Options evaluation - Black-Scholes model vs. Binomial options pricing model Article (PDF Available) November 2010 with 4,505 Reads How we measure 'reads'.
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The formula, developed by three economists—Fischer Black, Myron Scholes and Robert Merton—is perhaps the world's most well-known options pricing model. It was introduced in their 1973 paper, 'The Pricing of Options and Corporate Liabilities,' published in the Journal of Political Economy. Black passed away two years before Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives (the Nobel Prize is not given posthumously; however, the Nobel committee acknowledged Black's role in the Black-Scholes model).
Formula: C = SN(d 1)-Ke (-rt)N(d 2) where,C = Theoretical call premiumS = Current stock pricet = timeK = option striking pricer = risk free interest rateN = Cumulative standard normal distributione = exponential term (2.7183)d 1 = ( ln(S/K) + (r + (s 2/2))t ) / s√td 2 = d 1 - s√ts = standard deviation of stock returns Example:A company currently sells for $210.59 per share. The annual stock price volatility is 14.04%, and the annual continuously compounded risk-free interest rate is 0.2175%. Find the value of d1 in the Black-Scholes formula for the price of a call on a company's stock with strike price $205 and time for expiration of 4 days.
Given,S= $210.59,K= $205t = 4 daysr = 0.2175%s = 14.04% To Find,Call option priced1 Solution: Step 1:Substitute the given value in the formula,d1 = ( ln(210.59/205) + (0.002175+(0.1404 2) / 2)(0.01096) ) / 0.1404.√(0.01096)d1 = 1.8394 Step 2:d2 = 1.8394 - 0.1404.√(0.01096)d2 = 1.8247 Step 3:Substitute the value of d1 and d2 in the Call option (C) formulaC = 210.59. 205. SN(d1)-Ke(-rt)N(d2)C = -8.1313.
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