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11.1 The Pricing of Stock Options Using BlackScholes Chapter 11 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.2 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. Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Assumptions Underlying BlackScholes • • • 11.3 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 t Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.4 The Lognormal Property • These assumptions imply ln ST is normally distributed with mean: ln S ( 2 / 2) T and standard deviation: • Since the logarithm of ST T is normal, ST is lognormally distributed Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.5 The Lognormal Property continued ST 2 ln ( / 2)T , T S where m,s] is a normal distribution with mean m and standard deviation s Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.6 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. 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 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.7 11.8 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 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.9 The Volatility (cont.) • As a rough approximation, T 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. 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, . . . , Sn at intervals of years 2. Define the continuously compounded return as: Si ui ln Si 1 3. Calculate the standard deviation of the ui ´s (=s) s 4. The volatility estimate is s * Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.10 11.11 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 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull Computation of Volatility Using Historical data Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Closing Stock Price Price Relative Daily return 20 20 1/8 1.0063 0.00623 19 7/8 0.9876 -0.01250 20 1.0063 0.00627 20 1/2 1.0250 0.02469 20 1/4 0.9878 -0.01227 20 7/8 1.0309 0.03040 20 7/8 1.0000 0.00000 20 7/8 1.0000 0.00000 20 3/4 0.9940 -0.00601 20 3/4 1.0000 0.00000 21 1.0120 0.01198 21 1/8 1.0060 0.00593 20 7/8 0.9882 -0.01190 20 7/8 1.0000 0.00000 21 1/4 1.0180 0.01780 21 3/8 1.0059 0.00587 21 3/8 1.0000 0.00000 21 1/4 0.9942 -0.00587 21 3/4 1.0235 0.02326 22 1.0115 0.01143 Introduction to Futures and Options Markets, 3rd Edition © 1997 by John C. Hull 11.12