Modified Black-Scholes Option Valuation Methodology

Black Scholes is a calculation that uses 70% of the stated life which comes from the source and is usually 10 years.

The basic instrument used for the determination of a stock option's estimated value at grant was the Black-Scholes Option Pricing Model. That model features the input of six assumptions:

The first four assumptions are relatively straightforward:

The strike price per share was that specified by the company in its proxy statement.

The market price per share at the time of grant was assumed to be equal to the strike price per share, unless the company specified otherwise in its proxy statement.

The term of the grant was determined as follows:

Options were assumed to be granted on July 1st of the particular year for which data were reported. Since almost 75% of S&P 500 companies have a calendar year fiscal year, we used the July 1st data as a simplifying assumption.

The nominal term of the option was calculated as the time span between July 1st of the year of grant and the actual expiration date, as reported by the company in its proxy statement. Figures thus calculated were then rounded to the nearest whole year.

The term of the option was reduced by 30% to an amount of 70% of the actual term. We implement this reduction because executives rarely wait until the expiration date to exercise their options.

The risk-free rate of interest used was the approximate average yield that could have been earned in the particular year by investing in a U.S. Treasury bond carrying a seven-year term. The use of a Treasury bond satisfies the Black-Scholes condition that the interest rate be free of risk and that the compounded rate of interest that may be earned on the risk-free security be known at the time of grant. And the use of a security with a seven-year term first recognizes the fact that the overwhelming majority of executive stock option grants carry ten year terms and then goes on to reflect the earlier-mentioned assumption that an executive will exercise his option 70% of the way into its nominal term. The risk-free interest rates used for 1992 through 2003 were:
1992 6.43   1998 4.73
1993 5.53   1999 6.55
1994 7.84   2000 5.16
1995 5.49   2001 4.84
1996 6.34   2002 3.36
1997 5.77   2003 3.77

Estimated Future Stock Price Volatility
We use a 60 month volatility number. If a company is in the bottom or top 5% of volatilities, we increase or decrease its volatility to the 5th or 95th percentile values. This prevents us from using a volatility calculation that is so far outside of the norm that it will not likely repeat in the future. If a stock has traded for less than 60 months, use as many months as possible to do the calculation. If the stock has traded for less than one year, we input the average volatility value for the S&P 1500.

For the years 1992 through 2003, the mean stock price volatility so used was:
1992 0.313   1998 0.358
1993 0.312   1999 0.395
1994 0.357   2000 0.458
1995 0.331   2001 0.486
1996 0.319   2002 0.497
1997 0.319   2003 0.471

However, before inputting the statistics so obtained into the Black-Scholes model, we subjected them to two further adjustments, the first of which was determined as follows:

If the actual stock price volatility was higher than the 95th percentile of the volatility distribution, we lowered it to the 95% percentile value. For the years 1992 through 2003, the 95th percentile of volatility was:
1992 0.552   1998 0.645
1993 0.547   1999 0.706
1994 0.627   2000 0.85
1995 0.599   2001 0.913
1996 0.59   2002 0.916
1997 0.59   2003 0.881

If the actual stock price volatility was lower than the 5th percentile of the volatility distribution, we lowered it to the 5th percentile value. For the years 1992 through 2003, the 5th percentile of volatility was:
1992 0.154   1998 0.179
1993 0.15   1999 0.198
1994 0.168   2000 0.236
1995 0.157   2001 0.245
1996 0.155   2002 0.25
1997 0.163   2003 0.229

By making these adjustments, we avoided introducing volatility figures, that though currently correct, are so far outside the norm as to make it likely that they will be incorrect for the long-term future.

Estimated Future Dividend Yield
As with the stock price volatility series, we first assured that the initial input dividend yield was not outside the range of reasonableness. To measure past yields, we average dividend yields over a three-year period. Then, if the dividend yield was more than the amounts in the table below:
1992 7.586   1998 5.093
1993 6.819   1999 4.582
1994 5.721   2000 4.741
1995 5.851   2001 5.048
1996 5.826   2002 5.006
1997 5.51   2003 4.796
(values approximating the 95th percentile of the distribution of yields), we reduced it to the respective 95th percentile value.

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