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73323274-13-BSM-model - T he Black-ScholesMerton Model In...

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The Black-Scholes- Merton Model In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made a major breakthrough in the pricing of stock options. I This involved the development of what has become known as the Black-Scholes (or Black-Scholes-Merton) model. The model has had a huge influence on the way that traders price and hedge options. It has also been pivotal to the growth and success of financial engineering in the last 20 years. In 1997, the importance of the model was recognized when Robert Merton and Myron Scholes were awarded the Nobel prize for economics. Sadly, Fischer Black died in 1995, otherwise he too would undoubtedly have been one of the recipients of this prize. This chapter shows how the Black-Scholes model for valuing European call and put options on a non-dividend-paying stock is derived. It explains how volatility can be either estimated from historical data or implied from option prices using the model. It shows how the risk-neutral valuation argument introduced in Chapter 11 can be used. It also shows how the Black-Scholes model can be extended to deal with European call and put options on dividend-paying stocks and presents some results on the pricing of American call options on dividend-paying stocks. 13.1 LOGNORMAL PROPERTY OF STOCK PRICES The model of stock price behavior used by Black, Scholes, and Merton is the model we developed in Chapter 12. It assumes that percentage changes in the stock price in a short period of time are normally distributed. We define fJ..: Expected return on stock per year 0-: Volatility of the stock price per year The mean of the percentage change in the stock price in time .6.t is J.L.6.t and the 1 See F. Black and M. Scholes, "The Pricing of Options and Corporate Liabilities," JOllmal oj Political Economy, 81 (May/June 1973): 637-59; R.C. Merton, "Theory of Rational Option Pricing," Bell Journal oj Economics and Managemelll Science, 4 (Spring 1973): 141-83. 281
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282 CHAPTER 13 standard deviation of this percentage change is O'.;;s:i, so that (13.1) where /::;.S is the change in the stock price S in time /::;.t, and ¢(111, s) denotes a normal distribution with mean 111 and standard deviation s. As shown in Section 12.6, the model implies that InST -lnSo ~ ¢[ (tL - ~2)T, o'vT ] From this, it follows that and In ~: ~ ¢[ (tL - ~2) T, o'vT ] In ST ~ ¢[In So + (tL - ~2) T, o'vT ] (13.2) (13.3) where ST is the stock price at a future time T and So is the stock price at time O. Equation (13.3) shows that In ST is normally distributed, so that ST has a lognormal distribution. The mean of In ST is In So + (tL - O' 2 /2)T and the standard deviation is o'vT. Example 13.1 Consider a stock with an initial price of $40, an expected return of 16% per annum, and a volatility of 20% per annum. Fr()m equation (13.3), the probability distribution of the stock price ST in 6 months' time is given by In ST ~ ¢[ln40 + (0.16 - 0.2 2 /2) x 0.5, 0.2.JQ.5] InST ~ ¢(3.759, 0.141) There is a 95% probability that a normally distributed variable has a value within 1.96 standard deviations of its mean. Hence, with 95% confidence, 3.759 - 1.96 x 0.141 < In ST < 3.759 + 1.96 x 0.141 This can be written e3.759-1.96xO.141 < ST < e3.759+1.96xO.141 or 32.55 < ST < 56.56 Thus, there is
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73323274-13-BSM-model - T he Black-ScholesMerton Model In...

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