CC BY
65
UDC 336.764.2
TIMUR V. KRAMIN,
Doctor of Science (Economics), Professor
Institute of Economics, Management & Law, Kazan, Russia
STEPHEN D. YOUNG Senior Vice President and Managing Director
Evergreen Investments, Option Strategies Group, Charlotte, North Carolina, USA
SHARE RETURNS DISTRIBUTION: EMPIRICAL OBSERVATIONS AND IMPLICATIONS FOR OPTIONS PRICING
Underlying asset returns regularly departfrom normal. In a Black-Scholes economy, asset prices are assumed lognormal and subsequently returns are normal. The literature regarding asset return behavior is extensive. In addition, the number of options pricing models, which take into account non-normality, discontinuities, and stochastic variables, are also extensive. These models are an effort at reconciling real world option prices with the assumptions in the Black-Scholes paradigm. This article contains a review of the literature regarding asset returns; alternative option pricing parameterizations, and recovering the implied risk-neutral distribution from listed options. Following this, theory takes precedence as a ssimple plain-vanilla option-pricing model, which incorporates the first-four moments of the risk-neutral density is explored. In the results section, S&P 500 Index market daily log returns are explored and univariate properties lead to a rejection of the null hypothesis of normality. Then, using the simple four-moment model and skewness and kurtosis values consistent with those recoveredfrom listed options by numerous researchers we imply the risk neutral densities associated with actual option prices on the S&P 500 Index. The implications are profound. It is very clear that the implied risk neutral densities are significantly different from the normal distribution which forms the basis for the Black-Scholes model.
I. Introduction
It is well known that the value of a derivative security is simply the present value of the expected payoff calculated using risk-neutral probabilities and discounted at the risk-free rate of interest. Despite well-known deficiencies, the Black-Scholes (BS) (1973) framework has been the most widely used method for analytically deriving option prices. The foundation of these deficiencies lies directly in the assumptions surrounding the evolution of the stock price over the life of the option contract. The BS model assumes that prices are lognormally distributed and subsequently returns are normally distributed with constant variance over all times and market levels. This assumption cannot be satisfied by the empirical world and is not scientifically corroborated. Thus the model tends to be biased because it does not allow for asymmetries.
This paper is broken down into four sections with numerous parts. The first contains a literature review concerned with the evolution of asset prices, alternative option pricing models which address various issues related to the BS theory, model values, and inconsistencies with real world prices, and the implied risk-neutral density which may be obtained via listed options. In the second section, theory points to a simple option-pricing model, which allows for incorporating the first fourmoments ofthe risk-neutral density. The third section provides numerical results, which focus first on empirical observations related to daily log-share returns from the S&P 500 Index market. The null hypothesis of normal share returns is refuted. Following this, option values and hedge parameters, which incorporate the first four moments of the risk-neutral density, are compared to those calculated via Black-Scholes. Conclusions follow in the fourth and final section.
II. Literature review
Asset prices and non-normality
In many financial theories, an underlying return distribution is assumed. More often than not, it is the normal (Gaussian) distribution, which may be fully described by its mean (first moment) and variance (second moment). Another characteristic ofthe normal distribution is its symmetry and standard degree of peakedness. Under a null hypothesis of normality, measures of skewness and kurtosis are distributed with a mean of zero and three, respectively. The skewness and kurtosis characterize the third and fourth moments respectively and allow one to model asymmetric and peaked data.
