1. 11.1 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Pricing of Stock Options Using Black- Scholes Chapter 11

2. 11.2 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Black-Scholes Model Black-Scholes option pricing model was developed in 1970 by : Fischer Black Myron Scholes Robert Merton Their work has had huge influence on the way in which market participants price and hedge options.

3. 11.3 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Assumptions Underlying Black- Scholes Black-Scholes assume that stock prices follow a random walk. - This means that proportional changes in the stock price in a short period of time are normally distributed. Proportional change is the change in the stock price in time  t is  S. The return in time  t is  S/S This return is assumed to be normally distributed with mean  t and standard deviation

4. 11.4 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Lognormal Property These assumptions imply ln S T is normally distributed with mean: and standard deviation : Since the logarithm of S T is normal, S T is lognormally distributed  T

5. 11.5 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Lognormal Property continued where  m,s] is a normal distribution with mean m and standard deviation s

6. 11.6 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Problem Calculate the mean and standard deviation of the continuously compounded return in one one year for a stock with an expected retrun of 17 percent and volatility of 20 percent per annum.

7. 11.7 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Expected Return Two possible definitions:  is the arithmetic average of the returns realized in may short intervals of time  –  2 /2 is the expected continuously compounded return realized over a longer period of time  is an arithmetic average  –  2 /2 is a geometric average

8. 11.8 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Volatility The volatility of a stock, , is a measure of uncertainty about the return provided by the stock. - It is measured as the standard deviation of the return provided by the stock in one year when the return is expressed using continuous compounding. As an approximation it is the standard deviation of the proportional change in 1 year

9. 11.9 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Volatility (cont.) As a rough approximation, is the standard deviation of the proportional change in the stock price in time T. - Consider the situation, where  = 0.30 per annum standard deviation of the proportional change in: – six month – three month - Uncertainty about the future stock price increases with the square root of how far ahead you are looking.  T

10. 11.10 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Estimating Volatility from Historical Data 1. Take observations S 0, S 1,..., S n at intervals of  years 2. Define the continuously compounded return as: 3. Calculate the standard deviation of the u i ´s (=s) 4. The volatility estimate is

11. 11.11 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull The Concepts Underlying Black-Scholes The option price & the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock & the option which eliminates this source of uncertainty The portfolio is instantaneously riskless & must instantaneously earn the risk-free rate

12. 11.12 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Computation of Volatility Using Historical data