Let™s assume the mean expected stock price at the expiration of the

option is $100 per share. If the standard deviation is $1 per share, then

there is a 68% probability that the stock value will be between $99 and

$101 and a 95% probability that the stock value at expiration will be

between $98 and $102. That would be a tight distribution and would look

like a tall, thin bell-shaped curve. There would only be a 5% probability

that the price would be below $98 or above $102. Since the distribution

is symmetric, that means a 21„2% probability of being below $98 and a

21„2% probability of being above $102. The chances of hitting a jackpot on

this stock are very low.

Now let™s assume the standard deviation is $20 per share, or 20% of

the price. Now there is a 95% probability the price will be within $40 per

share (two standard deviations) of $100, or between $60 and $140. The

probability of hitting the jackpot is much higher.

We now have the background to understand how the stock volatility

is the main determinant of the value of the option. The more volatile the

stock, the shorter and fatter is the normal curve and the greater is the

probability of making a lot of money on the investment. If your stock

ends up on the right side of the curve, it does not matter how far up it

went”you will choose to not exercise the option and you lose only the

price of the option itself. In contrast to owning the stock itself, as an

option holder it matters not at all whether the stock ends up at $100 per

share or $140 per share”your loss is the same. Only the left side matters.

Therefore, a put option on a volatile stock is much more valuable than

one on a stable stock.

Black“Scholes Put Option Formula

The Black“Scholes options pricing model has the following forbidding

formula:

Rft

P EN( d2)e SN( d1)

where:

S stock price

N( ) cumulative normal density function

E exercise price

Rf risk-free rate, i.e., treasury rate of the same term as the option

t time remaining to expiration of the option

t 0.5]

d1 [ln(S/E) (Rf 0.5 variance) t]/[std dev

t0.5]

d2 d1 [std dev

PART 3 Adjusting for Control and Marketability

306

Chaffe™s Article: Put Options to Calculate DLOM of Restricted

Stock

David Chaffe (Chaffe 1973, p. 182) wrote a brilliant article in which he

reasoned that buying a hypothetical put option on Section 144 restricted

stock would ˜˜buy™™ marketability, and the cost of that put option is an

excellent measure of the discount for lack of marketability of restricted

stock.

Commentary to Table 8-2: Black“Scholes Calculation of DLOM

for ENCO, Inc.

Table 8-2 is the Black“Scholes put option calculation of the restricted stock

discount. We begin in row 5 with S, the stock price on the valuation date

of August 11, 1997, of $2.375. We then assume that E, the exercise price,

is identical (row 6).

Row 7 is the time in years from the valuation date to marketability.

According to SEC Rule 144, Robert Smith has a one-year period of re-

striction before he can sell all of his ENCO shares.

Row 8 shows the one year treasury bill rate as of August 11, 1997,

which was 5.32% (see note 1, Table 8-2 for the data source). Row 9 is the

square of row 10. Row 10 contains the annualized standard deviation of

ENCO™s continuously compounded returns, which we calculate in Table

8-2A to be 0.57.

Rows 11 and 12 are the calculation of the two Black“Scholes para-

meters, d1 and d2, the formulas of which appear in notes [2] and [3] of

Table 8-2. Rows 13 and 14 are the cumulative normal density functions

for d1 and d2.10 For example, look at cell B13, which is N( 0.380)

T A B L E 8-2

Black“Scholes Call and Put Options

A B

5 S stk price on valuation date $2.375

6 E exercise price $2.375

7 t time To expiration (yrs) 1.0

8 Rf risk-free rate [1] 5.32%

9 var variance 0.33

10 std dev standard deviation (Table 8-2A, C35) 0.57

11 d1 1st Black-Scholes Parameter [2] 0.380

12 d2 2nd Black-Scholes Parameter [3] (0.194)

13 N(-d1) cum normal density function 0.3521

14 N(-d2) cum normal density function 0.5771

[E * N(-d2)*e-Rft S * N(-d1) [4]

15 P $0.46

16 P/S 19.51%

[1] Source: 1 year secondary market Treasury Bill rate as of 8/11/97 from the Federal Reserve Bank of St. Louis, internet web site

http://www.stls.frb.org

[2] d1 [ ln (S/E) (Rf .5 * variance) * t ] / [ std dev * SQRT(t) ], where variance is expressed as an annual term.

[3] d2 d1 [ std dev * SQRT(t) ] , where std dev is expressed in annual terms.

[4] The put option formula can be found in Options Futures and Other Derivatives, 3rd Ed. by John C. Hull, Prentice Hall, 1997, pp.

241 and 242. The formula is for a European style put option.

10. We use d1 and d2 to calculate call option values and their negatives to calculate put option

values.

CHAPTER 8 Sample Restricted Stock Discount Study 307

0.3521. This requires some explanation. The cumulative normal table from

which the 0.3521 came assumes the normal distribution has been stan-

dardized to a mean of zero and standard deviation of 1.11 This means

that there is a 35.21% probability that our variable is less than or equal

to 0.380 standard deviations below the mean. In cell B14, N( d2)

N(0.194) 0.5771, which means there is a 57.71% probability of being

less than or equal to 0.194 standard deviations above the mean. For per-

spective, it is useful to note that since the normal distribution is sym-

metric, N(0) 0.5000, i.e., there is a 50% probability of being less than or

equal to the mean, which implies there is a 50% probability of being above

the mean.

In row 15 we calculate the value of the put option, which is $0.46

(B15) for the one-year option. In row 16 we calculate the ratio of the fair

market values of the put option to the stock price on the valuation date.