While most derivative pricing models rely on the normality assumption, strict observance ofthis is flawed. Returns often exhibit non-zero skewness and greater-than-three kurtosis. In fact, empirical findings reveal that log share returns typically have fat-tails. This implies that extreme asset returns occur more frequently than described by the normal distribution. Some of the earliest work, which supported this thesis, is Mandelbrot (1963) and Fama (1965). More recently, Singleton and Wingender (1986) document significantly positive skewness in a time series of monthly return data from the CRSP file over the period 1961-1980. Their work focuses on single equities and not indices. Aggarwal, Rao, and Hiraki (1989) examine the distribution of equity returns on the Tokyo Stock Exchange (TSE) over the period from 1965 to 1984. They compute skewness and kurtosis values for various sized portfolios and the entire time period as well as separate non-overlapping time periods. In short, they find that Japanese stock returns are characterized by significant departures fromnormality. Similarly, Aggarwal and Rao (1990) examine weekly return distributions for 1200 common stocks. Their examination centers on the effect of institutional ownership and higher order moments -skewness and kurtosis. In all the time periods they have examined - 1974, 1977, 1980, and 1983 - a large portion of the 300 randomly selected stocks exhibited positive skewness. They also document positive excess kurtosis. In short, they find an inverse relationship between the degree of institutional ownership and positive levels of skewness and excess kurtosis. More importantly, for this paper, Turner and Weigel (1992) document negative skewness and positive excess
kurtosis in the S&P 500 and Dow Jones indices over the 1928-1989 period. According to their research, returns in the 1980s had far more negative skewness and positive excess kurtosis than any other decade considered in the study. In addition, all decades in their study are consistent with "fat-tailed" distributions. Campbell and Hentschel (1992) report skewness and kurtosis for log excess stock returns on the Center for Research in Securities Prices (CRSP) value-weighted index of stocks from the New York Stock Exchange and American Stock Exchange for the period from 1926 to 1988. Consistent with previous research on stock indices, they find strong empirical support for the negative skewness and positive excess kurtosis for the CRSP Index.
Evidence ofnon-normality is not limited to equities and equity indices. Aggarwal (1990) examines the distributional aspects of weekly exchange rates for the floating rate period 1974-1984 for eight major currencies. The results indicate significant and persistent deviations from normality. Moreover, the author suggests that these deviations have implications for currency forecasting, hedging practices, portfolio analysis, and options pricing. This is not surprising as underlying return distributions typically form the basis for subsequent financial analyses and decision making. Sterge (1989) examines short and long-term fluctuations in the Treasury bond, 10-year Treasury note and Eurodollar futures price changes and concludes that the frequency of extreme observations is in excess of what is expected under the assumption of normality. These results are confirmed by statistical tests of goodness-of-fit.
Reconciling theory with practice - alternative option-pricing specifications
Numerous authors document pricing biases in the BS framework. Most notably, Macbeth and Merville (1979) report that the BS model undervalues in-the-money options and overvalues out-of-the money options using implied volatilities. Other research confirms biases related to strike price and time to maturity. Longstaff (1995) points to market frictions as a source of error in the valuation of derivative securities.
Deviations from normality and the implications for option pricing have not been lost on academics. One of the earliest attempts to correct for biases is in the jump- diffusion model of Merton (1976). This model
augments the BS return distribution with a Poisson-driven jump process, which allows for discontinuous stock prices. However, the model is difficult to parameterize as it calls for the estimation of two additional values: the number of jumps per annum and the percentage of the total volatility explained by the jumps.
Jarrow and Rudd (1982) incorporate deviations from normality by taking the correct value of an option to be the sum of the BS value plus adjustment terms, which depend on the second and higher order moments of the underlying diffusion process. Their approach approximates the distribution density of asset prices using empirical observationss of skewness and kurtosis. While it admits analytical option prices for densities with skewness and kurtosis, it requires the computation of derivatives ofthe log-normal distribution and is more complicated in its implementation and understanding. The pricing specification expressed in the Jarrow and Rudd paper uses a series of adjustment terms which represent the differences between the approximate and true distributions for the second, third, and fourth moments respectively. Thus the Jarrow and Rudd approach provides a methodology for approximating option values for arbitrary stochastic processes.
