Advances in Futures and Options Researchvolume 9 (1997), pp. 51-82.
The Skewness Premium:Option Pricing Under Asymmetric Processes
David S. Bates*
Finance DepartmentThe Wharton School
University of PennsylvaniaPhiladelphia, PA 19104-6367
March 1996
Abstract
This paper develops distribution-specific theoretical constraints on relative prices of out-of-the-moneyEuropean call and put options that are also valid for American options on futures. The constraints canbe used to identify which distributional hypotheses can and cannot explain observed moneynessbiases. An application to S&P 500 futures options over 1983-1993 illustrates the inability of standarddistributional hypotheses to explain observed moneyness biases -- especially those observed followingthe stock market crash of 1987. Some distribution-specific extensions to exotic option pricing arealso discussed.
*The Wharton School, University of Pennsylvania and National Bureau of Economic Research.
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Option pricing models such as Black and Scholes (1973) premised upon the underlying asset price following geometric
Brownian motion have been found to exhibit serious specification error when fitted to market data. A "moneyness"
or "striking price" bias is ubiquitous: model prices as a (monotone) function of the strike price are typically "tilted"
relative to observed options prices, so that the residuals for low strike prices tend to be opposite in sign from the
residuals for high strike prices. Such biases have been found in American call and put options on stocks traded on the
Chicago Board Options Exchange; in American calls and puts on S&P 500 futures traded on the Chicago Mercantile1
Exchange; in American foreign currency call options traded on the Philadelphia Stock Exchange; and in European2 3
Swiss franc-denominated call options on the dollar traded in Geneva. The biases are not in the same direction for all4
markets, nor are they constant over time. The biases have been especially egregious in stock and stock index options
since the stock market crash of 1987.
Observed moneyness biases and considerable time series evidence against the hypothesis that log-differenced
asset prices are homoskedastic and normally distributed have spurred the development of option pricing models for
alternative stochastic processes. The constant elasticity of variance (CEV) option pricing model of Cox and Ross
(1976) and Cox and Rubinstein (1985) relaxed the constant volatility assumption and allowed the instantaneous
conditional volatility of asset returns to depend deterministically upon the level of the asset price. Special cases include
both arithmetic and geometric Brownian motion. The recent “implied binomial tree” models of &WRKTG� ������
>OCP�CPF�-CPK��������CPF�4WDKPUVGKP�������can be viewed as flexible generalizations of the CEV model. The
stochastic volatility option pricing models of Hull and White (1987), Scott (1987), and Wiggins (1987) permitted more
general patterns of evolution for conditional volatility, parallelling the development of time series models such as
ARCH and GARCH that were concerned with the same issue. The recognition that asset returns are leptokurtic,
particularly for short holding periods, motivated Merton's (1976) model for pricing options on jump-diffusion
2
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processes. Finally, the recognition that options involve future payoffs has spurred examination of the impact of5
stochastic discount rates.
Which of these alternative factors and associated option pricing models can generate the moneyness biases
observed empirically? This question has motivated "horse race" papers such as Rubinstein (1985) and Shastri and
Wethyavivorn (1987), and is a relevant criterion not just of option pricing models but of time series models as well.
The central thesis of this paper is that most postulated processes can be ruled out a priori as inconsistent with observed
moneyness biases. Those biases constitute prima facie evidence that the distributions implicit in option prices are
markedly skewed relative to the benchmark lognormal distribution implied by geometric Brownian motion. By6
contrast, the above option pricing models under the parameter values typically used generate implicit distributions that
are essentially symmetric. Parameter values that are potentially consistent with observed moneyness biases are seldom
studied, and in fact are often ruled out to increase tractability. For instance, although Merton’s (1976) jump-diffusion
model allowed jumps to have non-zero mean, empirical tests by Ball and Torous (1983, 1985) assumed mean-zero
jumps -- precisely the parameterization incapable of eliminating observed moneyness biases. Stochastic volatility
models typically focus on the special case in which volatility evolves independently of the asset price -- a case equally
incapable of explaining moneyness biases.
The theoretical foundations for this central thesis rest on a method of measuring the moneyness bias that offers
valuable insights into which option pricing models can and cannot explain observed biases. The method is based upon
comparing the prices of out-of-the-money (OTM) calls and puts. Intuitively, since OTM calls pay out only if the asset
price rises above the call’s exercise price, while OTM puts pay off only if the asset price falls below the put’s exercise
price, call and put prices directly reflect the characteristics of the upper and lower tails of the distribution. Comparing
the two for OTM options is consequently a direct gauge of the relative thicknesses of the tails, and consequently a
3
measure of implicit skewness. More precisely, relative prices of OTM options will reflect the skewness of the "risk-
neutral" distribution generated from any given distributional hypothesis, and can therefore be used as a direct and easy
diagnostic of that underlying hypothesis.
Section I of the paper quantifies the relationship between distributional hypotheses and the relative prices of
European out-of-the-money calls and puts for the major families of option pricing models listed above. For the constant
elasticity of variance processes, OTM call and put prices are roughly equal under arithmetic Brownian motion and
diverge increasingly as one moves through intermediate CEV processes towards geometric Brownian motion. For the
last process, an "x% rule" holds: call options x% out-of-the-money are priced exactly x% higher than the corresponding
OTM puts. This "x% rule" also holds for the benchmark models in the other classes of option pricing models:
stochastic volatility models in which volatility evolves independently of the asset price, jump-diffusion models with
mean-zero jumps, and stochastic interest rates/dividend yields that follow Ornstein-Uhlenbeck processes. Equivalently,
the implicit volatility patterns are perfectly symmetric functions of the log of the strike price under these benchmark
models.
