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Black-Scholes Option Pricing Formula
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C = Sd -t N(x) Kr -t N(x - t) 1 with x = [log(Sd -t /Kr -t ) t] + t 2
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S= K= t = R= d = = current underlying asset price (in dollars) strike price (in dollars) current time to expiration (in years) riskless return (annualized) dividend yield (annualized) underlying asset volatility (annualized)
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CHAPTER FIVE Valuing Options
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world. Although there are conditions when the formula must be modified, empirical research has shown that the Black-Scholes formula closely approximates the price of traded options. 14 The Black-Scholes formula (and most other option pricing formulas and models) looks at the current stock price, the exercise price of the option, the time to expiration, the risk-free interest rate, the dividend rate on the stock, and the volatility of the underlying stock. Of these assumptions, the most important is volatility, meaning the swings or variation in price of the underlying stock. In general the more volatile a stock, the more likely it is that the market price will rise above the exercise price and the option will be in the money. Figure 5-2 shows that the greater the volatility of the underlying stock, the greater the value of the option. For the nonmathematician, a formula like Black-Scholes may look daunting. But that is not a reason to discount it. The fact is formulas, daunting or not, do exist and are used daily to evaluate options and complex transactions using options. It s not enough to say something is difficult and therefore can t be done. This is clearly a case of the Emperor has no clothes, but no one wants to admit it. There is an entire body of knowledge, as well as decades of practice, in option valuation, which can be adapted for use in calculating the
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FIGURE
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Black-Scholes Assumptions and Relations to Option Value
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Stock Option Estimated Fair Value Effect on Option Value Controllable Assumptions Expected Life of Stock Award Risk-Free Rate Exercise Price Dividend Yield Noncontrollable Assumptions Volatility of Stock Market Price If Assumption Increases If Assumption Decreases
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PART TWO Elements of the Solution
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value to the company of the options it grants (as well as the value to the employee, which is probably lower).