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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.

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85Skewness and kurtosis implied by stock option prices