Geometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc,

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1 12 Option Pricing We are now going to apply our continuous-time methods to the pricing of nonstandard securities. In particular, we will consider the class of derivative securities known as options in this section of the notes. We will derive the famous Black-Scholes option pricing model in this section, for which Scholes recently won the Nobel prize (Black died a few years ago). The original paper by Black and Scholes is the supplementary reading that goes with this lecture note. The paper is a model of how all good papers should be written. There is not a lot of extraneous information in there - it's pretty short. However, its inuence has been enormous. They seem to have maximized the ratio of economics to math in their paper. If you do not know much about options, you may want to talk to me about other readings. We will use a set of powerpoint notes to review some preliminary denitions and concepts The Black-Scholes Model We will assume that stock prices follow geometric Brownian motion, ds S = dt + dw (236) We also know that a particular call option's price is a function of time and its underlying stock price, C = C(S; t): (237) Since the call option's value is a function of time and the stock price, we know that it will follow a diusion process. We posit (and later verify) that the call option's value follows geometric Brownian motion, dc(s; t) C = ^dt +^dw: (238) 95

2 Geometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc, dc = C S ds + C t dt C SSdS 2 ; (239) and we can take the expectation of dc, E[dC] =C S Sdt + C t dt c SS 2 dt: (240) In order to verify that the call option's price follows Geometric Brownian motion we need to solve for ^ and ^. If we can nd expressions for both of these terms then the call price really does follow GBM. We solve for the terms using our expression for dc and E[dC]. Beginning with ^, ^ = E[dC] Cdt = C SS + C t + 1 C 2 SSS 2 2 : (241) C The term for ^ is ^ = dc, E[dC] CdW = C S(dS, E[dS]) + 1 C 2 SS(dS 2, E[dS 2 ]) ; (242) CdW which simplies to ^ = C SS C : (243) So we have shown that the call option price follows GBM with particular values for ^ and ^. Now we want to think about a portfolio, or a position that involves putting w dollars in the call option and putting (1, w) in the stock. The stochastic proces that will describe the value of our position, V, through time is another GBM 96

3 process, dv V = w dc C +(1,w)dS S : (244) Using the processes for dc and ds, we can be more specic about the process driving the value of our position, dv V =[w^+(1,w)]dt +[w^+(1,w)]dw: (245) We choose a value for w that eliminates all the risk in our position in the next instant by setting the coecient on the dw term above to zero, w =, ^ : (246) Choosing this particular w gives us the risk-free process dv V w =[w ^+(1,w )]dt = rdt (247) We have apparently done a very curious thing. By carefully choosing w, we can make the coecient on the dw term go to zero. If we set up our position just right, we can eliminate all of the risk in the portfolio. This is called a dynamic hedge. The risk-free position that we create will only be risk-free over the next instant - to maintain a risk-free position, we will have to constantly adjust w as both S and C move around through time. This is the continuous-time analog to one node of the binomial tree that we saw in the previous set of notes. The continuous-time hedge is the basis for most arbitrage pricing in continuous time. To eectively hedge continuously you must have very small transactions costs. Some researchers derive no-arbitrage results with transactions costs - they nd that rather than arriving at a unique no-arbitrage price, they get a range of possible option values that don't admit 97

4 arbitrage. This is a highly techincal research area that is dominated by statisticians and mathematicians. We have shown how to convert our position to a risk-free security through a dynamic hedge. Since this position is riskless over the next instant, dt, its payo must be equal to the payo of a risk-free security over the next instant (by the law of one price),, ^ ^, ^ = r: (248), ^ Doing a little bit of algebra, we can rearrange this to be, ^, r = ^ (, r): (249) Substituting the expressions for ^ and ^ into this expression, C S S + C t C SS 2 S 2 C, r = CSS C (, r); (250) which further simplies to C S S + C t C SS 2 S 2, rc = C S S(, r): (251) Dropping C S S from both sides, C S Sr + C t C SS 2 S 2, rc =0; (252) which gives us the equation that we have been solving for, (C S, C)r + C t C SS 2 S 2 =0; (253) which is a famous dierential equation. This is known as the heat transfer equation, 98

5 and its solution was derived many years ago. For the Black-Scholes formula, we solve the heat transfer equation subject to the following boundary conditions, Boundary conditions : C(0;t)= 0 C(S; T ) = max[s, X; 0] (254) C(S; t) S: Unfortunately, you will just have to take it on faith (for now) that the solution to the heat transfer equation is actually the Black-Scholes formula. For reference purposes, the equation can be written as C(S; t) =SN(d 1 ),Xe,r N (d 2 ); (255) where d 1 = ln(s=x)+(r+2 =2) p ; d 2 = d 1, p ; (256) and is the option's time to maturity, N () is the cumulative normal distribution, and X is the strike price. This is not the last time that we will see the Black-Scholes formula. We will derive it again with \change of measure" methods when we talk about the absence of arbitrage in continuous-time. Let's talk about the model a little more right now to get a little more understanding of how it works Solving for Delta An important quantity for the Black-Scholes model is what is commonly called, or the sensitivity of the option's call price to changes in the value of the option's underlying security. As we stated previously, in the Black-Scholes framework, a call 99