Rithchey (1990) delineates a mixture call option-pricing model to examine the impact of a non-normal density function. The model is taken as the weighted sums of BS solutions. Their research finds discrepancies between market prices and values derived via BS. In short, values of in-the-money options and options with longer maturities are closely approximated by the BS model. Low-priced, and out-of-the-money options are found to have significant mispricing using the BS framework.
In contrast to Jarrow and Rudd (1982), Corrado and Su (1996) approximate the asset price log-return (not price) distribution density. They apply a Gram-Charlier series expansion to the normal density function and provide skewness and kurtosis adjustment terms for the BS model. Their approach allows for an analytical solution. However, the Gram-Charlier and Edgeworth expansion, are infinite series. Thus, while their work admits an analytical solution via Hermite polynomials, if one chooses to include more terms in the series expansion beyond Gram-Charlier and Edgeworth this method becomes quite complicated.
Lastly, by minimizing the sum of squared errors, Corrado and Su solve for implied volatility, skewness, and kurtosis measures. Their findings are consistent with non-normal skewness and kurtosis implied in S&P 500 Index option prices.
Johnson, Pawlukiewicz, and Mehta (1997) present a binomial model for options pricing which incorporates asymmetric end-of-period return distributions. More specifically, this model presents a binomial specification that allows for non-normal skewness. Again, the results indicate that a failure to account for this deviation from normality results in mispricing.
Tian (1998) develops a trinomial option-pricing model that allows for the first four moments of the underlying asset distribution. This represents an improvement over the Johnson, Pawlukiewicz, and Mehta specification as it accounts for non-normal skewness and kurtosis, which allows one to address both "fatter tails" and "peakedness" in observed stock returns distributions. However, this parameterization admits a limited range of skewness and kurtosis values. In fact, because ofthis the numerical results omit many combinations of skewness and kurtosis values that are characteristic of real world market data.
Rubinstein (1998) significantly decreased the complexity of incorporating non-normal skewness and kurtosis by applying Edgeworth expansion in a binomial framework in order to incorporate these higher-order moments. There is one major drawback to this approach. The resulting skewness and kurtosis will only approximate the pre-specified levels. Further, similar to Tian's (1998) specification, this method only admits a limited locus of skewness and kurtosis parameters -under an Edgeworth expansion the locus is more constricted than if one reverts to a Gram-Charlier expansion. Outside this locus, one cannot ensure that the adjusted risk-neutral probabilities are everywhere on the interval between zero and one inclusive.
Corrado (2000) utilizes a generalized lambda distribution for generating option values where the underlying return distribution exhibits non-normal skewness and kurtosis. In contrast to the other methods, which augment the assumptions of BS by transforming the density function to account for higherorder moments, this approach relies on a completely different density. The generalized lambda distribution
is well known among statisticians, but has not yet been widely embraced by the financial community. As a consequence, it adds an additional layer of complexity to an already involved procedure.
In an attempt to reconcile real world option prices with theory, Das and Sundaram (1999) use two extensions of the BS model - introducing jumps into the return process and stochastic volatility - to examine the extent to which either of these methods is capable of resolving observed anomalies. In short, their research finds that neither approach is sufficient but the stochastic volatility model fares marginally better.
As the above literature suggests, over time it has become clear that the market does not price all options according to the BS paradigm. In the options market implied volatility frequently increases with decreasing strike. This is the familiar "negative" skew commonly observed in the market for S&P 500 Index options. Typically, traders adjust for this by quoting volatilities that correspond to the consensus of the market for a given maturity and strike. Yet only one index underlies all the options on a particular underlying asset and it can only have a unique volatility. The pattern of implied volatilities with variation in time is known as the "term structure of volatility" and the pattern associated with strike prices the "skew". Together, they combine to create what is called the "implied volatility surface". This variation with maturity and strike does imply that volatility may be some complicated function of both spot and time and has been explored by numerous researchers. The existence of the "implied volatility surface" is evidence that the market does not rely solely on the standard Black-Scholes assumptions but rather attaches different probabilities to terminal prices than those consistent with the lognormal price distribution. In fact, as a result, there has been a proliferation of alternative volatility specifications since the original constant volatility posited by BS.