Section II extends the results to American options. Although the above results for European options will hold
only approximately for American options in general, most of the results will hold exactly in the case of American
options on futures. Furthermore, distribution-specific theoretical relationships between the immediate-exercise critical
values for American calls and puts on futures can be derived. These results follow from the key insight that the put
valuation problem is a distribution-specific transformation of variables of the call valuation problem for American
options on futures. Based on these results, the "skewness premium" metric is defined as the percentage deviation of
OTM call prices from the prices of puts correspondingly out-of-the-money. Section III applies the metric to American
options on S&P 500 futures options traded on the Chicago Mercantile Exchange over 1983-1993, illustrating the
substantial and persistent negative implicit skewness present since the crash of 1987.
4
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Many of the results in the first two sections have been reported previously, without proof. This article7
provides the proofs, and some extensions. These proofs are of interest not only for confirming the validity of previous
claims, but also for the insights they provide, and for potential generalizations to other contingent claims. For instance,
the key insight that the put valuation problem is a distribution-specific transformation of variables of the call valuation
problem for European options in general and for American options on futures is largely unprecedented in the option
pricing literature. Some applications to exotic option pricing are given in the concluding Section IV.
Ft, t%T ' St e bT .
5
(1)
I. Asymmetry and European Option Pricing
A. Assumptions and Notation
Throughout this paper, the following standard assumptions will typically be maintained:
A1) Markets are frictionless: there are no transactions costs or differential taxes, trading can take placecontinuously, there are no restrictions on borrowing or selling short.
A2) There exists a "risk-neutral" probability measure under which options can be priced at their expecteddiscounted future payoff.
A3) The instantaneous risk-free interest rate r is known and constant.
A4) The "cost of carry" to maintaining a position in the underlying asset is a constant proportion b of theasset price.
Standard specifications of the "risk-neutral" probability measure will be examined below. Specific extensions of the
results to stochastic interest rates and cost of carry will also be examined.
The following notation will be used:
S (or S) = the current price on the underlying asset;t
F (or F) = the forward price on the asset for delivery T periods from now;t,t+T
T = the time to maturity of the option;
X = the strike price of the option;
c(S ) [C(S )] = European [American] call option price conditional upon current information S ;t t t
p(S ) [P(S )] = European [American] put option price conditional upon current information S .t t t
When the cost of carry is constant, the forward and spot prices are related by
Without loss of generality, option prices will be specified in terms of the forward price rather than the spot price. Call
options will be referred to as out-of-the-money (OTM) if the strike price X exceeds the forward price (X > F), and in-
the-money (ITM) if the reverse is true. OTM put options will be those for which X < F, while ITM puts are those for
which X > F.
c (St ) ' e &rT E ( [ max(St%T & X, 0 ) * St ]
' e &rT E ( [ St%T & X * St%T > X ; St ] Prob ( [St%T > X * St ]
p (St ) ' e &rT E ( [ max( X & St%T , 0 ) * St ]
' e &rT E ( [X & St%T * St%T < X ; St ] Prob ( [St%T < X * St ] ,
6
(2)
(3)
In general, only a subset of the information set S will be highlighted: the current underlying forward assett
price F and other state variables, the time to maturity T, and the strike price X. For instance, American call options will
have prices written as C = C(S ) = C(F, T; X). Model-specific parameters will be included in the list of argumentst t
when option pricing formulas depend in interesting fashions upon those parameters.
I.B Symmetric and log-symmetric processes
When interest rates are constant, European call and put prices are given by
where E* is the expectation under the risk-neutral probability measure. The risk-neutral expected future spot price
under nonstochastic interest rates is the forward price on the asset: '*(S |S ) = F = S e . t+T t t,t+T tbT
A key insight from (2) and (3) is that calls and puts as functions of the exercise price X are symmetric
contingent claims. Call prices reflect conditions in the upper tail of the risk-neutral distribution, whereas put prices
reflect conditions in the lower tail. If the strike prices of the put and call are spaced symmetrically around the mean
E*(S |S ) = F , symmetries or asymmetries in the risk-neutral distribution will be directly reflected in the relativet+T t t,t+T
prices of these out-of-the-money (OTM) calls and puts. Since different distributional hypotheses typically imply
different risk-neutral distributions of S , options data can be used to distinguish amongst those hypotheses. The majort+T
theoretical contribution of this paper is to prove theoretical propositions that quantify the relationship between OTM
calls and puts for different distributional hypotheses, thereby providing a metric for easily judging which hypotheses
are consistent with observed option prices.
c (F, T ; F % x ) ' p (F, T ; F & x) for *x* < F.
