...determinants of the option price in the Black-Scholes option pricing model for European options is likely to change the price of a call option. A derivative is a financial instrument that has a value determined by the price of something else, such as options. The crucial idea behind the derivation was to hedge perfectly the option by buying and selling the underlying asset in just the right way and consequently "eliminate risk" (Ray, 2012). The derivative asset we will be most interested in is a European call option. A call option gives the holder of the option the right to buy the underlying asset by a certain date for a certain price, but a put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The date in the contract is known as the expiration date or maturity date; the price in the contract is known as the exercise price or strike price. The market price of the underlying asset on the valuation date is spot price or stock price. Intrinsic value is the difference between the current stock market price and the exercise price or simply higher of zero. American options can be exercised at any time up to the expiration date. European options can be exercised only on the expiration date itself. (Hull, 2012). For example, consider a July European call option contract on XYZ with strike price $70. When the contract expires in July, if the price of XYZ stock is $75 the owner will exercise the option and realize a profit...
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...Three different methods of option pricing The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three different methods of option pricing are: The Black-Scholes model, binomial trees and Monte Carlo Simulation. The three...
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...The Black–Scholes /ˌblæk ˈʃoʊlz/[1] or Black–Scholes–Merton model is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options. The formula led to a boom in options trading and legitimised scientifically the activities of the Chicago Board Options Exchange and other options markets around the world.[2] lt is widely used, although often with adjustments and corrections, by options market participants.[3]:751 Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices, although there are well-known discrepancies such as the "option smile".[3]:770–771 The Black–Scholes was first published by Fischer Black and Myron Scholes in their 1973 paper, "The Pricing of Options and Corporate Liabilities", published in the Journal of Political Economy. They derived a stochastic partial differential equation, now called the Black–Scholes equation, which estimates the price of the option over time. The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way, and consequently "eliminate risk". This hedge is called delta hedging and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds. The hedge implies that there is a unique price for the option and this is given by the...
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...Flaws with Black Scholes & Exotic Greeks Treasury Perspectives Flaws with Black Scholes & Exotic Greeks 1 Flaws with Black Scholes & Exotic Greeks 2 Flaws with Black Scholes & Exotic Greeks Dear Readers:It’s been a difficult and volatile year for companies across the Globe. We have seen numerous risk management policies failures. To name a few... UBS, JPM Morgan, Libor manipulations by European, US and Japanese banks and prominent accounting scandals like Lehman… As rightly said by Albert Einstein “We can't solve problems by using the same kind of thinking we used when we created them.” and when you can't solve the problem, then manage it and don’t be dependent upon science as Science is always wrong, it never solves a problem without creating ten more. The same is the case with Foreign Exchange Risk Management Policies (FXRM) which if can’t be managed properly then would lead to either systematic shocks or negative implications at the bottom line of the corporate, banks, FI and trading houses P&L A/cs. That is something risk management struggles with, say the experts. In Richard Meyers’ estimation, risk managers or traders do not socialize enough. “It’s all about visibility,” he said. Meyers, chairman and CEO of Richard Meyers & Associates, a talent acquisition and management firm in New Jersey, relates the story of a firm that decided to adopt an Enterprise Risk Management (ERM) strategy. Instead of appointing its risk manager to head...
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...JØRGENSEN Author: QIAN Zhang (402847) Pricing of principle protected notes embedded with Asian options in Denmark ---- Using a Monte Carlo Method with stochastic volatility (the Heston Model) Aarhus School of Business and Social Science 2011 2 Acknowledgements My gratitude and appreciation goes to my supervisor Peter Lø chte Jø rgensen, for his kind and insightful discussion and guide through my process of writing. I was always impressed by his wisdom, openness and patience whenever I wrote an email or came by to his office with some confusion and difficulty. Especially on access to the information on certain Danish structured products, I have gained great help and support from him. 3 Abstract My interest came after the reading of the thesis proposal on strucured products written by Henrik, as is pointed out and suggested at the last part of this proposal, one of the main limitations of this thesis may be the choice of model. This intrigues my curiosity on pricing Asian options under assumption of stochstic volatility. At first, after the general introduction of strucutred products, the Black Scholes Model and risk neutral pricing has been explained. The following comes the disadvanges of BS model and then moves to the stochastic volatility model, among which the Heston model is highlighted and elaborated. The next part of this thesis is an emricical studying of two structured products embbeded with Asian options in Danish market and follows with a conclusion...