Most notably, Cox and Ross (1976) delineate the option pricing formula for the deterministic case where there is a constant elasticity relationship between volatility and the underlying security price . In this model, volatility is proportional to the level of the stock price raised to a power. Other methods of deterministic volatility structures include autoregressive conditional heteroscedasticity (ARCH), generalized autoregressive conditional heteroscedasticity
(GARCH), the exponential GARCH (EGARCH), and exponentially weighted moments models.
Stochastic volatility posits that the future level of volatility cannot be perfectly predicted from currently available information and is typically some complicated function ofspot and time . From a stochastic perspective, among others, Hull and White (1987), Scott (l987), and Wiggins (1987) were among the first to develop models and explore this issue. Stein and Stein (1991) and Heston (1993), also worked on the stochastic volatility issue but with models that are simpler and computationally more efficient than their predecessors.
From implied volatility to implied risk-neutral densities
The earliest work related to recovering the implied risk neutral densities (IRND) from option-prices is based on the theory of state-contingent securities. A state contingent claim, or elementary claim, reflect investors' assessments of the probabilities of particular states occurring in the future conditional on a particular information set. State-contingent securities find their genesis in the works of Arrow (1964) and Debreu (1959), and, as such, they are justifiably known as "Arrow-Debreu" securities. Their work, and the concept of state-prices, is directly related to the more modern methods ofrecovering risk-neutral probability distributions from option prices. In short, there is a direct relationship between Arrow-Debreu securities and the risk-neutral probabilities used for pricing options and popularized via lattice methods.
Breeden and Litzenberger (1978) capitalized on the concept of state-prices and demonstrated that if the underlying price at a specific time has a continuous probability distribution, then one can recover the discounted IRND function of the underlying asset conditioned on the price and time. This distribution is determined by the second derivative of the call price with respect to the strike. More succinctly, in options terminology, one makes use ofthe notion of a "butterfly-spread" applied across a continuum of possible stock price values to infer the IRND.
Recently, academics and practitioners have elaborated "implied" lattice methods - see Derman and Kani (1994), Dupire (1994), and Rubinstein (1994). If one knows the standard index option prices for all strikes and maturities, then, the local volatility surface can be determined. The essence of this approach is that
the market is correct and that one can both recover the implied risk-neutral probabilities and implied volatilities from the liquid options market. Recovering the risk-neutral probabilities proves very powerful as these probabilities are immutable with respect to option type and so this method allows one to price not only plain vanilla but also exotic options in a manner consistent with the liquid market. Jackwerth and Rubinstein (1995) demonstrate how to recover risk-neutral probabilities from the market prices of traded options on the S&P 500 Index. In examining the period from April 1986 through December 1993, they find that the probability distributions in the pre-crash (October 1987) period were "somewhat left-skewed and platykurtic". After the crash (post October 1987), their research points to more left-skewed distributions with consistent levels of negative skewness and excess kurtosis. In a later paper, Jackwerth and Rubinstein (1996) corroborate their earlier findings with respect to the S&P 500 Index.
In summary, there are a few approaches that researchers have employed to recover the IRND. One simple approach relies on the risk-neutral histogram generated via "butterfly spreads". The histogram may then be smoothed via a kernel density so that the tails are more fully described. Alternatively, a parametric assumption is followed by minimizing the distance between observed option prices and those generated by the option-model functional specification. This minimization may take many forms with the most common applied to the sum of the squared deviations. Other approaches include non-parametric methods.
Consistent across all of the approaches is the underlying belief that the density is some complicated function not fully captured in the standard Black-Scholes framework with its assumption of normality. The real phenomenon underlying term structures, smiles, and surfaces may be that either market forces prevent prices from taking their true BS values or that share returns are not normally distributed.
