资源描述
Click to edit Master Title Style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,*,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,*,Chapter 15: Markets for Options and Contingent Claims,Objective,Options,Pricing relationships,Pricing models,Financial decisions,analyzed through options,1,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Chapter 15 Contents,15.1 How Options Work,15.2 Investing with Options,15.3 The Put-Call Parity Relationship,15.4 Volatility & Option Prices,15.5 Two-State (Binomial) Option Pricing,15.6 Dynamic Replication & the Binomial Model,15.7 The Black-Scholes Model,15.8 Implied Volatility,15.9 Contingent Claims Analysis of Corporate Debt and Equity,15.10 Credit Guarantees,15.11 Other Applications of Option-Pricing Methodology,2,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Objectives,How to use options to modify ones exposure to investment risk.,To understand the pricing relationships that exist among calls, puts, stocks, and bonds.,To explain the binomial and Black-Scholes option-pricing models and apply them to the valuation of corporate bonds and other contingent claims.,To explore the range of financial decisions that can be fruitfully analyzed in terms of options.,3,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Introduction,This chapter explores how option prices are affected by the volatility of the underlying security,Exchange traded options appeared in 1973, enabling us to determine the markets estimate of future volatility, rather than relying on historical values,4,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Ubiquitous Options,This chapter focuses on traded options, but it would be a mistake to believe that the tools we will be developing are restricted to traded options,Some examples of options are given on the next few slides,6,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Government Price Supports,Governments sometimes provide assistance to farmers by offering to purchase agricultural products at a specified support price,If the market price is lower than the support, then a farmer will exercise her right to put her crop to the government at the higher price,7,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Old Mortgage,Traditional US mortgages give the householder the right to call the mortgage at a strike equal to the outstanding principle,If interest rates have fallen below the notes rate, then the home owner will consider refinancing the mortgage,8,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Tenure and Seniority,In a company that has a policy of last-in first-out, a worker with seniority may forego a higher salary in another company because of the loss of job security,The worker has been given the right, but not the obligation, to have work under a set of adverse economic conditions,10,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Copper Pennies, Silver Coin,Silver and copper coinage has been replaced by zinc and cooper alloys/ composites to reduce their minting costs,The old coins are often legal currency, and so contain an option feature:,If the price of the underlying metal falls below its legal value, I have the right to return the coin into circulation,11,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Supply Contracts,A nuclear power plant supplier once got into serious trouble by guaranteeing to supply enriched uranium at a fixed price,The market price of enriched uranium rose precipitously,13,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Limited Liability,The owners of a limited liability corporation have the right, but not the obligation, to put the company to the corporations creditors and bondholders,Limited liability is, in effect, a put option,15,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Trading on Commission,You are a trader with a contract giving you a commission of 20% of each months trading profits,If you make a loss, then you walk away, but if you make a profit, you stay,You may be tempted to increase your volatility to boost the value of your option,16,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,European Option: an option that can only be exercised on the maturity date,Tangible Value: The hypothetical value of an option if it were exercised immediately,At-the-Money: an option with a strike price equal to the value of the underlying asset,Out-of-the-Money: an option thats not at-the-money, but has no tangible value,In-the-Money: an option with a tangible value,Time Value: the difference between an options market value and its tangible value,Exchange-Traded Option: A standardized option that an exchange stands behind in the case of a default,Over the Counter Option: An option on a security that is not an exchange-traded option,18,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,19,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,20,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,15.2 Investing with Options,The payoff diagram (terminal conditions, boundary conditions) for a call and a put option, each with a strike (exercise price) of $100, is derived next,21,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Option Payoff Diagrams,The value of an option at expiration follows immediately from its definition,In the case of a call option with strike of $100, if the stock price is $90,($110), then exercising the option results purchasing the share for $100, which is $10 more expensive,($10 less expensive),than buying it, so you wouldnt,(would),exercise your right,22,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Put Option Payoff Diagram,24,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Payoff Diagrams for Alternative Bullish Stock Strategies,25,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,26,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Payoff Diagram for Pure Discount Bond Plus Call,28,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,29,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Observation,The most important point to observe is that the value of the “call + bond” strategy, is