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Implied volatility skews and stock return
skewness and kurtosis implied by stock
option prices
C. J. CORRADO and TIE SU1
Department of Finance, 214 Middlebush Hall, University of Missouri, Columbia,
MO 65211, USA and 1Department of Finance, 514 Jenkins Building, University of
Miami, Coral Gables, FL 33124, USA
The Black–Scholes* option pricing model is commonly applied to value a wide range of
option contracts. However, the model often inconsistently prices deep in-the-money
and deep out-of-the-money options. Options professionals refer to this well-known
phenomenon as a volatility ‘skew’ or ‘smile’. In this paper, we examine an extension of
the Black–Scholes model developed by Corrado and Su‡ that suggests skewness and
kurtosis in the option-implied distributions of stock returns as the source of volatility
skews. Adapting their methodology, we estimate option-implied coefficients of skewness
and kurtosis for four actively traded stock options. We find significantly nonnormal
skewness and kurtosis in the option-implied distributions of stock returns.
Keywords: Stock options, implied volatility, skewness, kurtosis
1. INTRODUCTION
The Black–Scholes (1973) option pricing model is commonly applied to value a
wide range of option contracts. Despite this widespread acceptance among
practitioners and academics, however, the model has the known deficiency of
often inconsistently pricing deep in-the-money and deep out-of-the-money
options. Options professionals refer to this well-known phenomenon as a
volatility ‘skew’ or ‘smile’. A volatility skew is the pattern that results from
calculating implied volatilities across the range of strike prices spanning a given
option class. Typically, the skew pattern is systematically related to the degree
to which the options are in- or out-of-the-money. This phenomenon is not
predicted by the Black–Scholes model, since, theoretically, volatility is a
property of the underlying instrument and the same implied volatility value
should be observed across all options on the same instrument.
The Black–Scholes model assumes that stock log-prices are normally dis-
tributed over any finite time interval. Hull (1993) and Nattenburg (1994) point
* Black and Scholes (1973)
‡ Corrado and Su (1996)
1351–847X © 1997 Chapman & Hall
The European Journal of Finance 3, 73–85 (1997)
out that stock returns exhibit nonnormal skewness and kurtosis and that
volatility skews are a consequence of empirical violations of the normality
assumption. In this paper, we examine the empirical distribution of stock
returns implied by option prices and the resulting volatility skews. We use the
method suggested by Corrado and Su (1996) to extend the Black–Scholes
formula to account for nonnormal skewness and kurtosis in stock return
distributions. This method is based on fitting the first four moments of a
distribution to a pattern of empirically observed option prices. The mean of this
distribution is determined by option pricing theory, but an estimation pro-
cedure yields implied values for the variance, skewness and kurtosis of the
distribution of stock returns.
The paper is organized as follows. In the next section, we show how
nonnormal skewness and kurtosis in stock return distributions give rise to
volatility skews. This includes a review of Corrado and Su’s (1996) development
of a skewness- and kurtosis-adjusted Black–Scholes option price formula. We
then describe the data sources used in our empirical analysis. In the subsequent
empirical section, we compare the performance of the Black–Scholes model
with that of a skewness- and kurtosis-adjusted extension to the Black–Scholes
model. The final section summarizes and concludes the paper.
2. DERIVATION OF A SKEWNESS- AND KURTOSIS-ADJUSTED BLACK–
SCHOLES MODEL
Corrado and Su (1996) develop a method to incorporate option price adjust-
ments for nonnormal skewness and kurtosis in an expanded Black–Scholes
option pricing formula. Their method adapts a Gram–Charlier series expansion
of the standard normal density function to yield an option price formula that is
the sum of a Black–Scholes option price plus adjustment terms for nonnormal
skewness and kurtosis. Specifically, the density function g(z) defined below
accounts for nonnormal skewness and kurtosis, denoted by m3and m4,
respectively, where n(z) represents the standard normal density function.
g(z)=n(z)
3
1+m
3
3!(z3–3z)+m
4–3
4! (z4–6z
2+3)
4(1)
where
z=ln(St/S0)–(r–s
2
/2)t
sÎt
and
S0is a current stock price;
Stis a random stock price at time t;
ris the risk free rate of interest;
tis the time remaining until option expiration, and
sis the standard deviation of returns for the underlying stock.