III. Non-normality and a simple option-pricing specification
In the case of a European call option, the final payoff is - X )+ ,vith the option value given by the expected present value payoff, in a risk-neutral economy:
rjT / \
Vc — e-rT E
[ -X)+],
(1)
where r is the continuously compounded risk-free rate of interest, Tthe time to expiration in years, E is the expectation operator in a risk-neutral economy, is the underlying asset value at maturity, X is the strike price, and+is the "if positive" operator. Under a Geometric Brownian Motion diffusion process and a risk-neutral economy, our model for the evolution of asset prices is given by:
dS = (r - d )Sdt + oSdB, (2)
where d the continuously compounded dividend yield, the instantaneous volatility of the asset price, dt is an infinitely small increment of time, and B is Brownian motion. By using Ito's Lemma and substituting V TZ for B where Z is a standard normal random variable, we have the following expression for ST:
St — Sq<
(r-d )--y T+o4tz
(3)
where S0 is the current underlying asset value. By substituting for ST in Equation (1):
A
V — e -rT E
( 02 ^
( - d)- — t+o4Tz 2
- X
. (4)
And, by the definition of the expected value, one may write:
V —
_c
•-d )- — T + cVTu 2
- X
1 2 —u
e 2 du.
Introducing m — ln(s0 ) + (r - d)• T - -^j-T and
=cvr; one can simplify:
V —
e-rT œ T + I —w e- j |em+su - x I e 2 du.
-c
(5)
The term em+su - X in Equation (5) is positive if u > a =ln(X^ m. Therefore, V may be written as:
-rT œ . .1
e 'I <-s+m
BS, c
/ \ -—t 2 j(+m - X)• e 2 • dt. (6)
+
+
-rT œ
Similarly, with a change in the limits ofintegration, and the boundary condition to (X-ST)+, the value of a European put is given by:
-rT
V
BS, p
_
X _ e
t ■ s + m
-It 2 2
dt (7)
with a, s, and m previously defined. Equations (6) and (7) represent the BS formula for European calls and puts in integral form.
In general, for a wide class of continuous distributions it is possible to adjust the cumulants by applying a specific operator to the probability density function (PDF). To adjust the integrals given by Equations (6) and (7) for higher-order terms, one can
apply a series expansion. The cumulants of an adjusted
t r
density can be determined as coefficients of — in the expansion of the following function:
r!
:(x ) = ln
| etxg( x )dx
( ) =
1
V2ñ
■ f (t ),
where f(t) represents an adjustment to the initial standard normal density function.
Usually, only the first four moments are matched, and the Gram-Charlier (three-term) or Edgeworth (four-term) expansions are applied. While the importance of the additional fourth factor depends on a particular assumption about the orders ofmagnitude
of successive cumulants, this assumption may not be consistent with reality. Thus one can not expect either regularity in convergence of these series or conclude the theoretical superiority of the Edgeworth expansion over the Gram-Charlier. As an example, in this work, a generalized Edgeworth expansion is applied, transforming the standardized density given a prespecified skewness and kurtosis.
The adjustmentf(t) is defined accordingly (see Balakrishnan, Johnson and Kotz (1970)):
(8)
f (t )=i+6 3 - 3t )+( - 3)4 - 6t2+3)+ +72 S2 ( t5 - 1Qt3+151 j ,
where is skewness and represents kurtosis. Thus for a European call and put options, under an Edgeworth expansion, the theoretical option values are given by thefollowing:
_rT œ
It should be noted that not for all x. Moreover, the multimodality in the above function represented by "humps" in the tails becomes pronounced ifthe number of terms in the expansion increases. Therefore, an adjusted density will not, in general, represent a proper probability density function. Nevertheless, due to the orthogonality of Hermite polynomials, the adjusted density recovers the correct values for any desired finite number ofmoments and satisfies the unit condition:
j g(x )dx = 1.