St%T
R ' G &T6' ( =:R& 5
V%6* 5
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ª 2TQD ( =5V%6� : ?
c ' e &rT E ([St%T & Xc * St%T > Xc ]
× Prob ([St%T > Xc ]
Prob ((St%T < Xp ) Prob ((St%T > Xc )
Xp Xc
E ( [Xp & St%T * St%T < Xp ] E ( [Xp & St%T * St%T < Xp ]
7
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(4)
Risk-neutral probabilitydensity function of
The simplest relationship between out-of-the-money calls and puts is for those distributions that generate
symmetric risk-neutral distributions for S .t+T
Proposition 1 (symmetric processes): If the terminal risk-neutral distribution of S is symmetric around its meant+T
F = S e , and the strike prices of the call and put are spaced symmetrically around F, then European call and puttbT
prices as functions of the spot price, time to maturity, and strike price are related by
Proof: Obvious from Figure 1. O
Proposition 1 says that a call option with strike price 5% above the forward price (x = 0.05F) will be priced
identically to a put with strike price 5% below the forward price, if the distribution is symmetric. One historically
dS ' µS dt % FdZ .
c (F, T; X ) ' p (F, T; X ) % e &rT (F & X ) ,
c (F, T ; F % x ) > p (F, T ; F & x ) for 0 < x < F.
8
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(6)
(7)
important example of a symmetric distribution is Bachelier’s (1900) Brownian motion process with zero drift and no
absorbing barrier, generalized by Cox and Ross (1976) to processes of the form
While admittedly invalid given limited liability, Bachelier option prices are indistinguishable when bankruptcy is8
unlikely from those generated by the Cox and Ross (1976) model with µ = 0 and absorbing barrier at S=0.
Consequently, Proposition 1 holds virtually exactly for processes of the form (5) when bankruptcy risk is negligible,
and serves as a useful bound in other cases.
It is important to realize that (4) is unrelated to the European put-call relationship
which is an arbitrage-based relationship between puts and call with identical strike prices, and holds regardless of
distribution. Proposition 1 is a statement about calls and puts with different strike prices, and holds only for symmetric
terminal risk-neutral distributions. If the distribution is asymmetric, prices of calls and puts symmetrically out of the
money relative to the forward price will diverge. For the major distributions studied hitherto, that divergence takes9
systematic forms. The positively skewed log-normal distribution used in the Black-Scholes model has a positive
divergence between out-of-the-money (OTM) calls and puts:
The degree of divergence can be quantified for geometrically spaced strike prices.
c (F, T ; Fk ) ' k p (F, T ; F /k ) for any k > 0.
c BS (F, T ; X ) ' e &rT [ FN (d1 ) & XN (d2 )]
p BS (F, T ; X ) ' e &rT [ XN (&d2 ) & FN (&d1 )]
d1 'ln(F /X ) % ½F²T
F T, d2 ' d1 & F T ,
9
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(8)
(9)
(10)
(11)
Proposition 2 (Black-Scholes): If the underlying asset price follows geometric Brownian motion, then European call
and put prices are related by
Proof: The Black-Scholes formulas are
where
and N(@) is the cumulative normal. Plugging X = Fk into (9) and X = F/k into (10) yields (8). Oc p
Proposition 2 states that given geometric Brownian motion, if the call and put options have strike prices 5%
out-of-the-money relative to the forward price (and geometrically symmetric), then the call should be priced 5% higher
than the put. If the asset price followed arithmetic Brownian motion, then from Proposition 1 the call and put options
would have about the same price.10
Proposition 2 also holds for the benchmark stochastic volatility and jump-diffusion models, which can be
written as mean-preserving randomizations of the Black-Scholes formula.11
dS /S ' b dt % Ft dZ
dFt ' µ(Ft , t ) dt % v (Ft , t ) dZF
Cov (dS/S, dF) ' Ddt
c HW(F, T ; Fk ) ' m4
0c BS (F, T ; Fk, V ) f(V ) dV
' k m4
0p BS (F, T ; F /k, V ) f(V ) dV
' k p HW(F, T ; F /k) ,
V f (V )
c(F, T ; Fk )
k p(F, T ; F/k ) DsF
10
(12)
(13)
Corollary #1 (stochastic volatility) If the risk-neutral representation of the underlying asset price is of the form
then (8) holds if D = 0.sF
Proof: Under the independent-volatility assumptions of Hull and White (1987), option prices are
where is the realized average variance per unit time and is its risk-neutral density. O
The corollary does not hold if the correlation D between volatility shocks and price shocks is non-zero. AssF
noted by Hull and White (1987), a positive D biases OTM call option prices upward and ITM call options downwardsF
relative to Black-Scholes prices, whereas D < 0 induces the reverse biases. The implication in this framework is that sF
depending on whether D 0. Figure 2 illustrates the impact of varying for the square root>< sF><
variance process used inter alia by Hull and White (1988) and Heston (1993).
-0.5 0.5
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
3-month options
1-month options
4%
DsF
Dsp
dS /S ' Ft dZ
dF2t ' 6ln2(.152 & F2
t )dt % .15Ft dZv
F0 Xcall /F ' 1.04 ' F /Xput
dS /S ' (b & 8(k( ) dt % F dZ % k ( dq (
ln (1% k ( ) - N ln (1% k( ) & ½*2, *2
k(
11
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(14)
Corollary #2 (jump-diffusions) If the risk-neutral representation of the underlying asset price is a jump-diffusion
as in Merton (1976),
where
8* is the risk-adjusted jump frequency,
q* is a Poisson counter with intensity 8*, and
,
then Proposition 2 holds if = 0 and 8* is constant.