identical (at maturity) with the protective-put strategy “put + share”,So, if the put and the call have the same strike price, we obtain the put-call parity relationship: put + share = call + bond,31,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Technical Note,The above relationship is true, in general, for dividend-less European options, but the actual proof requires taking expectations of the option and security boundary conditions,The argument is therefore a heuristic for remembering and seeing the relationship,The full proof is left for your investment class,32,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Put-Call Parity for American and European Options,A European option that pays no dividend during its life fully satisfies the requirements of put-call parity,In the case of American options, the relationship is fully accurate only at maturity, because American puts are sometimes exercised early,33,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Put-Call Parity Equation,34,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Synthetic Securities,The put-call parity relationship may be solved for any of the four security variables to create synthetic securities:,C=S+P-B,S=C-P+B,P=C-S+B,B=S+P-C,35,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Synthetic Securities,C=S+P-B and P=C-S+B may be used by floor traders to flip between a call and a put,S=C-P+B may be used by short-term traders wishing to take advantage of lower transaction costs,B=S+P-C may be used to create a synthetic bond said to pay a slightly higher return than the physical bond,36,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Options and Forwards,We saw in the last chapter that the discounted value of the forward was equal to the current spot,The relationship becomes,37,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Implications for European Options,If the forward price of the underlying stock is equal to the strike price, then the value of the call is equal to the value of the put,This relationship is so important, that some option traders define at-the-money in terms of the forward rather than the spot,38,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Implications for European Options,If (F E) then (C P),If (F = E) then (C = P),If (F E) then (C P),E is the common strike price,F is the forward price of underlying share,C is the call price,P is the put price,39,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Strike = Forward,Call = Put,40,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,41,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,PV Strike,Strike,42,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,15.4 Volatility and Option Prices,We next explore what happens to the value of an option when the volatility of the underlying stock increases,We assume a world in which the stock price moves during the year from $100 to one of two new values at the end of the year when the option matures,Assume risk neutrality,43,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,44,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Illustration Explained,The stock volatility in the second scenario is higher, and the expected payoffs for both the put and the call are also higher,This is the result of truncation, and holds in all empirically reasonable cases,Conclusion: Volatility increases all option prices,45,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,15.5 Two-State (Binomial) Option-Pricing,We are now going to derive a relatively simple model for evaluating options,The assumptions will at first appear totally unrealistic, but using some underhand mathematics, the model may be made to price options to any desired level of accuracy,The advantage of the method is that it does not require learning stochastic calculus, and yet it illustrates all the key steps necessary to derive any option evaluation model,46,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model Assumptions,Assume:,the exercise price is equal to the forward price of the underlying stock,option prices then depend only on the volatility and time to maturity, and do not depend on interest rates,the put and call have the same price,47,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model Assumptions,More specifically we assume:,share price = strike price = $100,time to maturity = 1 year,dividend rate = interest rate = 0,stock prices either rise or fall by 20% in the year, and so are either $80 or $120 at yearend,48,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Call,Strategy:,replicate the call using a portfolio of,the underlying stock,the riskless bond,by the law of one price, the price of the actual call must equal the price of the synthetic call,49,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Call,Implementation:,the synthetic call, C, is created by,buying a fraction x of shares, of the stock, S, and simultaneously selling short risk free bonds with a market value y,the fraction x is called the,hedge ratio,50,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Call,Specification:,We have an equation, and given the value of the terminal share price, we know the terminal option value for two cases:,By inspection, the solution is x=1/2, y = 40,51,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Call,Solution:,We now substitute the value of the parameters x=1/2, y = 40 into the equation,to obtain:,52,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Put,Strategy:,replicate the put using a portfolio of the underlying stock and riskless bond,by the,law of one price,the price of the actual put must equal the price of the synthetic put replicated above,Minor changes to the call argument are made in the next few slides for the put,53,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Put,Implementation:,the synthetic put, P, is created by,sell short a fraction x of shares, of the stock, S, and simultaneously buy risk free bonds with a market value y,the fraction x is called the,hedge ratio,54,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Decision