Thus, one can obtain a useful approximate representation of a distribution with known moments. The most often used initial distribution is the normal distribution, therefore the expansion can be represented as follows:
V
Edge, c
42K
/ \ _-12 |(+m _ X)■ e 2 ■ f (t )■ dt, (9)
e_rt a¡ \ _112
VEdge,„ = V |((_et s+m )■ e 2 ■ f (t )■ dt. (10) v2n _c
If £=0 and k=3 thenf(t) = 1, the density function corresponds to standard normal distribution N(0,1) and option values are consistent with the BS paradigm. The above integrals are easily evaluated via numerical integration - a trapezoidal or parabolic (Simpson's rule) method works well.
Analogously, for European calls and puts, the following partial derivatives (the "Greeks" or hedge parameters) (delta, gamma, vega, theta, and rho):
A dV d 2V 3V dV dV A = —, T = —v = —, © = —, p = — dS dS2 da dt dr
may be calculated numerically (the partial derivatives
with respect to skewness ^Z and dV kurtosis may
d£ 3k
also be numerically obtained). In addition, one may also determine the marginal change in value due to skewness by setting the kurtosis value equal to three and comparing this value to the BS output:
Si; = VEdge (S,X,a,T,r,d= 3)— VBS (S,X,a,T,r,d ).
OO
OO
— 00
The marginal change in value due to kurtosis may similarly be determined by setting the skewness parameter equal to zero and comparing the resulting value to the BS output.
§K = VEdge (S.X. ^T.r>d& =
= 0,k )_ VBS (S, X,a,T,r,d ).
The model values, as well as the Greeks, provide insight into the pricing and hedging biases introduced in an environment characterized by non-normal underlying returns.
IV. Results
Asset prices and non-normality - S&P 500 index daily log returns
For this paper, in examining the phenomenon of S&P 500 Index return data and non-normality, the historical data over the period from 12/29/89 - 11/02/ 01 is considered. Daily returns are calculated as the natural logarithm ofthe price relatives:
/; = ln(p /^- ). (11)
The sample mean, standard deviation, skewness, and excess kurtosis are as follows:
" (12)
í \2 2 n Í V
n £ r. _ £ \r.)
\ =
K
Excess
-1 i -1 i - 1
I n(n _l)
n 2 / r. 1
(n _ ixn _ 2)i 2i V
n(n +1) 2 (
_l)n _ 2Xn _ 3)i -1{
3(n _ 1)2
■ \3
(13)
(14)
r - r
V /
(15)
(n _ 2)n _ 3)
where r. , the daily return, is given by equation (11), r is the mean of the return series defined by equation (12), n is the number of observations, and s is the sample standard deviation of returns (equation (13)). The Jarque-Bera (JB) test statistic which is used to test for normality given by:
where n is the number of observations, the sample value of skewness, and the sample value of kurtosis (not excess kurtosis). The JB test statistic follows a chi-square distribution with two degrees of freedom. If the JB test statistic is greater than the critical value of the chi-square distribution, we may reject the null hypothesis of normality. Figure 1 below displays a histogram of the log returns series as well as summary statistics including skewness and kurtosis (not excess) over the entire sample period.
Both from a graphical and statistical sense - as evidenced by the test statistic - we may reject the null hypothesis of normality. A histogram is a simple non-parametric representation yet is insightful as Figure 1 displays peakedness and fatter tails than are consistent with the normal distribution. Using the full data set, Figure 2 presents a kernel density ofthe daily log returns. The kernel density represents a smoothed distribution and is generated by putting less weight on observations, which are further from the point being evaluated. Formulaically, the kernel density estimate ofthe series X (daily log returns) at a point x is estimated by:
1 N ( x _ X \ f (x )=_L E K
Nh i=1 h
(17)
JB -
[n / 6][ 2 +(k _ 3)
_ 3)2 / 4
where N is the number of observations, h is the smoothing parameter, and K is a kernel weighting function that integrates to one. And for Figure 2, the kernel weighting function (K) is normal (Gaussian) and given by:
1 ( "2 ) K (u J=-j= el 2
As with the simple histogram, it is clear that the actual daily log returns distribution is more peaked and has fatter tails than the normal distribution. And lastly, from a graphical perspective, a tool for examining departure from normality is the QQ (quantile - quantile) plot. The QQ plot presented below further confirms the evidence ofnon-normality.
