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# lecture12 - Lecture 12 The Black-Scholes Model Steven...

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Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794–4400 http://www.cs.sunysb.edu/ skiena

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The Black-Scholes-Merton Model Analyzing the Binomial tree model with infinitely time small steps gives the Black-Scholes option pricing model, which says the value of a stock option is determined by six factors: S, the current price of the underlying stock y, the dividend yield of the underlying stock K, the strike price specified in the option contract r, the risk-free interest rate over the life of the option contract T, the time remaining until the option contract expires σ , the price volatility of the underlying stock.
The Pricing Formula The price of a call option on a single share of common stock is: C = Se yT N ( d 1 ) - Ke rT N ( d 2 ) The price of a put option on a single share of common stock is: P = Ke rT N ( d 2 ) - Se yT N ( d 1 ) and d 1 = ln( S/K ) + ( r - y + σ 2 / 2) T σ T d 2 = d 1 - σ T

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Formulae Details Three common functions are used to price call and put option prices: e - rt , or exp( - rt ) , is the natural exponent of the value of rt (in common terms, it is a discount factor) ln( S/K ) is the natural log of the “moneyness” term, S/K . e = 2 . 71828 is the base of the natural log N ( d 1 ) and N ( d 2 ) denotes the standard cumulative normal probability for the values of d 1 and d 2 . It is the probability that a random draw from a normal dist. will be < d .
Pricing Example Suppose you are given the following inputs: S = \$50 (current stock price) y = 2% (dividend yield) K = \$45 (strike price) T = 3 months (or 0.25 years) s = 25% (stock volatility) r = 6% (risk-free interest rate)

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## This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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lecture12 - Lecture 12 The Black-Scholes Model Steven...

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