74 Corrado and Su
Notice that skewness m3and kurtosis m4for the density g(z) are explicit
parameters in its functional form. Under a normal specification, these co-
efficients are m3= 0 and m4= 3 (Stuart and Ord, 1987: 222–3).
Applying the density g(z) in equation (1) to derive a theoretical call price as
the present value of an expected payoff at option expiration yields the following
option price expression, where z(St) = (logSt– m)/sÎt, m= logS0+ (r–s2/2)tand
Kis the option’s strike price.
CGC =e
–rt e
∞
K(St–K)g(z(St))dz(St) (2)
Evaluating this integral yields the following formula for an option price based on
a Gram–Charlier density expansion, here denoted by CGC.
CGC =CBS +m3Q3+(m
4–3)Q
4(3)
where CBS =S0N(d)–Ke–rtN(d–sÎt) is the Black–Scholes option price formula,
and
Q3=1
3!S0sÎt((2sÎt–d)n(d)–s
2
tN(d))
Q4=1
4!S0sÎt((d2–1–3sÎt(d–sÎt))n(d)+s
3
t
3/2N(d))
d=ln(S0/K)+(r+s
2
/2)t
sÎt
In equation (3) above, the terms m3Q3and (m4–3)Q
4measure the effects of
nonnormal skewness and kurtosis on the option price CGC.
Nonnormal skewness and kurtosis give rise to implied volatility skews. To
illustrate these effects, option prices are generated according to equation (3)
based on parameter values m3= –0.5, m4= 4, S0= 50, s= 30%, t= 3 months,
r= 4% and strike prices ranging from 35 to 65. Implied volatilities are then
calculated for each skewness and kurtosis impacted option price using the
Black–Scholes formula. The resulting volatility skew is plotted in Fig. 1, where
the horizontal axis measures strike prices and the vertical axis measures
implied standard deviation values. While the true volatility value is s= 30%, Fig.
1 reveals that implied volatility is greater than true volatility for deep out-of-the-
money options, but less than true volatility for deep in-the-money options.
Figure 2 contains an empirical volatility skew obtained from Telephonos de
Mexico (TMX) call option price quotes recorded on 2 December 1993 for options
expiring in May 1994. In Fig. 2, the horizontal axis measures option moneyness
as the percentage difference between a discounted strike price and a dividend-
adjusted stock price level. Negative (positive) moneyness corresponds to in-the-
money (out-of-the-money) options with low (high) strike prices. The vertical
axis measures implied standard deviation values. Each solid black marker
represents an implied volatility calculated using the Black–Scholes formula.
75Skewness and kurtosis implied by stock option prices
+
Each hollow marker represents an implied volatility calculated from a skewness-
and kurtosis-adjusted option price. The actual number of price quotes on this
day was 217, but because many quotes are unchanged updates made through-
out the day the number of visually distinguishable dots is smaller than the
actual number of quotes.
Figure 2 reveals that Black–Scholes implied volatilities range from about 36%
for deep in-the-money options (negative moneyness) to about 29% for deep out-
of-the-money options (positive moneyness). By contrast, the skewness- and
kurtosis-adjusted prices yield essentially the same implied volatility of about
33% regardless of option moneyness. Comparing Fig. 2 with Fig. 1 reveals that
the implied volatility skew for TMX options is consistent with negative skewness
and positive excess kurtosis in the distribution of TMX stock returns.* In the
empirical results section of this paper, we examine the economic impact of
these volatility skews.
* Excess kurtosis is defined as (m423), which is the difference between actual kurtosis of m4and
normal distribution kurtosis of 3.