Further statistics are found below in Table 1 which contains S&P 500 Index rolling values for means, maximum, and minimum values of skewness (14) and excess kurtosis (15) for 1 - month, 3 - month, and 6 -month periods. For each of the rolling periods, the Jarque-Bera test statistic strongly suggests that we may reject the null hypothesis of normality.
(16)
Histogram andSummary StatisticsofDailyLogReturns - SPX 12/89 - 11/01 800
Series: SPX Daily Log Returns - 12/89 - 11/01 Observations 2989
Mean
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Probability
0.000376 0.009877 -0.256033 7.337063 2375.299 0.000000
Histogram Bars of Daily Log Returns - SPX 12/89 - 11/01
The above histogram and summary statistics are for daily log returns for the S&P 500 Index over the period 12/29/89 - 11/02/01.
Figure 1
Kernel Density of Daily Log Returns - SPX 12/89 - 11/01
A smoothed representation of daily log returns for the S&P 500 Index over the period 12/29/89 - 11/02/01.
Figure 2
QQ Plot of Daily Log Returns - SPX 12/89 -11/01
Daily Log Returns - SPX 12/89 -11/01
Figure 3
Table 1
S&P 500 Index Returns Skewness and Kurtosis Values 12/29/89 - 11/02/01
1 - Month Rolling Skewness and Excess Kurtosis (24 observations per period)
Mean Max
Skewness -0.0151 1.8921
Excess Kurtosis 0.5608 11.0215
Total Observations in Rolling Period: 2,966
Jarque-Bera Test Statistic Value 38.98
Min -2.8161 -1.3175
3 - Month Rolling Skewness and Excess Kurtosis (64 observations per period)
Mean Max
Skewness -0.1217 0.9686
Excess Kurtosis 1.1549 10.2886
Total Observations in Rolling Period: 2,926
Jarque-Bera Test Statistic Value 169.85
Min -1.9138 -0.7653
6 - Month Rolling Skewness and Excess Kurtosis (127 observations per period)
Mean Max
Skewness -0.1847 0.6683
Excess Kurtosis 1.6842 10.6264
Total Observations in Rolling Period: 2,863
Jarque-Bera Test Statistic Value 354.64
Min -2.0355 -0.6273
The above table depicts the mean, maximum, and minimum values for skewness and excess kurtosis over 1, 3, and 6 month rolling periods. The Jarque-Bera test statistic is calculated using the mean values for skewness and kurtosis for each of the respective time periods.
The above conclusions regarding non-normal returns are consistent with the research on indices by Turner and Weigel (1992) and Campbell and Hentschel (1992), which cover the periods from 1928 through 1989 and from 1926 through 1988 respectively.
Pricing options in the presence of non-normal share returns - values and comparative statics
The S&P 500 Index options market (these options are of European style) is characterized by academics and practitioners as having a negative skew. Perhaps, the most noticeable implication from a trading perspective is that out-of-the-money puts typically trade at high-implied volatilities relative to other options. However, if one uses a constant volatility as assumed by the BS model, a graphic of strike versus implied volatility would show a flat line. Typically, traders adjust the valuation of options for this negative skew by applying different volatilities across strikes and time. Some solve for the volatility surface by observing prices and inverting the BS equation to derive volatility
measures while others have a particular view of the market. The non-constancy of volatility across strike and time is often linked to a violation of the Black-Scholes assumptions of which normality is one.