-5% 5%
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
4%
1-month options
3-monthoptions
k*
k(
8( * ' .02F2 % 8({ [ln(1%k() & ½*2 ]2 % *2 } ' .152
c Merton (F, T ; X ) ' j 4
0 wn c BS Fe n ln(1%k( ) & 8k(T, T ; X, F2 %n*2
T
' j 4
0 wn c BS F, T ; X, F2 %n*2
T, wn / e &8(T (8(T )n
n!
k(
k(
12
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(15)
Proof: The mean-zero jump assumption implies call option prices of the form
with a similar expression for puts. Since (8) constrains Black-Scholes call and put prices conditional on the number
of jumps n, it also holds for the weighted sum across all n. O
Positive (negative) raises (lowers) the skewness of the distribution relative to the lognormal, increasing
(decreasing) OTM call option prices relative to OTM put option prices. Figure 3 illustrates the impact of varying
k(
k
dS/S ' (r & r ( )dt % F(t)dZ
dr ' ["(t) & $(t)r ]dt % F1(t)dZ1
dr ( ' ["((t) & $((t)r ]dt % F2(t)dZ2
c (F, T ; X ) ' B (r, T ) [FN(d1 ) & XN(d2 ) ]
F2 % 8({ [ ln(1%k() & ½*2 ] 2 % *2}
d1 ' [ ln(F/X) % ½Vavg T ] / Vavg T
d2 ' d1 & Vavg T
13
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upon the deviation of calls from puts when the average variance per unit time
is held constant.12
Corollary #3 (Ornstein-Uhlenbeck interest rates/dividend yields) If the underlying asset price has a risk-neutral
representation of the form
where r* is the foreign interest rate (for foreign currency options) or stock dividend yield (for stock options), then
Proposition 2 holds regardless of the correlations between innovations in asset prices, interest rates, and dividend
yields.
Proof: Ornstein-Uhlenbeck interest rate processes with time-dependent parameters imply modified Black-Scholes
foreign currency option prices of the form13
whereB(r, T ) is the price of a discount bond with time T to maturity;
B*(r*, T ) is the price of a foreign discount bond with time T to maturity;
F = SB*/B is the forward price on the asset;
;
; and
dS ' bS dt % FS DdZ
Vavg T ' mT
0Var (dF /F )
Var (dF/F )
ln(X /F )
ln(Xc /F ) ' ln k ' &ln (Xp /F )
ln(X /F )
DsF < 0 k
14
���
,
where is a deterministic function of time under processes of the form (16). Consequently, Proposition
2 applies directly. O
Corollary #4 (volatility smile) The Black-Scholes implicit volatility patterns inferred from observed European option
prices will be symmetric in moneyness for distributional hypotheses satisfying equation (8); e.g., Black-
Scholes, stochastic volatility models with zero correlation between asset and volatility shocks, jump-diffusions with
mean-zero jumps, and Ornstein-Uhlenbeck stochastic interest rates.
Proof: Since Black-Scholes option prices satisfy (8), any other model satisfying (8) must have identical implicit
volatilities for calls and puts with strike prices such that for k>0. By put-call parity,
