Chapter 16 Option Valuation Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin

16-2 Call Option Price 1. At the expiration date – –C T = Max (0, S T – X)Why? –C o = Max (0, S o – X* ) –If you capture the possibility of call option useful in the future or not useful, you can rewrite the whole thing like the Black-Scholes equation in the next page

16-3 Ln=natural log. You can find this button on your regular calculator. Ln(x) => enter x (number) first, then press Ln button. C 0 = Current call option value. X = Exercise Price S 0 = Current stock price e = , the base of the natural log. For, you can get (-rT) first, then press “exp” function on the calculator r = Risk-free interest rate (annualized with continuous compounding) T = Time until expiration in years  = Annual standard deviation of stock returns N(d) = probability that a random draw from a normal distribution will be less than d Once you get the current value of the Call option using the B-S model, you need to compare the value against the current price (premium) of the call. If the current price is higher than the value => Don’t buy (even sell), Otherwise, buy the call option!! Black-Scholes Option Valuation

16-4 A Standard Normal Curve

16-5 Value of a Call Option X Stock Price t Value $0 Prior to expiration Value at expiration or “Exercise” or “Intrinsic” Value 6 mo 2 mo Difference is the “Time Value” of the option What is the time value of a call option? The time value of a call incorporates the probability that S will be in the money at period T given S 0, time to T,  2 stock,X, and the level of interest rates 

16-6 Using the Model S 0 =$100; X = $95 T = 3 months or 0.25 r = 5.2%  = 40% $11.23 Cumulative Normal Cumulative Normal Call data

16-7 Aside) Replicating Portfolio, No Arbitrage Pricing, Riskless Portfolio Security A pays $3 if sunny tomorrow(t=1), $0 if not, Security B pays $0 if sunny, $2 if not. If I want to have $6 tomorrow for sure, what can I do? You can create a replicating portfolio with 2A+3B package. If A and B are priced today(t=0) at $1.40 and $0.95 respectively, what should be the cost of such a portfolio? $5.65. If such a portfolio package is traded for $5.85, you are not in equilibrium as you can make an arbitrage profits by doing Buy Low and Sell High. The equilibrium can be achieved only when the price of the package to be $5.65 (if the prices of A and B do not change) where no arbitrage exists. This portfolio is riskless, since you get $6 regardless of future state. If the risk-less T-Bill interest rate (one day) is 7%, are you in equilibrium? Since PV of $6 tomorrow = $6/(1.07) = $5.61, you would Buy Low T-Bill at $5.61 and Sell High the package for $5.65 (2A+3B and deliver $6 regardless of the weather tomorrow) and make arbitrage profits of $0.04. To have no arbitrage, the price of the package has to be $5.61 or the interest has to be 6.19%.

16-8 Problem $8.13