6 option's is just equal to N (d 1 ). We should prove this rigorously. We start by taking the derivative of the Black-Scholes call option = N(d 1), : (257) Remembering that we can express d 1 as, d 1 = ln(s=x) p + r p p ; (258) we can take the derivative ofd 1 with respect to = 1 S p : (259) Now using the chain (d 1 = f(d 1) S p : (260) Since d 2 is just d 1 minus p,asimilar result holds for N (d 2 (d 2 = f(d 1, p ) S p : (261) In both of these expressions, f() is just the standard normal distribution, 2 =2 f(x) = e,x p : (262) 2 Returning to our expression for, we can now state = N(d 1)+ f(d 1) p,xe,r S f(d 1, p ) p : (263) 100

7 Therefore, our result ( = N (d 1 )) will hold as long as f(d 1 )= Xe,r S f(d 1, p ): (264) To see that this condition is true, we express the ratio of f(d 1 ) to f(d 2 )as e,d2 1 =2 e,(d 1, p ) 2 =2 =Xe,r S : (265) Next, we express the denominator as e,(d 1, p )=2 = e,(d2 1,2d 1 p + 2 )=2 ; (266) which yields the condition e,d 1 p + 2 =2 = Xe,r S : (267) Finally, substituting the denition of d 1 into our condition, e, ln(s=x),(r+2 =2) + 2 =2 = Xe,r S ; (268) which we can verify as true. Thus, we have shown = N(d 1): (269) What do we use this term for? For one thing, tells us how to perform our continuous-time hedge. As we stated in the notes on the binomial tree pricing model, we always want to hold shares of stock for each written call option in a risk-free portfolio. As the components of d 1 change, will change. We can update our position by constantly checking if we have shares of stock per short call. 101

8 A second reason for caring about is that it tells us about the expected return on an option. To see this, we will dene another quantity known as omega, C = SN(d 1 ) SN(d 1 ), Xe,r N (d 2 ) > 1: (270) Omega is the elasticity ofthe call option's price to changes in the value of the call's underlying security. Thus, If the underlying asset's value changes by 3% because the market moves 3%, the call option's price should move by times 3%. In other words, it can be shown that C = S : (271) Since is always greater than one (as demonstrated above), the expected return of a call option written on an asset with a positive beta is always higher than the expected return of the underlying asset in a Black-Scholes/CAPM world. There are, of course, much more general things that can be said about option returns. However, the intuition behind options returns arguments is the Black-Scholes/CAPM case. We have spent a great deal of time discussing option pricing. This may seem unwarranted because options are just one type of nancial instrument available to market participants. However, Black and Scholes point out at the end of their article that lots of securities actually have option-like characteristics. Consider, for example, the stock of a company that has some level of debt outstanding that must be paid in a lump sum. If the company plans to liquidate immediately after paying o its debt then the company's stock can be thought of as an option on the value of the rm. If the rm is suciently protable then the option will expire in the money and stockholders will be paid. Otherwise, all the rm's value will go to bondholders and the option (equity) will expire worthless. 102

9 12.3 Homework Problems 1. Assume that returns are driven by a k-factor model, so that ds i =S i = i dt + i1 df 1t + :::+ ik df kt + i dw i (272) and assume that each factor follows geometric Brownian motion with drift fj and volatility fj for the jth factor. Let the Wiener process of each factor (dw fj ) be independent of the corresponding process of each other factor and let it be independent ofdw i 8i. Assume also all of the assumptions that make the Black-Scholes formula valid. Show that the jth factor beta of a call option on stock i is equal to i times the stock's factor beta, ij. 2. Suppose you buy one share of stock, write a one-year call option on the stock with a strike price (X) of10, and buy a one-year put with X = 10. Your total outlay for this position is $9.50. what is the risk-free rate? 3. Show that an at-the-money call option must cost more than an at-the-money put on the same stock and with the same maturity. 4. Show that the dierence between the prices of two European call options with strike prices X 1 and X 2 (where X 1 < X 2 ) is less than or equal to the present value of (X 2, X 1 ). 5. In the numerical example of a two-stage binomial tree in the powerpoint notes, solve for the value of a call option with a strike price of

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