Fig. 1. Implied volatility skew
Fig. 2. Implied volatilities for Telephonos de Mexico (TMX)
76 Corrado and Su
3. DATA SOURCES
We base this study on the Chicago Board Options Exchange (CBOE) market for
four actively traded stock option contracts: International Business Machines
(IBM), Paramount Communications Inc. (PCI), Micron Technology (MU) and
Telephonos de Mexico (TMX). Intraday price data come from the Berkeley
Options Data Base of CBOE options trading. Stock prices, strike prices and
option maturities also come from the Berkeley database. To avoid bid–ask
bounce problems in transaction prices, we take option prices as midpoints of
CBOE dealers’ bid–ask price quotations. The risk-free rate of interest is taken as
the US Treasury bill rate for a bill maturing closest to option contract
expiration. Interest rate information is culled from the Wall Street Journal.
CBOE stock options are American style and may be exercised anytime before
expiration. To justify the Black–Scholes formula for American-style options, our
data sample includes only call options for which either (1) no cash dividend was
paid during the life of the option, or (2) if a dividend was paid, it was so small
that early exercise was never optimal. The first condition is embedded in the
second, since both conditions are summarized by the following inequality:
D,K(1–e
–rt) (4)
where Dis a dividend payment, Kis the strike price, ris the risk-free rate on the
ex-dividend date and tis the length of time between the ex-dividend date and
option expiration. Merton (1973) shows that an American-style option is never
optimally exercised before expiration where this inequality holds. When a
dividend payment is made, however, we use the method suggested by Black
(1975) and adjust the stock price by subtracting the present value of the
dividend. Stock dividend information is extracted from the Daily Stock Price
Record published by Standard and Poor’s Corporation.
Following data screening procedures in Barone-Adesi and Whaley (1986), we
delete all option prices less than $0.125 and all transactions occurring before
9:00 a.m. Obvious outliers are also purged from the sample; including recorded
option prices lying outside well-known no-arbitrage option price boundaries
(Merton, 1973).
4. EMPIRICAL RESULTS
In this section, we first assess out-of-sample performance of the Black–Scholes
option pricing model. Specifically, we estimate implied standard deviations on a
daily basis for call options on each of four underlying stocks, where on the day
prior to a given current day we obtain a unique implied standard deviation from
all bid-ask price midpoints for a given option maturity class using Whaley’s
(1982) simultaneous equations procedure. This prior-day out-of-sample implied
standard deviation becomes an input used to calculate current-day theoretical
Black–Scholes option prices for all price observations within the same maturity
class. We then compare these theoretical Black–Scholes prices with their
corresponding market-observed prices.
Next, we assess the out-of-sample performance of the skewness- and kurtosis-
adjusted Black–Scholes option pricing formula developed in Corrado and Su
77Skewness and kurtosis implied by stock option prices
(1986). Following their methodology, on the day prior to a given current day we
simultaneously estimate implied standard deviation (ISD), implied skewness
(ISK) and implied kurtosis (IKT) parameters using all bid–ask midpoints for a
given option maturity class. These prior-day out-of-sample parameter estimates
provide inputs used to calculate current-day theoretical option prices for all
options within the same maturity class. We then compare theoretical skewness-
and kurtosis-adjusted Black–Scholes option prices with their corresponding
market-observed prices.
4.1 The Black–Scholes option pricing model
The Black–Scholes formula specifies five inputs: a stock price, a strike price, a
risk-free interest rate, an option maturity and a return standard deviation. The
first four inputs are directly observable market data. The return standard
deviation is not directly observable. We estimate return standard deviations
from values implied by options using Whaley’s (1982) simultaneous equations
procedure. This procedure yields a value for the argument BSISD that minimizes
the following sum of squares.
min
BSID O
N
j=1 FCOBS.j–CBS.j(BSISD)G2(5)
In equation (5) above, Ndenotes the number of price quotations available on a
given day for a given maturity class, COBS represents a market-observed call
price, and CBS (BSISD) specifies a theoretical Black–Scholes call price based on
the parameter BSISD. Using prior-day values of BSISD, we calculate theoretical
Black–Scholes option prices for all options in a current-day sample within the
same maturity class. We then compare these theoretical Black–Scholes option
prices with their corresponding market-observed prices.