That ratio is our calculation of the restricted stock discount using Black“

Scholes. Thus, our calculation of the restricted stock discount is 19.51%

(B16) for the one-year period of restriction.

Commentary to Table 8-2A: Annualized Standard Deviation of

Continuously Compounded Returns

In Table 8-2A we calculate the annualized standard deviation of contin-

uously compounded returns for use in Table 8-2. Column A shows the

date, Column B shows the closing price, and Columns C and D show the

continuously compounded returns.

We calculated continuously compounded returns over 10 trading

days intervals for ENCO stock. In column C we start with the 1/23/97

closing price and in column D we start with the 1/30/97 closing price.

For example, the 10-trading-day return from 1/23/97 (A5) to 2/6/97 (A7)

is calculated as follows:

return Ln(B7/B5) Ln(3.75/4.25) 0.12516 (cell C7)

In cells C33 and D33 we get two measures of standard deviation of

0.09414 and 0.13500 respectively. To get the annualized standard deviation

we must multiply each interval standard deviation by the square root of

the number of intervals which would occur in a year. The equation is as

follows:

SQRT (# of interval returns in sample period

annualized interval returns

365 days/days in sample period)

For example, the sample period in column C is the time period from the

close of trading on 1/23/97 to the close of trading on 7/31/97 or 189

days, and the number of calculated returns is 13. Therefore the annualized

standard deviation of returns is:

0.09414 SQRT(13 365/189) 0.47169 (cell C34)

annualized

Similarly, the annualized standard deviation of returns in column D is

11. One standardizes a normal distribution by subtracting the mean from each value and dividing

by the standard deviation.

PART 3 Adjusting for Control and Marketability

308

T A B L E 8-2A

Standard Deviation of Continuously Compounded Returns

A B C D

4 Date Closing Price Interval Returns

5 1/23/97 4.2500

6 1/30/97 4.1250

7 2/6/97 3.7500 0.12516

8 2/13/97 3.6250 0.12921

9 2/21/97 3.2500 0.14310

10 2/28/97 3.8750 0.06669

11 3/7/97 3.7500 0.14310

12 3/14/97 3.3750 0.13815

13 3/21/97 3.2500 0.14310

14 3/31/97 2.8750 0.16034

15 4/7/97 2.7500 0.16705

16 4/14/97 2.7500 0.04445

17 4/21/97 2.7500 0.00000

18 4/28/97 2.1875 0.22884

19 5/5/97 2.7500 0.00000

20 5/12/97 2.6250 0.18232

21 5/19/97 2.3125 0.17327

22 5/27/97 2.0625 0.24116

23 6/3/97 2.0625 0.11441

24 6/10/97 2.2500 0.08701

25 6/17/97 2.1250 0.02985

26 6/24/97 2.3750 0.05407

27 7/2/97 2.0625 0.02985

28 7/10/97 2.1875 0.08224

29 7/17/97 1.9375 0.06252

30 7/24/97 2.1250 0.02899

31 7/31/97 1.9375 0.00000

32 8/7/97 2.3750 0.11123

33 Interval std deviation 0.09414 0.13500

34 Annualized std deviation 0.47169 0.67644

35 Average of 2 std deviations 0.57406

0.67644 (D34), while the average of the two is 0.57406 (C35), which trans-

fers to Table 8-2 cell B10.

The reason that we use 10-day intervals in our calculation instead of

daily intervals is that the bid ask spread on the stock may create apparent

volatility that is not really present. This is because the quoted closing

prices are from the last trade. In Nasdaq trading, when one sells to a

dealer it is at the bid price, but when one buys it is at the ask price. If

the last price of the day is switching randomly from a bid to an ask price

and vice versa, this can cause us to measure an apparent volatility that

is not really there. By using 10-day intervals, we reduce any measurement

effect caused by the spread.

Commentary to Table 8-3: Final Calculation of Discount

Table 8-3 is our ¬nal calculation of the restricted stock discount. We use

a weighted average of the two valuation approaches discussed earlier in

the report.

According to the multiple regression analysis in Table 8-1, cell C93,

the discount should be 21.41%. We show that in Table 8-3 in cell C6. In

CHAPTER 8 Sample Restricted Stock Discount Study 309

T A B L E 8-3

Final Calculation of Discount

A B C D E

4 Weighted

5 Method Source Table Discount Weight Discount

6 Multiple regression analysis 8-1, C93 21.41% 50% 10.7%

7 Black-Scholes put option 8-2, B16 19.51% 50% 9.8%

8 Total 100% 20.5%

10 Freely trading closing price, 8/11/97 [1] $ 2.375

11 Less discount for lack of marketability-20.5% $ (0.486)

12 Fair market value of restricted stock $ 1.889

13 Number of shares 500,000

14 FMV of restricted shares (rounded) $945,000

Source: America Online, Prophet Line.

C7 we show the Black“Scholes calculation of 19.51%, which we calculated

in Table 8-2, B16. We weight the two approaches equally, which results

in a discount of 20.5% (E8). The closing price of ENCO, Inc. common

stock on August 11, 1997, was $2.375 (E10) per share.12 The 20.5% discount

is $0.486 (E11) per share, leaving the fair market value of the restricted

stock on that date at $1.889 per share (E12). Multiplying that by 500,000

shares (E13), the fair market value of the ENCO stock received by Robert

Smith is $945,000 (E14).

Conclusion of Discount for Lack of Marketability

It is our opinion, subject to this report and the statement of limiting con-

ditions, that the proper discount to fair market value of the restricted

shares from the traded price of ENCO, Inc. stock on August 11, 1997, is