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viations on either side includes 95% of the entire population.
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

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