Table 1 summarizes calculations for Telephonos de Mexico (TMX) call option
prices observed during December 1993 for options maturing in May 1994. To
maintain table compactness, column 1 lists only even-numbered dates within
the month and column 2 lists the number of price quotations available on each
of these dates. Black–Scholes implied standard deviations (BSISD) for each date
are reported in column 3. To assess the economic significance of differences
between theoretical and observed prices, column 6 lists the proportion of
theoretical Black–Scholes option prices lying outside their corresponding bid–
ask spreads, either below the bid price or above the asked price. In addition,
column 7 lists the average absolute deviation of theoretical prices from bid–ask
boundaries for only those prices lying outside their bid–ask spreads. Specific-
ally, for each theoretical option price lying outside its corresponding bid–ask
spread, we calculate an absolute deviation according to the following formula.
max(CBS(BSISD)–Ask, Bid–CBS(BSISD))
This absolute deviation statistic is a measure of the economic significance of
deviations of theoretical option prices from observed bid–ask spreads. Finally,
column 4 lists day-by-day averages of observed call prices and column 5 lists
day-by-day averages of observed bid-ask spreads.
78 Corrado and Su
In Table 1, the bottom row lists column averages for all variables. For
example, the average number of daily price observations is 225 (column 2), with
an average option price of $9.96 (column 4) and an average bid–ask spread of
$0.24 (column 5). The average implied standard deviation is 28.87% (column 3).
Regarding the ability of the Black–Scholes model to describe observed option
prices, the average proportion of theoretical Black–Scholes prices lying outside
their corresponding bid–ask spreads is 78% (column 6), with an average
deviation of $0.12 (column 7) for those observations lying outside a spread
boundary.
The average price deviation of $0.12 for observations lying outside a spread
boundary is equivalent to about a one-eighth price tick. While informative,
an overall average deviation understates the pricing problem since price
deviations are larger for deep in-the-money and deep out-of-the-money options.
For example, Table 1 shows that the Black–Scholes implied standard deviation
(BSISD) value for TMX options on 2 December was 30.80%, while Fig. 2 reveals
that Black–Scholes implied volatilities for TMX range from about 36% for deep
in-the-money options to about 29% for deep out-of-the-money options. Based on
2 December TMX input values, i.e. S= 57, r= 3.2%, T= 169 days, a deep in-the-
money option with a strike price of 45 yields call prices of $13.63 and $13.25,
respectively, from volatility values of 36% and 30.8%. Similarly, a deep out-of-the-
money option with a strike price of 65 yields call prices of $2.04 and $2.29,
respectively, from volatility values of 29% and 30.8%. Since a standard stock
option contract size is 100 shares, these prices correspond to contract price
Table 1. Comparison of Black–Scholes prices and observed prices of Telephonos de
Mexico (TMX) options
Date
Number
of price
observations
Implied
standard
deviation
(%)
Average
call price
($)
Average
bid–ask
spread
($)
Proportion of
theoretical
prices outside
the bid–ask
spread
Average
deviation of
theoretical prices
from spread
boundaries ($)
2/12/93 217 30.80 6.77 0.23 0.76 0.15
6/12/93 171 30.45 10.30 0.24 0.71 0.20
8/12/93 366 28.66 8.89 0.23 0.88 0.13
10/12/93 266 28.76 9.23 0.25 0.76 0.13
14/12/93 149 28.79 8.73 0.23 0.83 0.13
16/12/93 189 28.81 9.60 0.25 0.81 0.12
20/12/93 185 28.18 10.09 0.23 0.77 0.09
22/12/93 184 28.45 11.89 0.24 0.68 0.11
28/12/93 208 27.78 11.25 0.24 0.81 0.09
30/12/93 315 28.01 12.84 0.24 0.75 0.08
Average 225 28.87 9.96 0.24 0.78 0.12
On each day indicated, a Black–Scholes implied standard deviation (BSISD) is estimated from current price
observations. Theoretical Black–Scholes option prices are then calculated using BSISD. All observations
correspond to call options traded in December 1993 and expiring in May 1994.