European calls and puts of identical strike prices and maturities must have identical implicit volatilities. O
The Black-Scholes and Ornstein-Uhlenbeck stochastic interest rate models imply identical implicit volatilities
for all strike prices of a given maturity. The benchmark stochastic volatility and jump-diffusion models are leptokurtic,
and imply U-shaped implicit volatility patterns or “volatility smiles” that are perfectly symmetric in .
Conversely, implicit volatility patterns will be a “tilted” U-shape for non-benchmark stochastic volatility and jump-
diffusion models. Implicit volatilities from ITM calls/OTM puts will be higher than those from correspondingly OTM
calls/ITM puts for stochastic volatility models with and jump-diffusions with * < 0, and lower when the
relevant parameters are positive.
I.C. CEV processes
Both arithmetic and geometric Brownian motions are special cases of the broader class of EQPUVCPV� GNCUVKEKV[�QH
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c(F, T ; Fk, F, D, b) ' k p(F, T; F /k, FF 2(1&D(), D(, b ( ) ,
c(F, T ; Fk, Fk 1&D, D, b) ' k p(F, T; F /k, F , D, b) for k > 0 and any D,
c(F, T ; F% x , F, D, b) < p(F, T; F& x, F , D, b ) for 0 < x < F only if D < 0
c(F, T ; F% x , F, D, b) > p(F, T; F& x, F , D, b ) for 0 < x < F, D > 0
15
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QH�D��VJGTG�KU�C�TGNCVKQPUJKR�DGVYGGP�16/�ECNNU�CPF�RWVU�KPVGTOGFKCVG�DGVYGGP�VJQUG�QH�2TQRQUKVKQPU���CPF���
.GOOC�� � #� 'WTQRGCP� ECNN� YKVJ� %'8� RCTCOGVGT� D��� EQTTGURQPFU� VQ� C� 'WTQRGCP� RWV� YKVJ� %'8� RCTCOGVGT
D��������D� ����EQUV�QH�ECTT[�D�����D��CPF�C�OQFKHKGF�XQNCVKNKV[�RCTCOGVGT�
YJGTG�VJG�CTIWOGPVU�CTG�VJG�HQTYCTF�RTKEG��VJG�VKOG�VQ�OCVWTKV[��VJG�UVTKMG�RTKEG��VJG�XQNCVKNKV[�RCTCOGVGT��VJG�%'8
RCTCOGVGT��CPF�VJG�EQUV�QH�ECTT[��TGURGEVKXGN[�
2TQQH���5GG�CRRGPFKZ�
2TQRQUKVKQP���%'8�RTQEGUUGU����+H�VJG�TKUM�PGWVTCN�CUUGV�RTKEG�HQNNQYU�VJG�EQPUVCPV�GNCUVKEKV[�QH�XCTKCPEG�RTQEGUU
�����VJGP�ECNN�CPF�RWV�RTKEGU�CTG�TGNCVGF�D[
CPF�UCVKUH[�VJG�HQNNQYKPI�KPGSWCNKVKGU�
c(F, T ; Fk , F, D, b) < k p(F, T; F /k , F , D, b ) for k > 1, D < 1
c(F, T ; Fk , F, D, b) > k p(F, T; F /k , F , D, b ) for k > 1, D > 1 ,
16
����� (QT�D���VJG�XQNCVKNKV[�QH�CUUGV�RTKEG�KPETGOGPVU�TKUGU�CU�VJG�CUUGV�RTKEG�HCNNU��YJKEJ�VGPFU�VQ�KPFWEG�PGICVKXG��
UMGYPGUU���6JKU�ECP��JQYGXGT��DG�OQTG�VJCP�QHHUGV�D[�VJG�CDUQTDKPI�DCTTKGT�CV�5����YJKEJ�VTWPECVGU�VJG�NQYGT�VCKNQH�VJG�FKUVTKDWVKQP�
� � � � � 6JG� CDQXG� CUUGTVKQPU� HQT� VJG� EQORQWPF� CPF� FKURNCEGF� FKHHWUKQP� QRVKQP� RTKEKPI�OQFGNU� CTG� DCUGF� WRQP��
EQORWVCVKQP�QH�VJG�TGURGEVKXG�QRVKQP�RTKEKPI�HQTOWNCU�
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YKVJ�KPGSWCNKVKGU�TGXGTUGF�HQT�Z���CPF�M�����KP�VJG�OQPG[�QRVKQPU��
2TQQH� 'ZRTGUUKQP�����KU�RTQXGF�KP�#RRGPFKZ�+�HQT�D���XKC�OCPKRWNCVKQP�QH�VJG�%'8�QRVKQP�RTKEKPI�HQTOWNC���6JCV