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...financial risk as much as possible when do financial decision. Derivative financial instruments such as options, futures and others have been introduced and more commonly used to manage financial risk for improving decision making in this dynamic competitive environment. Options are defined as securities which one party has the right (no obligation) to buy or sell underlying assets with certain price within a certain/specific period of time (Hull, 2012). The option can be either call (right to buy) or put (right to sell) in the form of American options (exercised any time until expiry date) or European options (exercised on expiry date) as either traded options (standard option contracts) or overt-the-counter options (tailor made options). Due to various choices of options, different option pricing models such as Put-Call Parity, Black-Scholes, Cox-Rubenstein Binominal, Risk-Neutral valuation, the Greeks and others has been developed and applied in current financial market. Black-Scholes Option Pricing Model (BS) BS is designed and introduced by Fisher Black and Myron Scholes in 1973 with the assumptions of the market is efficient, returns are lognormal distributed, no commission or transaction cost is charged, no dividend is paid, no penalties to short selling, terms of European option is used, interest rate is remained constant and known rate (Black & Scholes, 1973). Thereafter, the assumption of no dividends has been relaxed by Robert Merton in the same year...
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...Explain how the option pricing formula developed by black and scholes can be used for common stock and bond valuation. Include in your discussion the consequences of using variance applied over the option instead of actual variance. Its generally known that Black and Scholes model became a standard in option pricing methods , with almost everything from corporate liabilities and debt instruments can be viewed as option (except some complicated instruments), we can modify the fundamental formula in order to fit the specifications of the instrument that will be valued. An argument done by Black and Scholes which was based on the past proposition of Miller and Modigliani a well as assuming some ideal conditions, States that value of the firm is a sum of total value of debt plus the total value of common stock. As well as the fact that in the absence of taxes, the value of the firm is independent of its leverage and the change of debt has no effect on the firm value. V = E + Dm V: value of the firm. E: shareholders right (common stock values). Dm: market value of the debt. As the above equation impose that Equity (common stock values) can be viewed as a call option on the firm value (due to the shareholders limited liability and with consideration that firm debt can be represent as a zero-coupon bond), where exercising the option means that equity holders buy the firm at the face value of debt (which is in this case will be the exercise price of the option), on the liquidation...
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...Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price – exercise price for a put option: exercise price – stock price the intrinsic value for out-the-money or at-themoney options is equal to 0 time value of an option = difference between actual call price and intrinsic value as time approaches expiration date, time value goes to zero 21-2 Determinants of Option Values Call + – + + + – Put – + + + – + Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend rate of stock 21-3 Binomial Option Pricing consider a stock that currently sells at S0 the price an either increase by a factor u or fall by a factor d (probabilities are irrelevant) consider a call with exercise price X such that dS0 < X < uS0 hence, the evolution of the price and of the call option value is uS0 Cu = (uS0 – X) C S0 dS0 Cd = 0 21-4 Binomial Option Pricing (cont.) now, consider the payoff from writing one call option and buying H shares of the stock, where Cu − Cd uS0 − X H= = uS0 − dS0 uS 0 − dS0 the value of this investment at expiration is Up Down Payoff of stock HuS0 HdS0 Payoff of calls –(uS0 – X) 0 Total payoff HdS0 HdS0 21-5 Binomial Option Pricing (cont.) hence, we obtained a risk-free investment with end value HdS0 arbitrage argument: the current value of this investment should...
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...Accounting for Stock Options http://www.nysscpa.org/printversions/cpaj/2005/805/p30.htm Print Accounting for Stock Options Update on the Continuing Conflict By Nicholas G. Apostolou and D. Larry Crumbley AUGUST 2005 - In December 2004, a decade after bending to Congressional pressure and backing away from requiring the expensing of options on financial statements, FASB issued a revised standard to recognize stock-option compensation as an expense on income statements. Many in Congress may try to thwart the proposal before it becomes effective. A bill by Representative Richard Baker of Louisiana that would require expensing the cost of stock options for only the top five executives of a company has drawn the support of those groups still resolutely opposed to expensing. This time, however, FASB is likely to prevail. Investors are demanding tougher accounting standards, and the International Accounting Standards Board (IASB) has already passed rules requiring the expensing of options. Many large U.S. corporations have already voluntarily agreed to expense options. Finally, there is more concern about, and less support for, Congressional interference in FASB’s standards-setting process. History of the Debate Accounting for stock options has been one of the most controversial topics in accounting during the last decade. The principal debate is whether compensation expense should be recognized for stock options and, if so, the periods over which it should be allocated. Before 1995...
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...Jon M. Huntsman School of Business Master of Science in Financial Economics August 2013 Pricing and Hedging Asian Options By Vineet B. Lakhlani Pricing and Hedging Asian Options Table of Contents Table of Contents 1. Introduction to Derivatives 2. Exotic Options 2.1. Introduction to Asian Options 3.1. Binomial Option Pricing Model 3.2. Black-Scholes Model 3.2.1. Black-Scholes PDE Derivation 3.2.2. Black-Scholes Formula 1 2 3 4 4 5 6 7 3 3. Option Pricing Methodologies 4. Asian Option Pricing 4.1. 4.2. 4.3. 4.4. Closed Form Solution (Black-Scholes Formula) QuantLib/Boost Monte Carlo Simulations Price Characteristics 8 8 10 11 14 5. Hedging 5.1. Option Greeks 5.2. Characteristics of Option Delta (Δ) 5.3. Delta Hedging 5.3.1. Delta-Hedging for 1 Day 5.4. Hedging Asian Option 5.5. Other Strategies 6. Conclusion 16 17 17 19 20 22 25 26 27 32 34 Appendix i. ii. iii. Tables References Code: Black-Scholes Formula For European & Asian (Geometric) Option 1 Pricing and Hedging Asian Options 1. Introduction to Derivatives: Financial derivatives have been in existence as long as the invention of writing. The first derivative contracts—forward contracts—were written in cuneiform script on clay tablets. The evidence of the first written contract was dates back to in nineteenth century BC in Mesopotamia...