79Skewness and kurtosis implied by stock option prices
deviations of $38 for deep in-the-money options and $25 for deep out-of-the-
money options.
Price deviations of the magnitude described above indicate that CBOE market
makers quote deep in-the-money (out-of-the-money) call option prices at a
premium (discount) compared to prices that can be rationalized by the Black–
Scholes formula. Nevertheless, the Black–Scholes formula does provide a first
approximation to deep in-the-money or deep out-of-the-money option prices.
Immediately below, we examine the improvement in pricing accuracy obtainable
by adding skewness- and kurtosis-adjustment terms to the Black–Scholes
formula.
4.2 Skewness- and kurtosis-adjusted Black–Scholes model
In the second set of estimation procedures, on a given day within a given option
maturity class we simultaneously estimate return standard deviation, skewness
and kurtosis parameters by minimizing the following sum of squares with
respect to the arguments ISD, ISK and IKT, respectively.
min
ISD,ISK,IKT O
N
j=1 FCOBS.j–(C
BS.j(ISD)+ISKQ3+(IKT–3)Q4)G2(6)
The resulting values for ISD, ISK and IKT represent estimates of implied standard
deviation, implied skewness and implied kurtosis parameters based on Nprice
observations. Substituting ISD, ISK and IKT estimates into equation (3) yields
the following skewness- and kurtosis-adjusted Black–Scholes option price:
CGC =CBS(ISD)+ISKQ3+(IKT –3)Q
4(7)
Equation (7) yields theoretical skewness- and kurtosis-adjusted Black–Scholes
option prices from which we calculate deviations of theoretical prices from
market-observed prices.
Table 2 summarizes calculations for the same Telephonos de Mexico (TMX)
call option prices used to compile Table 1. Consequently, column 1 in Table 2
lists the same even-numbered dates and column 2 lists the same number of
price quotations listed in Table 1. To assess out-of-sample forecasting power of
skewness and kurtosis adjustments, the simultaneously estimated implied
standard deviations (ISD), implied skewness coefficients (ISK) and implied
kurtosis coefficients (IKT) are all estimated from prices observed on trading
days immediately prior to dates listed in column 1. For example, the first row of
Table 2 lists the date 2 December 1993, but columns 3, 4 and 5 report standard
deviation, skewness and kurtosis values obtained from 1 December prices.
Thus, out-of-sample parameters ISD, ISK and IKT reported in columns 3, 4 and 5,
respectively, correspond to one-day lagged estimates. We use these one-day
lagged values of ISD, ISK and IKT to calculate theoretical skewness- and kurtosis-
adjusted Black–Scholes option prices according to equation (7) for all price
observations on the even-numbered dates listed in column 1. In turn, these
theoretical prices based on out-of-sample ISD, ISK and IKT values are then used
to calculate daily proportions of theoretical prices outside bid–ask spreads
(column 6) and daily averages of deviations from spread boundaries (column 7).
80 Corrado and Su
Like Table 1, column averages for Table 2 are reported in the bottom row of the
table.
As shown in Table 2, all skewness coefficients in column 4 are negative, with
a column average of –0.55. All kurtosis coefficients in column 5 are greater than
3, with a column average of 4.92. By comparison, normal distribution skewness
and kurtosis values are 0 and 3, respectively. Column 6 of table 2 lists the
proportion of skewness- and kurtosis-adjusted prices lying outside their
corresponding bid–ask spread boundaries. The column average proportion is
19%. Column 7 lists average absolute deviations of theoretical prices from bid–
ask spread boundaries for only those prices lying outside their bid–ask spreads.
The column average price deviation is $0.05, which is about one-fifth the size of
the average bid–ask spread of $0.24 reported in Table 1. Moreover, Fig. 2 reveals
that implied volatilities from skewness- and kurtosis-adjusted option prices
(hollow markers) are unrelated to option moneyness. In turn, this implies that
the corresponding price deviations are also unrelated to option moneyness.
