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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).
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THE FOUR GUIDING PRINCIPLES
As we move toward more accurate evaluation of the cost of options, I invite companies to follow the Four Guiding Principles for Evaluating Executive and Employee Stock Options. All the implications of each point may not be known at this time. Nonetheless the criteria serve as a powerful guide to more accurate and responsible evaluation of the costs and risks involved in option grants. Four Guiding Principles for Evaluating Executive and Employee Stock Options 1. Use the correct mathematics. 2. Input the correct assumptions or boundary conditions. 3. Consider options both as an expense and as a contingent liability. 4. Acknowledge that options are an investment in human capital, and calculate a return on that investment. Let s take a look at these principles one by one. 1. Use the correct mathematics. There is a very sophisticated, robust science dedicated toward option valuation. The financial services industry knows how to apply it, as do numerous academics and economists. These mathematical formulas have simply not yet been applied thoroughly to executive and employee options. 2. Input the correct assumptions or boundary conditions. Using the right assumptions in the option methodology will require companies to take an actuarial approach. Given all the options it grants, what is the percentage that will be exercised early What percentage will be forfeited What percentage will never be exercised for whatever reason How does stock performance affect the timing of
CHAPTER FIVE Valuing Options
option exercises Just as an insurance company studies the demographics of large populations to make assumptions about incidents of accident or illness and life expectancy, companies can apply this approach to option grants. Companies already have years of actual experience with which they can make assumptions and predictions, based upon things such as the age, sex, salary level, and years of service of the option-holder population. The Delves Group, in partnership with Chicago Consulting Actuaries, has developed a comprehensive economic model to determine the total economic cost of options. For example what assumptions and predictions can be derived from the previous experience of companies when it comes to early exercise of options that would impact their value What are the impacts on cash flow and EPS The better the assumptions used in the valuation model, the more fair the fair value of an option will become. This is of vital importance to major corporations, which might be looking at an option expense of $200 million to $300 million per year. The quality and accuracy of the mathematical model and inputs to the model could potentially save tens of millions of dollars per year in expense. 3. Consider options both as an expense and as a contingent liability. As mentioned in 4 and as I ll explore in detail later in this chapter, options should be thought of as a liability on the balance sheet with an offsetting investment in human capital. This will also be a complex actuarial determination no less important than the magnitude of the company s pension expense. To do this accurately, companies will have to look at the actual experience of options granted in the past. For example, after what percentage increase in the stock price do people tend to exercise their options As depicted in Figure 5-3, companies will most likely construct their own bell curve of experience, looking at the percentage
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