KV�CNUQ�JQNFU�HQT�D ��HQNNQYU�HTQO�VJG�NGOOC�������CPF�����HQNNQY�HTQO������UKPEG�QRVKQP�RTKEGU�CTG�KPETGCUKPI
HWPEVKQPU�QH�VJG�XQNCVKNKV[�RCTCOGVGT�F�������CNUQ�HQNNQYU�HTQO�����HQT�D ��UKPEG�(�� Z�� �(��Z��CPF�R@��KU
KPETGCUKPI�KP�VJG�UVTKMG�RTKEG���CPF�JCU�DGGP�EQPHKTOGF�PWOGTKECNN[�HQT���D�����%QORWVCVKQP�CNUQ�KPFKECVGU�VJCV
D���KU�C�PGEGUUCT[�DWV�PQV�C�UWHHKEKGPV�EQPFKVKQP�HQT�16/�ECNNU�VQ�DG�EJGCRGT�VJCP�16/�RWVU���#P�CFFKVKQPCN
EQPFKVKQP�VJCV�GPUWTGU�VJG�KPGSWCNKV[�KP�����KU�VJCV�VJG�RTQDCDKNKV[�QH�JKVVKPI�VJG�CDUQTDKPI�DCTTKGT�CV�5���DG�UOCNN���
(KIWTG���KNNWUVTCVGU�VJG�KORCEV�QP�TGNCVKXG�QRVKQP�RTKEGU�QH�XCT[KPI�D� O
1VJGT�OQTG� GZRNKEKV�OQFGNU� QH� HKPCPEKCN� NGXGTCIG� KPENWFG�)GUMGU� ������EQORQWPF�QRVKQP�OQFGN� CPF
4WDKPUVGKPU�������FKURNCEGF�FKHHWUKQP�OQFGN���)GUMGU�OQFGN�TGEQIPK\GU�VJCV�FGDV�KORNKGU�VJCV�VJG�WPFGTN[KPI�UVQEM
KU�KVUGNH�C�ECNN�QRVKQP�QP�VJG�XCNWG�QH�C�HKTO���+H�VJG�HKTOU�XCNWG�HQNNQYU�IGQOGVTKE�$TQYPKCP�OQVKQP��CU�KP�)GUMG
�������VJGP�GSWKV[�QRVKQPU�CTG�UKOKNCT�VQ�C�%'8�OQFGN�YKVJ���D���CPF�UCVKUH[�����CPF�������4WDKPUVGKP������
OQFGNU�VJG�HKTO�CU�C�RQTVHQNKQ�QH�IGQOGVTKE�$TQYPKCP�OQVKQP�KPXGUVOGPVU�CPF�TKUMNGUU�CUUGVU��YJGTG�VJG�NCVVGT�OC[
DG�PGICVKXG���4WDKPUVGKPU�OQFGN�KU�UKOKNCT�VQ�C�%'8�OQFGN�YKVJ���D���YJGP�VJG�HKTO�KU�C�PGV�FGDVQT��CPF�CNUQ
UCVKUHKGU�����CPF�������9JGP�VJG�HKTO�KU�C�PGV�JQNFGT�QH�TKUMNGUU�CUUGVU��VJG�FKURNCEGF�FKHHWUKQP�QRVKQP�OQFGN�KU
UKOKNCT�VQ�C�%'8�OQFGN�YKVJ�D ���CPF�UCVKUHKGU�������
-3 -2 -1 1 2 3 D
-40%
-30%
-20%
-10%
0%
10%
20%
30%
40%
arithmeticallysymmetricstrikes
geometricallysymmetricstrikes
4%
Xcall /F ' 1.04 ' F /Xput Xcall /F '1.04, Xput /F ' .96
ln(X /F ) > 0
ln(X /F ) < 0
17
(KIWTG�����4GNCVKXG�RTKEGU�QH����16/�VJTGG�OQPVJ�'WTQRGCP�ECNN�CPF�RWV�QRVKQPU��CU�CHWPEVKQP�QH�VJG�%'8�RCTCOGVGT�D���%QUV�QH�ECTT[�D�������)GQOGVTKECNN[�U[OOGVTKE�UVTKMGU�
��CTKVJOGVKECNN[�U[OOGVTKE�� �
'SWCVKQPU�����CPF�����ECP�QH�EQWTUG�DG�VTCPUNCVGF�KPVQ�VJG�EQTTGURQPFKPI�KORNKEKV�XQNCVKNKV[�RCVVGTPU�CETQUU
FKHHGTGPV�UVTKMG�RTKEGU���#P�GNCUVKEKV[�RCTCOGVGT�D ��KPFWEGU�OQTG�RQUKVKXG�UMGYPGUU�VJCP�VJG�NQIPQTOCN��CPF�KORNKGU
JKIJGT� KORNKEKV� XQNCVKNKVKGU� HQT� 16/� ECNNU�+6/� RWVU� YKVJ� � VJCP� HQT� +6/� ECNNU�16/� RWVU� YKVJ
���#P�GNCUVKEKV[�RCTCOGVGT�D���JCU�VJG�QRRQUKVG�GHHGEV�����
'SWCVKQPU�����CPF������JCXG�PQ�UKORNG�RCTCNNGNU�KP�KORNKEKV�XQNCVKNKV[�RCVVGTPU��CPF�CTG�KORQTVCPV�FKCIPQUVKEU
QH�YJGVJGT�NGXGTCIG�OQFGNU�ECP�QT�ECPPQV�GZRNCKP�GORKTKECNN[�QDUGTXGF�OQPG[PGUU�DKCUGU����6JWU��4WDKPUVGKP U
������CPF�5JGKMJ U�������ENCKO�VJCV�XQNCVKNKV[�RCVVGTPU�KP�UVQEM�CPF�UVQEM�KPFGZ�QRVKQPU�CTG�CV�VKOGU�EQPUKUVGPV�YKVJ
NGXGTCIG�OQFGNU�KU�DCUGF�GUUGPVKCNN[�WRQP�QRVKQP�RTKEGU�UCVKUH[KPI�������9JGVJGT�VJG[�CNUQ�UCVKUH[�����YCU�PQV�VGUVGF�
&CKN[�GUVKOCVGU�QH�KORNKEKV�%'8�RCTCOGVGTU�D[�/CE$GVJ�CPF�/GTXKNNG�������CPF�'OOCPWGN�CPF�/CE$GVJ������
QHVGP�NC[�YGNN�QWVUKFG�VJG�=����?�TCPIG�TGSWKTGF�D[�UVCPFCTF�NGXGTCIG�OQFGNU�
++���#U[OOGVT[�CPF�#OGTKECP�1RVKQP�2TKEKPI
6JG�TGNCVKQPUJKRU�FGTKXGF�CDQXG�HQT�'WTQRGCP�QWV�QH�VJG�OQPG[�TGNCVKXG�ECNN�CPF�RWV�RTKEGU�YKNN�PQV�JQNF�HQT
#OGTKECP�16/�ECNN�CPF�RWV�RTKEGU� KP�IGPGTCN�� �6JG�RTQDNGO� KU� VJCV� VJG��GCTN[�GZGTEKUG�RTGOKWO��OCTMWR�QH
18
#OGTKECP�QXGT�'WTQRGCP�QRVKQP�RTKEGU�KU�CHHGEVGF�RTKOCTKN[�D[�VJG�ECUJ�HNQYU�QH�VJG�WPFGTN[KPI�CUUGV��CPF�QPN[
UGEQPFCTKN[�D[�VJG�UMGYPGUU�QH�VJG�TKUM�PGWVTCN�FKUVTKDWVKQP���(QT�GZCORNG��CP�#OGTKECP�QRVKQP�QP�C�PQP�FKXKFGPF
RC[KPI�UVQEM�YKNN�PGXGT�DG�GZGTEKUGF�GCTN[�TGICTFNGUU�QH�YJCV�VJG�WPFGTN[KPI�FKUVTKDWVKQP�KU��YJGTGCU�KV�YKNN�DG�QRVKOCN
VQ�GZGTEKUG�CP�#OGTKECP�RWV�GCTN[�KH�VJG�CUUGV�RTKEG�HCNNU�NQY�GPQWIJ���5KPEG�KP�VJKU�ECUG�#OGTKECP�ECNN�RTKEGU�GSWCN
'WTQRGCP�ECNN�RTKEGU�YJKNG�#OGTKECP�RWV�RTKEGU�GZEGGF�'WTQRGCP�RWV�RTKEGU��CP[�FKUVTKDWVKQP�URGEKHKE�TGNCVKQPUJKR
DGVYGGP� RTKEGU� QH� 'WTQRGCP� ECNNU� CPF� RWVU� QP� PQP�FKXKFGPF� RC[KPI� UVQEMU� ECPPQV� CNUQ� JQNF� GZCEVN[� HQT� VJG
EQTTGURQPFKPI� #OGTKECP� RTKEGU�� � 6JG� TGNCVKQPUJKRU� OC[� JQNF� CRRTQZKOCVGN[�� JQYGXGT�� HQT� QWV�QH�VJG�OQPG[
#OGTKECP�QRVKQPU�YKVJ�PGINKIKDNG�GCTN[�GZGTEKUG�RTGOKWOU�
6JG�KPHNWGPEG�QH�VJG�ECUJ�HNQYU�QH�VJG�WPFGTN[KPI�CUUGV�WRQP�VJG�OCTMWR�QH�#OGTKECP�QXGT�'WTQRGCP�RTKEGU
KU�ECRVWTGF�KP�VJG�EQUV�QH�ECTT[�RCTCOGVGT�D���5KIPKHKECPVN[�RQUKVKXG�XCNWGU�QH�D�G�I���VJG�D���T�ECUG�QH�C�PQP�FKXKFGPF
RC[KPI�UVQEM��KORN[�EGVGTKU�RCTKDWU�C�NCTIG�OCTMWR�HQT�RWVU�CPF�UOCNN�OCTMWR�HQT�ECNNU��YJKNG�UKIPKHKECPV�PGICVKXG
XCNWGU�KORN[�VJG�TGXGTUG����#P�KNNWUVTCVKQP�QH�VJKU�KU�KP�5JCUVTK�CPF�6CPFQP��������YJQ�GZCOKPG�VJG�FKHHGTGPEG
DGVYGGP�#OGTKECP�CPF�'WTQRGCP�ECNN�CPF�RWV�RTKEGU�WPFGT�IGQOGVTKE�$TQYPKCP�OQVKQP�HQT�HQTGKIP�EWTTGPE[�QRVKQPU�
HQT�YJKEJ�VJG�EQUV�QH�ECTT[�D���T���T��ECP�DG�RQUKVKXG�QT�PGICVKXG�
+P�VJG�ECUG�QH�QRVKQPU�QP�HWVWTGU��VJG�HCEV�VJCV�VJG�EQUV�QH�ECTT[�GSWCNU�\GTQ�ETGCVGU�C�MPKHG�GFIG�ECUG�KP�YJKEJ
VJG�OCTMWR�HQT�#OGTKECP�QXGT�'WTQRGCP�ECNN�QRVKQPU�RCTCNNGNU�VJG�OCTMWR�HQT�EQTTGURQPFKPI�RWV�RTKEGU���9KVJ�\GTQ
EQUV�QH�ECTT[��VJG�TKUM�PGWVTCN�RTQEGUU�WUGF�KP�RTKEKPI�QRVKQPU�DGEQOGU�C�OCTVKPICNG��CPF�VJG�GCTN[�GZGTEKUG�FGEKUKQP
HQT�ECNNU�CPF�HQT�RWVU�KU�CU�U[OOGVTKE�QT�NQI�U[OOGVTKE�CU�VJG�VGTOKPCN�'WTQRGCP��XCNWCVKQP�RTQDNGO���%QPUGSWGPVN[�
2TQRQUKVKQPU�����HQT�'WTQRGCP�QRVKQPU�YKVJ�CTDKVTCT[�EQUV�QH�ECTT[�YKNN�CNUQ�JQNF�HQT�#OGTKECP�QRVKQPU�QP�HWVWTGU�
(WTVJGTOQTG��TGNCVKQPUJKRU�DGVYGGP�VJG�QRVKOCN�GCTN[�GZGTEKUG�RQNKEKGU�QH�#OGTKECP�ECNNU�CPF�RWVU�ECP�DG�FGTKXGF
WPFGT�VJG�XCTKQWU�FKUVTKDWVKQPCN�J[RQVJGUGU�
C(F, T ; F% x ) ' P(F, T ; F& x ) for * x* < F and any T ,
(F & X )(c ' & (F & X )(p .
c (F, T ; Fk ) ' k p (F, T ; F /k ) for k > 0 and any T ,
(F / X )(c ' 1/ (F / X )(p .