Overall, we conclude that skewness- and kurtosis-adjustment terms added to
the Black–Scholes formula yield significantly improved pricing accuracy for
deep in-the-money or deep out-of-the-money stock options. Furthermore, these
improvements are obtained from out-of-sample estimates of skewness and
kurtosis. There is an added cost, however, in that two additional parameters
must be estimated. But the added cost is a fixed startup cost, since once the
computer code is in place the added computation time is trivial on modern
personal computers.
Table 2. Comparison of skewness- and kurtosis-adjusted Black–Scholes prices and
observed prices of Telephonos de Mexico (TMX) options
Date
Number
of price
observations
Implied
standard
deviation
(%)
Implied
skewness
(ISK)
Implied
kurtosis
(IKT)
Proportion of
theoretical
prices outside
the bid–ask
spread
Average
deviation of
theoretical prices
from spread
boundaries ($)
2/12/93 217 32.99 –0.71 4.41 0.08 0.07
6/12/93 171 30.59 –1.00 4.07 0.54 0.10
8/12/93 366 30.40 –0.66 4.68 0.15 0.04
10/12/93 266 30.62 –0.57 4.96 0.14 0.03
14/12/93 149 30.95 –0.60 5.15 0.14 0.04
16/12/93 189 30.87 –0.51 5.28 0.16 0.04
20/12/93 185 30.37 –0.32 5.38 0.17 0.03
22/12/93 184 28.89 –0.64 4.29 0.18 0.07
28/12/93 208 29.85 –0.28 5.55 0.16 0.03
30/12/93 315 29.94 –0.20 5.46 0.14 0.03
Average 225 30.55 –0.55 4.92 0.19 0.05
On each day indicated, implied standard deviation (ISD), skewness (ISK), and kurtosis (IKT) parameters are
estimated from one-day lagged price observations. Theoretical option prices are then calculated using these
out-of-sample implied parameters. All observations correspond to call options traded in December 1993
and expiring in May 1994.
81Skewness and kurtosis implied by stock option prices
4.3 Higher order moment estimates
It might appear that price adjustments beyond skewness and kurtosis could add
further improvements to the procedures specified above. For example, higher
order analogues to the terms Q3and Q4, say Q5and Q6, could be used to
augment the estimation procedure specified in equation (6). Unfortunately,
including additional terms creates severe collinearity problems since all even-
numbered subscripted terms, e.g. Q4and Q6, are highly correlated with each
other. Similarly, all odd-numbered subscripted terms, e.g. Q3and Q5, are also
highly correlated. Consequently, adding higher order terms leads to severely
unstable parameter estimates.
4.4 Further empirical results
We also applied all procedures leading to Tables 1 and 2 to options data for
three other stocks: International Business Machines (IBM), Micron Technology
(MU) and Paramount Communications Inc. (PCI). Table 3 summarizes results
obtained from option price data for these three stocks by reporting monthly
averages for all variables reported in Tables 1 and 2. Specifically, panel A in
Table 3 summarizes results obtained from the Black–Scholes formula. Similarly,
panel B in Table 3 summarizes results obtained from the skewness- and kurtosis-
Table 3. Comparison of theoretical option prices and observed option prices
Panel A: Black–Scholes model
Ticker
Number
of price
observations
Implied
standard
deviation
(%)
Average
call price
($)
Average
bid–ask
spread
($)
Proportion of
theoretical
prices outside
the bid–ask
spread
Average
deviation of
theoretical prices
from spread
boundaries ($)
IBM 276 30.89 6.13 0.29 0.57 0.11
MU 342 59.32 7.42 0.24 0.54 0.09
PCI 114 24.11 6.41 0.32 0.76 0.23
Panel B: Skewness- and kurtosis-adjusted model
Date
Number
of price
observations
Implied
standard
deviation
(%)
Implied
skewness
(ISK)
Implied
kurtosis
(IKT)
Proportion of
theoretical
prices outside
the bid–ask
spread
Average
deviation of
theoretical prices
from spread
boundaries ($)
IBM 276 32.39 –0.52 4.14 0.35 0.06
MU 342 61.58 –0.32 3.44 0.37 0.06
PCI 114 26.07 –1.09 5.27 0.48 0.13
Monthly averages of Black–Scholes implied standard deviation (BSISD), and implied standard deviation
(ISD), skewness (ISK), and kurtosis (IKT) parameters estimated daily from intraday price observations.