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...Aswath Damodaran Stern School of Business 44 West Fourth Street New York, NY 10012 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...
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...Second Order Moment Approach to Real Options Analysis Submitted as a Component of Required Courses for the Award of Bachelor of Engineering (Civil) Honours School of Civil Engineering University of New South Wales Author: Ariel Hersh October 2010 Supervisor: Professor David G. Carmichael i ORIGINALITY STATEMENT ‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’ Signed …………………………………………….............. Date …………………………………………….............. ii 1. ABSTRACT Real options analysis can be used by investors to determine the value of potential investments that offer an owner the right but not the obligation to exercise a strategic decision at a predetermined time and price. Tools which are popular for valuing financial ...
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...BEA3001 Financial Management 2012-2013 Option Pricing Dr Bill Peng, CFA Today • Describe the basic characteristics of financial options • Develop the Binomial Option Pricing Model • Discuss the Put-Call Parity theorem • Introduce and apply Black-Scholes Option Pricing Model BP BEA3001 Financial Management 2 Coursework Test 1 Directions • Reminder: CW Test 2 [4pm Wed 20th Mar 2013] • CW Test 1: 6pm on Monday 26th November • Students entitled to extra time: STC/C • Surnames starting with letters “A” to “K” (inclusive): Amory Moot Room • Surnames starting with letters “L” to “W” (inclusive): STC/A • Surnames starting with letters “X” to “Z” (inclusive): STC/B BP BEA3001 Financial Management 3 Coursework Test 1 Directions cont’d • Everyone should arrive in the corridor outside either Amory Moot Room or STC A/B/C by 6:15pm and wait QUIETLY to be called into a test room • We will aim to get you settled and started as soon as possible and you will be free by 8:00pm at the latest, if all of you could kindly follow the instructions • Fair-play: breach of exam rules will be punished • You should have with you your student ID card, a note of your 2012-2013 candidate number, pencils, erasers, pens and a calculator BP BEA3001 Financial Management 4 Coursework Test 1 Directions cont’d • You should take a seat where a test paper (together with an answer booklet) has been placed, but DO NOT touch the test paper or the answer booklet until told to do...
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...Chapter 12 & 20 Chapter 21 The Black-Scholes Formula and Option Greeks Adapted from Black & Scholes (1973), The Pricing of Options and Corporate Liabilities, The Journal of Political Economy, Vol. 81, No. 3., pp. 637-654. 2 Black-Scholes Assumptions • Assumptions about stock return distribution Continuously compounded returns on the stock are normally distributed and there is no jumps in the stock price The volatility is a known constant Future dividends are known, either as discrete dollar amount or as a fixed dividend yield • Assumptions about the economic environment The risk-free rate is a known constant There are no transaction costs or taxes It is possible to short-sell costlessly and to borrow at the risk-free rate 3 Black-Scholes Assumptions • The original paper by Black and Scholes begins by assuming that the price of the underlying asset follows a process like the following dS (t ) ( )dt dZ (t ) S (t ) where (20. 1) S(t) is the stock price dS(t) is the instantaneous change in the stock price is the continuously compounded expected return on the stock δ is the dividend yield on the stock is the continuously compounded standard deviation (volatility) Z(t) is the standard Brownian motion dZ(t) is the change in Z(t) over a short period of time 4 Black-Scholes Assumptions • There are 2 important implications of equation (20.1) Suppose the stock price now is S(0). If the stock...
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...Major Findings: This paper examines the problem of pricing a European call on an asset (Stock) that has a stochastic or variable volatility. Addressing this problem was done by investigating two cases: the first case is to determine the option price when the stochastic volatility is independent of stock price. The second case is to determine the option price when the stochastic volatility is correlated with the stock price. This paper provides a solution in series form for the stochastic volatility option, in addition to a discussion about the numerical methods that are used to examine pricing biases, and an investigation about the occurrence of the biases in the case of stochastic volatility. As for the results obtained, this paper presents interesting results for each of the two cases. When the stochastic volatility is independent of stock price, the results show that the price calculated using Black-Scholes equation is overestimated for at-the-money options and underestimated for deep in-and out-of-the-money options. This overpricing takes place for stock prices within about ten percent of the exercise price. Moreover, it is shown that the degree of the pricing bias can be up to five percent of the Black-Scholes price. For the second case when the stock price is positively correlated with the volatility, the results show that the Black-Scholes formula overprices in-the-money options and underprices out-of-the-money options. On the other hand, when the stock price is negatively...
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