C(F, T ; Fk, Fk 1&D, D, b) ' k P(F, T; F /k, F , D, b) for k > 0 and 0 # D < 2
C(F, T ; F% x , F, D, b) < P(F, T; F& x, F , D, b ) for 0 < x < F only if D < 0
C(F, T ; F% x , F, D, b) > P(F, T; F& x, F , D, b ) for 0 < x < F, 0 < D < 2
k(
ln (X /F )
19
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(27)
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2TQRQUKVKQP�����(QT�#OGTKECP�QRVKQPU�QP�HWVWTGU�YJGP�KPVGTGUV�TCVGU�CTG�PQPUVQEJCUVKE��
��������� +H�VJG�TKUM�PGWVTCN�HWVWTGU�RTKEG�HQNNQYU�CTKVJOGVKE�$TQYPKCP�OQVKQP�YKVJ�PQ�CDUQTDKPI�DCTTKGT�VJGP
CPF�VJG�ETKVKECN��OCVWTKV[�FGRGPFGPV�HWVWTGU�RTKEG�GZGTEKUG�RTKEG�FKHHGTGPVKCNU�CDQXG�DGNQY��YJKEJ�VJG�ECNN
RWV��YKNN�DG�GZGTEKUGF�KOOGFKCVGN[�CTG�U[OOGVTKE�
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������C� IGQOGVTKE�$TQYPKCP�OQVKQP��������D� C�UVQEJCUVKE�XQNCVKNKV[�RTQEGUU�YKVJ�KPFGRGPFGPV�GXQNWVKQP�QH�XQNCVKNKV[��QT������E� C��NQI�U[OOGVTKE��LWOR�FKHHWUKQP�YKVJ� ���
VJGP
CPF�VJG�ETKVKECN��OCVWTKV[��CPF�UVCVG�FGRGPFGPV�HWVWTGU�RTKEG�GZGTEKUG�RTKEG�TCVKQU�CDQXG�DGNQY��YJKEJ�VJG
ECNN�RWV��YKNN�DG�GZGTEKUGF�KOOGFKCVGN[�CTG�IGQOGVTKECNN[�U[OOGVTKE�
(WTVJGTOQTG��KORNKEKV�XQNCVKNKV[�RCVVGTPU�KPHGTTGF�WUKPI�CP�#OGTKECP�HWVWTGU�QRVKQP�RTKEKPI�XCTKCPV�QH�$NCEM�5EJQNGU�YKNN�DG�RGTHGEVN[�U[OOGVTKE�HWPEVKQPU�QH�OQPG[PGUU� �
��������� +H�VJG�TKUM�PGWVTCN�HWVWTGU�RTKEG�HQNNQYU�C�%'8�RTQEGUU��VJGP
CPF�VJG�HQNNQYKPI�KPGSWCNKVKGU�CTG�UCVKUHKGF�
C(F, T ; Fk , F, D, b) < k P(F, T; F /k , F , D, b ) for k > 1, 0 < D < 1
c(F, T ; Fk , F, D, b) > k p(F, T; F /k , F , D, b ) for k > 1, 1 < D < 2 ,
C(F, T ; Fk, F, D, b) ' k P(F, T; F /k, FF 2(1&D(), D(, b ( ) ,
20
����� #OGTKECP�%'8�QRVKQP�RTKEGU�YGTG�ECNEWNCVGF�WUKPI�C�XCTKCPV�QH�VJG�%QZ�4WDKPUVGKP�������DKPQOKCN�QRVKQP��
RTKEKPI�OGVJQFQNQI[���1RVKQP�RTKEGU�YGTG�GZRTGUUGF�KP�VGTOU�QH�VJG�VTCPUHQTOGF�UVCVG�XCTKCDNG�
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CPF� LWOR�FKHHWUKQPU� KU� VJCV� VJG� #OGTKECP� RWV� XCNWCVKQP� RTQDNGO� HQT� VJGUG� FKUVTKDWVKQPU� KU� VJG� UCOG� WPFGT� C
FKUVTKDWVKQP�URGEKHKE�VTCPUHQTOCVKQP�QH�XCTKCDNGU�CU�VJG�#OGTKECP�ECNN�XCNWCVKQP�RTQDNGO��UGG�VJG�CRRGPFKZ�HQT�FGVCKNU�
� � 6JCV� VJG� TGUWNVU� HQT�'WTQRGCP�%'8�QRVKQPU� CNUQ� CRRN[� VQ�#OGTKECP� QRVKQPU� QP� HWVWTGU� JCU� DGGP� EQPHKTOGF
PWOGTKECNN[�QXGT�VJG�RCTCOGVGT�TCPIG��#D���CPF�KP�TGIKQPU�QH�D���YJGTG�DCPMTWRVE[�TKUM�KU�PGINKIKDNG� ��6JG��
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UJQWNF� DG� RTKEGF� KFGPVKECNN[�� � 6JKU� KU� CP� KPVGTGUVKPI� VJGQTGVKECN� TGUWNV� IKXGP� VJCV� VJGTG� CTG� PQ� CTDKVTCIG�DCUGF
TGUVTKEVKQPU�GSWCVKPI�RTKEGU�QH�CV�VJG�OQPG[�#OGTKECP�ECNNU�CPF�RWVU�UKOKNCT�VQ�VJQUG�HQT�'WTQRGCP�QRVKQPU�����QPN[
TCVJGT�YGCM�KPGSWCNKV[�EQPUVTCKPVU�FKUEWUUGF�KP�5VQNN�CPF�9JCNG[�������NKOKVKPI�JQY�HCT�VJG[�ECP�FGXKCVG���6JG
TGUWNVU�FQ�FGRGPF�WRQP�VJG�CUUWORVKQP�QH�PQPUVQEJCUVKE�KPVGTGUV�TCVGU��JQYGXGT����+V�FQGU�PQV�CRRGCT�VJCV�VJG�TGUWNVU
21
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20%
30%
40%
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standard error
Crash of ’87
25
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-40%
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0%
20%
40%
0%
10%
20%
830131 850314 861029 880712 900515 920701
4%
standard error
crash of ’87
26
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0%
10%
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30%
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830131 850315 861030 880714 900517 920706
Implicit volatilities:ATM options
volatility spread:4% OTM puts - ATM options
volatility spread:4% OTM calls - ATM options
27
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