Model 17 A.1. -
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. -
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 -
Unified Pricing of Asian Options
Unified Pricing of Asian Options Jan Veˇceˇr∗ ([email protected]) Assistant Professor of Mathematical Finance, Department of Statistics, Columbia University. Visiting Associate Professor, Institute of Economic Research, Kyoto University, Japan. First version: August 31, 2000 This version: April 25, 2002 Abstract. A simple and numerically stable 2-term partial differential equation characterizing the price of any type of arithmetically averaged Asian option is given. The approach includes both continuously and discretely sampled options and it is easily extended to handle continuous or dis- crete dividend yields. In contrast to present methods, this approach does not require to implement jump conditions for sampling or dividend days. Asian options are securities with payoff which depends on the average of the underlying stock price over certain time interval. Since no general analytical solution for the price of the Asian option is known, a variety of techniques have been developed to analyze arithmetic average Asian options. There is enormous literature devoted to study of this option. A number of approxima- tions that produce closed form expressions have appeared, most recently in Thompson (1999), who provides tight analytical bounds for the Asian option price. Geman and Yor (1993) computed the Laplace transform of the price of continuously sampled Asian option, but numerical inversion remains problematic for low volatility and/or short maturity cases as shown by Fu, Madan and Wang (1998). Very recently, Linetsky (2002) has derived new integral formula for the price of con- tinuously sampled Asian option, which is again slowly convergent for low volatility cases. Monte Carlo simulation works well, but it can be computationally expensive without the enhancement of variance reduction techniques and one must account for the inherent discretization bias result- ing from the approximation of continuous time processes through discrete sampling as shown by Broadie, Glasserman and Kou (1999). -
New IRS Rules Would Impose U.S. Withholding Tax on Many Derivatives and Other Financial Transactions Linked to U.S
December 19, 2013 clearygottlieb.com New IRS Rules Would Impose U.S. Withholding Tax on Many Derivatives and Other Financial Transactions Linked to U.S. Stock I. OVERVIEW On December 5, 2013, the U.S. Department of the Treasury (“Treasury”) and the Internal Revenue Service (“IRS”) published proposed regulations that would impose U.S. withholding tax on a wide range of financial instruments linked to stock issued by U.S. issuers, including most equity swaps and many other equity derivatives, and many mandatory convertible bonds and structured notes that provide for the delivery of, or payment by reference to, U.S. equities. The withholding tax also will apply to certain conventional convertible bonds acquired in the secondary market. As drafted, the rules appear to apply to certain kinds of U.S. equity-based deferred compensation, M&A and joint venture transactions involving the acquisition of a minority stake and certain types of insurance although it is not clear that all of those applications were intended. The withholding tax would apply to payments and deemed payments by U.S. and non- U.S. payors on financial instruments held by nonresident aliens and legal entities organized outside the United States that are “long” the underlying stock, including individuals, hedge funds, activist shareholders, special purpose vehicles, and dealers trading for their own account, starting January 1, 2016. The rules generally would apply to equity swaps outstanding on that date, and would apply to other equity-linked instruments acquired on or after March 5, 2014, regardless of when they were issued or entered into. -
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. -
11 Option Payoffs and Option Strategies
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expira- tion for various stock prices. 73 74 CHAPTER 11 OPTION PAYOFFS AND OPTION STRATEGIES 2. Consider a put option with an exercise price of $80 and a cost of $4. Graph the profits and losses at expiration for various stock prices. ANSWERS TO QUESTIONS AND PROBLEMS 75 3. For the call and put in questions 1 and 2, graph the profits and losses at expiration for a straddle comprising these two options. If the stock price is $80 at expiration, what will be the profit or loss? At what stock price (or prices) will the straddle have a zero profit? With a stock price at $80 at expiration, neither the call nor the put can be exercised. Both expire worthless, giving a total loss of $9. The straddle breaks even (has a zero profit) if the stock price is either $71 or $89. 4. A call option has an exercise price of $70 and is at expiration. The option costs $4, and the underlying stock trades for $75. Assuming a perfect market, how would you respond if the call is an American option? State exactly how you might transact. How does your answer differ if the option is European? With these prices, an arbitrage opportunity exists because the call price does not equal the maximum of zero or the stock price minus the exercise price. To exploit this mispricing, a trader should buy the call and exercise it for a total out-of-pocket cost of $74. -
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.