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  PFE Chapter 25, The Black-Scholes formula page 1 CHAPTER 25: OPTION PRICING—THE BLACK-SCHOLES FORMULA *   Current version: July 18, 2005 Chapter contents  Overview..............................................................................................................................2 25.1. The Black-Scholes Model..........................................................................................5 25.2. Historical volatility: Computing σ    from stock prices...............................................8 25.3. Implied volatility: Calculating σ    from option prices..............................................11 25.4. An Excel Black-Scholes function............................................................................14 25.5. Doing sensitivity analysis on the Black-Scholes formula.......................................16 25.7. Does the Black-Scholes model work? Applying it to Microsoft options...............19 25.8. Real options (advanced topic)..................................................................................24 Summary............................................................................................................................29 Exercises............................................................................................................................30 Appendix: Getting Option Information from Yahoo........................................................35 *  This is a preliminary draft of a chapter of Principles of Finance with Excel . © 2001 – 2005 Simon Benninga ( benninga@wharton.upenn.edu ).  PFE Chapter 25, The Black-Scholes formula page 2 Overview In the two previous chapters on option pricing, we’ve discussed some facts about options,  but we haven’t discussed how to determine the price of an option . In this chapter we show how to price options using the Black-Scholes formula. The Black-Scholes formula is the most important option pricing formula. The formula is in wide use in options markets. It has also achieved a certain degree of notoriety, in the sense that even non finance people (lawyers, accountants, judges, bankers . . . ) know that options are priced using Black-Scholes. They may not know how to apply it, and they certainly wouldn’t know why the formula is correct, but they know that it is used to price options. In our discussion of the Black-Scholes model, we’ll make no attempt whatsoever to give a theoretical background to the model. It’s hopeless, unless you know a lot more math than 99% of all beginning finance students will ever know. 1  The next chapter discusses the other major model for pricing options, the binomial option  pricing model . The binomial model gives some insights into how to price an option, and it’s also used widely (though not as widely as the Black-Scholes equation). Most books discuss the  binomial model—which, in a theoretical sense, underlies the Black-Scholes formula—first and then discuss Black-Scholes. However, since we have no intention of making the theoretical connection between the binomial model and Black-Scholes, we’ve chosen to reverse the order and deal with the more important model first. 1  A bitter truth, perhaps. But get this—your professor probably can’t prove the Black-Scholes equation either (don’t ask him, he’ll be embarrassed). On the other hand—you know how to drive a car but may not know how an internal combustion engine works, you know how to use a computer but can’t make a central processing unit chip, ....  PFE Chapter 25, The Black-Scholes formula page 3 What does “pricing an option” mean? Suppose we’re discussing a call option on Microsoft stock which is sold on 8 February 2002. On this date, Microsoft’s stock price is S  0  = $60.65. Suppose that the call option has an exercise price  X = $60  and expires on July 19, 2002. Here’s what you’ve learned so far: ã   From Chapter 23, you know the basic option terminology. You know what an exercise  price  X   is, you know the difference between a call and a put, etc. ã   From Chapter 23, you also know what the  payoff     pattern  and  profit pattern  of the call option looks like—by itself and in combination with other assets ã   From Chapter 24, you know that there are some pricing restrictions  on the call option. A simple restriction (“Fact 1” from Chapter 23, page000) says that ( ) 00 ,0 CallMaxSPVX  > ⎡ − ⎤⎣ ⎦ . A more sophisticated restriction (“Fact 3,” put-call  parity, page000) says that—once we know the price of Microsoft stock, the call price, and the interest rate—the put price is determined by the relation ( ) 000 PutSCallPVX  + = + . All of these facts are –by themselves—interesting. However, they don’t tell us what the  price  of the call option should be. This is the subject of this chapter—the Black-Scholes formula tells us how what the market price of the option should be. Chapter notation We recall the notation we’re using throughout Chapters 23-27:  PFE Chapter 25, The Black-Scholes formula page 4 Notation   Throughout the chapter we use the following notation S   = The price of the stock. When we want to be precise about the price of the stock on a specific date, we will sometimes write S  0  for the price of the stock today (time 0) and S  T   for the price of the stock on the option exercise date T  .  X   = The option exercise price r   = The interest rate C   = The call option price. When we want to be precise about the call price on a specific date, we will sometimes write C  0  for the price of the stock today (time 0) and C T   for the  price of the stock on the option exercise date T  . Occasionally we will even use the full word, writing Call 0  . P  = The put option price. When we want to be precise about the put price on a specific date, we will sometimes write P 0  for the price of the stock today (time 0) and P T   for the  price of the stock on the option exercise date T  . Occasionally we will even use the full word, writing Put  0  . Finance concepts in this chapter ã   Black-Scholes formula ã   Put-call parity ã   Stock price volatility ã   Implied volatility ã   Real options
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