428_lecture21_BSoption_strategy_S09

428_lecture21_BSoption_strategy_S09 - Valuation Lecture 21...

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Unformatted text preview: Valuation Lecture 21 Black-Scholes and Option Trading Strategies Materials from Smart, et al. Ch. 18 & Grinblatt & Titman Ch.7 Black-Scholes Model Reasoning behind Black and Scholes is similar to that of the binomial model: Does a combination of option and shares exist that provides a risk-free payoff? Assumption s of the model • Stock prices can move at every moment in time. • Movements of stock prices are random, and therefore unpredictable. • Volatility (standard deviation) of stock’s movements is known. September 23, 2009 Calculating B-S Call Options S = current market price of underlying stock X = strike price of option t = amount of time before option expires (in years) r = annual risk-free interest rate = annual standard deviation of underlying stock’s returns e = 2.718 N(X) = probability of drawing a value less than or equal to X from standard normal distribution σ t d d t t r X S d σ σ σ- = + + = 1 2 2 1 2 ln C = SN(d 1 ) – Xe-rt N(d 2 ) September 23, 2009 An Example Price of Stock Holm Inc. is currently $28 A European call option on Stock Holm Inc. has expiration date three months in the future and strike price of $25. Estimate of standard deviation on Stock Holm Inc. is 40% and the risk-free rate is 3%. 5041 . 7041 . 4 / 1 4 . 4 1 2 4 . 03 . 25 28 ln 1 2 2 1 =- = = + + = t d d d σ What should the call price be? N(0.7041) = 0.7593 N(0.5041) = 0.6929 C = 28(0.7593) – 25( 2.718 -(.03)(0.25) )(0.6929) = $4.07 September 23, 2009 B-S Call Values for Stock Holm Inc. 0 15 28 45 60 5 10 15 20 25 30 35 40 45 50 Stock Price ($) Value of Call Option ($) Call Price Now Call Payoff at expiration C = SN(d 1 ) – Xe-rt N(d 2 ) September 23, 2009 Option’s Delta Measures how much the call price changes as the underlying stock price changes: Delta is equal to the slope of the solid line in previous graph. Delta equals the value N(d1) from B-S formula. When call is far out of the money, delta is close to zero....
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This note was uploaded on 09/23/2009 for the course AEM 4280 taught by Professor Ng,d. during the Spring '08 term at Cornell.

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428_lecture21_BSoption_strategy_S09 - Valuation Lecture 21...

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