Tree for Dynamic Replication of Call Option,55,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Put,Specification:,We have an equation, and given the value of the terminal share price, we know the terminal option value for two cases:,By inspection, the solution is x=1/2, y = 60,56,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Binary Model: Put,Solution:,We now substitute the value of the parameters x=1/2, y = 60 into the equation,to obtain:,57,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,15.6 Dynamic Replication and the Binomial Model,We now take the next step towards greater realism by dividing the year into 2 sub-periods of half a year each. This gives 3 possible outcomes,Our first task is to find a,self-financing investment,strategy that does not require injection or withdrawal of new funds during the life of the option,We first create a,decision tree,:,58,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Decision Tree for Dynamic Replication of a Call Option,($120*100%) + (-$100) = $20,59,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Reading the Decision Tree,The tree is constructed backwards because we know only the future contingent call prices,For Example, when constructing the weights for time 6-months, the option prices for 12-months are used,For consistency with the next model, the discrete stock prices are usually fixed ratios, i.e. 121, 110, 100, 90.91, 82.64,60,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Power of Lattice Models,Lattice models, of which the binary model is the simplest, are very important to traders because they may be modified to handle different distributions, the possibility of early exercise, and discrete dividend payments,To see how easy it is to change the distributional assumption, the above illustration results in stock prices being normally distributed, and the modification results in a lognormal distribution of prices,61,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,15.7 The Black-Scholes Model,The most widely used model for pricing options is the Black-Scholes model,This model is completely consistent with the binary model as the interval between stock prices decreases to zero,The model provides theoretical insights into option behavior,The assumptions are elegant, simple, and quite realistic,62,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Black-Scholes Model,We will work with the generalized form of the model because the small additional complexity results in considerable additional power and flexibility,First, notation:,63,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Black-Scholes Model: Notation,C = price of call,P = price of put,S = price of stock,E = exercise price,T = time to maturity,ln(.) = natural logarithm,e = 2.71828.,N(.) = cum. norm. distn,The following are annual, compounded continuously:,r = domestic risk free rate of interest,d = foreign risk free rate or constant dividend yield, = volatility,64,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Normal Problem,It is not unusual for a student to have a problem computing the cumulative normal distribution using tables,table structures vary, so be careful,using standard-issue normal tables degrades computed option values because of errors caused by catastrophic subtraction,Many professionals use Hastings formula as reported in Abramowitz and Stegun as equation 26.2.19 (never, never use 26.2.18). Its certificate valid in 0=xInf, so use symmetry to get -Infx0,65,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Normal Problem,The functions that come with Excel have adequate accuracy, so consider using Normsdist() in the statistical functions (note the,s,in Norm,s,dist),66,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Black-Scholes Model: Whats missing,There are no expectations about future returns in the model,The model is preference-free,-risk, not,b,-risk, is the relevant risk,67,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Black-Scholes Model: Equations,68,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Black-Scholes Model: Equations (Forward Form),69,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,The Black-Scholes Model: Equations (Simplified),70,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,So What Does it Mean?,You can now obtain the value of non-dividend paying European options,With a little skill, you can widen this to obtain approximate values of European options on shares paying a dividend, and to some American options,71,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,72,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,73,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Implied Volatility SPX(July 2005-June 2006),74,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,75,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Observable Variables,All the variables are directly observable, excepting the volatility, , and possibly, the next cash dividend, d,We do not have to delve into the psyche of investors to evaluate options,We do not forecast future prices to obtain option values,76,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,15.8 Implied Volatility,The following slides show how to estimate volatility using Excel,The option most commonly used to estimate volatility is the one closest to the present value of the strike price: That is, the option that is has a strike closest to the forward price,This option has the most “oomph”,77,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Insert any number to start,Formula for option value minus the actual call value,78,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,79,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Pats Plan,Pat has a plan to get rich with no risk:,Set up special portfolio, (Pat calls this a “self financing, delta neutral portfolio with positive curvature,” but Pat has this thing with words),80,Copyright 2009 Pearson Education, Inc. Publishing as Prentice Hall,Pat, the Strategist,Short some shares, and off-set small price changes about the current price with some call
展开阅读全文