1. 1 The Black-Scholes Model Chapter 13 (7 th edition) Ch 12 (pre 7 th edition) + Appendix (all editions)

2. 2 Black-Scholes 1970s: Fischer Black, Myron Scholes and Robert Merton developed their, now famous, option pricing model 1997: Nobel prize in economics to Merton and Scholes; Black died in 1995

3. 3 Black and Scholes Model assumptions The stock pays no dividends during the option's life; most companies pay dividends, thus, subtract the discounted value of a future dividend from the stock price. European exercise terms are used; this limitation is not a major concern because very few calls are ever exercised before the last few days of their life. This is true because when you exercise a call early, you forfeit the remaining time value on the call and collect the intrinsic value. Markets are efficient; this assumption suggests that people cannot consistently predict the direction of the market or an individual stock. No commissions are charged; usually market participants do have to pay a commission which can often distort the output of the model. Interest rates remain constant and known; the Black and Scholes model uses the risk-free rate to represent this constant and known rate returns lognormally distributed

4. 4 The Black-Scholes Formulas European calls and puts on non-dividend paying stocks Where N(x) is the cumulative probability function for a standardized normal variable or the probability that a variable with a standard normal distribution will be less than x; shaded area represents N(x)

5. 5 Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes option price equals the market price Calculated by trial and error Basically, implied volatility tells an investor what the market thinks about the riskiness of a stock

6. 6 Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes Only dividends with ex-dividend dates during life of option should be included

7. 7 American Calls An American call on a non-dividend-paying stock should never be exercised early due to the time value implication An American call on a dividend-paying stock, however, should only ever be exercised immediately prior to an ex-dividend date; the reason is that a sufficiently large dividend will make the call option less valuable even when considering the remaining time value of the option

8. 8 Black’s Approach to Dealing with Dividends in American Call Options Set the American price equal to the maximum of two European prices: 1.The 1st European price is for an option maturing at the same time as the American option 2.The 2nd European price is for an option maturing just before the final ex- dividend date

9. 9 7 th edition: 13.4, 13.13, 13.14, 13.18, 13.25, 13.26, 13.27 5 th & 6 th edition: 12.4, 12.13, 12.14, 12.18, 12.25, 12.26, 12.27