Theoretical option prices are calculated using out-of-sample implied parameters.
82 Corrado and Su
adjusted Black–Scholes formula. Each row in Table 3 reports monthly averages
for each stock.
Empirical results reported in Table 3 are qualitatively similar to results
reported in Tables 1 and 2. In particular, option-implied estimates of skewness
are all negative, ranging from –0.32 to –1.09, and estimates of kurtosis are all
greater than 3, ranging from 3.44 to 5.27. Table 3 also reports that adjustments
for skewness and kurtosis reduce substantially the proportions of theoretical
option prices lying outside observed bid–ask spreads and the average devia-
tions of theoretical prices from bid–ask spread boundaries.
Figures 3, 4 and 5 contain volatility skews for International Business Machines
(IBM), Micron Technology (MU) and Paramount Communications Inc. (PCI),
respectively. In all figures, horizontal axes measure option moneyness where
negative (positive) moneyness corresponds to in-the-money (out-of-the-money)
options with low (high) strike prices and vertical axes measure implied
Fig. 3. Implied volatilities for International Business Machines (IBM)
Fig. 4. Implied volatilities for Micron Technology (MU)
83Skewness and kurtosis implied by stock option prices
standard deviation values. Solid black markers represent Black–Scholes implied
volatilities and hollow markers represent implied volatilities calculated from
skewness- and kurtosis-adjusted option prices. Figures 3, 4 and 5 all reveal that
CBOE market makers quote deep in-the-money (out-of-the-money) call option
prices at a premium (discount) to Black–Scholes formula prices. Moreover,
implied volatilities from skewness- and kurtosis-adjusted option prices (hollow
markers) are unrelated to option moneyness, implying that corresponding price
deviations are also unrelated to option moneyness.
5. SUMMARY AND CONCLUSIONS
We have empirically tested an expanded version of the Black–Scholes (1973)
option pricing model suggested by Corrado and Su (1996) that accounts for
skewness and kurtosis deviations from normality in stock return distributions.
The expanded model was applied to estimate coefficients of skewness and
kurtosis implied by stock option prices. Relative to a normal distribution, we
found significant negative skewness and positive excess kurtosis in the
distributions of four actively traded stock prices. In summary, we conclude that
skewness- and kurtosis-adjustment terms added to the Black–Scholes formula
yield significantly improved accuracy for pricing deep in-the-money or deep out-
of-the-money stock options.
REFERENCES
Barone-Adesi, G. and Whaley, R.E (1986) The valuation of American call options and the
expected ex-dividend stock price decline, J. Financial Economics, 17, 91–111.
Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities, J.
Political Economy, 81, 637–59.
Black, F. (1975) Fact and fantasy in the use of options, Financial Analysts Journal, 31,
36–72.
Fig. 5. Implied volatilities for Paramount Communications Inc. (PCI)
84 Corrado and Su
Corrado, C.J. and Su, T. (1996) Skewness and kurtosis in S&P 500 Index returns implied
by option prices, J. Financial Research, 19, 175–92.
Hull, J.C. (1993) Options, Futures, and Other Derivative Securities. Englewood Cliffs, NJ.
Prentice Hall.
Merton, R.C. (1973) The theory of rational option pricing, Bell Journal of Economics and
Management Science, 4, 141–83.
Nattenburg, S. (1994) Option Volatility and Pricing, Chicago: Probus Publishing.
Stuart, A. and Ord, J.K. (1987) Kendall’s Advanced Theory of Statistics, New York: Oxford
University Press.
Whaley, R.E. (1982) Valuation of American call options on dividend paying stocks, J.
Financial Economics, 10, 29–58.
85Skewness and kurtosis implied by stock option prices