In depicting the skews which are observed in the market and to recover the implied risk neutral densities associated with actual option prices, we collected market data on S&P 500 Index options as of the close of business January 30, 2009. These option values are as per Bloomberg and are for a range of strike prices (i.e. low to high strike) and for maturities of22, 50, and 78 days. Shorter dated options tend to be significantly liquid as are options which are closer to the at-the-money.
Figure 4 below depicts the skews (some refer to this pattern as a "smile") present in the market for S&P 500 Index options. It is clear from the data that implied volatilities differ significantly for options with the same maturity but with different strikes. Further, it is also clear that the skew effect diminishes or "flattens" out as the
S&P 500 Index Implied Volatility Skews
As one can see from the above, the shorter maturity options have a more pronounced skew which "flattens" out as a function of option maturity. The implied volatilities are most certainly not constant but rather a function of strike and maturity. This is a clear violation of Black-Scholes which posits a constant volatility. In graphing the skews the industry standard practice is using put volatilities for all strikes below the current index level and call volatilities for strikes above the current index level.
Figure 4
expiration of the option increases but that we still observe a pronounced pattern even at the 78 day expiration. It is important to note that Black-Scholes posits a constant volatility where some have taken this in the strict sense to be for all maturities and others would suggest that it is perfectly viable to observe a different implied volatility for varying tenors within a Black-Scholes world.
By minimizing the mean sum of squared errors between the observable S&P 500 Index option prices and model values produced via the Edgeworth model, we can solve for the time-dependent skewness and kurtosis values implied by the market. This allows us to plot the implied Edgeworth density (i.e. the implied risk neutral density) consistent with the observable market data. In a Black-Scholes world we would observe normality with zero skewness and zero excess kurtosis.
In Figure 5. we plot the implied risk neutral densities for S&P 500 Index options with maturities of22, 50, and 78 days and we superimpose this on a normal distribution which is consistent with the Black-Scholes paradigm. As observed from the graph, the densities are distinctly non-normal with "fat tails" and negative skewness which both imply that movements down in the underlying index are more likely than one would predict based on a normal distribution. It is important to note that option prices are driven by supply and demand and therefore these prices and the subsequent densities can be interpreted as the markets aggregate outlook regarding the potential future movements in the underlying index. Further, the shape of the underlying densities is consistent with what researchers have found which is that low strike options are under-priced by the Black-Scholes model.
3,00E-02
1,00E-02
5,00E-03
0,00E+00
Normal versus Edgeworth Implied Risk Neutral Densities
^ J* n® n® ^ # J? ^ ^ Nf „6? S
>" y > > > > >" >' o- O- O" O1 Cr V V "V "V 'V- 'V k"
— Normal —♦—22 Day IRND —o—50 Day IRND —A—78 Day IRND
The above figure plots the normal density versus those implied by the Edgeworth option model where the skewness and kurtosis values are solved for by minimizing the mean squared error between Black-Scholes option values and those from the Edgeworth model. Black-Scholes posits that log returns are normally distributed and that the diffusion which underlies the model has constant volatility for a specified horizon. Observed implied volatilities clearly contradict this. The above implied densities are effectively the markets outlook as to the distribution of returns.
Figure 5
V. Conclusions
In general, models are simplifications of reality typically based on assumptions with varying degrees of truth. One such model is the BS option-pricing model. This paper revisits the assumption of normally distributed returns consistent with the BS option-pricing paradigm. In observing daily log returns for the S&P 500 Index over the sample period (12/89 -11/01), the null hypothesis of normality is rejected.
While numerous researchers have proposed alternatives option-pricing specifications to accommodate non-normal share returns, practitioners still rely heavily on BS because of its simplicity. The proposed model, represents a simple alternative that addresses higher order moments in the context of options pricing. The model is simple to implement and affords one the ability to value options consistent with the first four moments of the risk-neutral distribution. The results developed in the tables, demonstrate that values and hedge parameters calculated via the proposed model, are significantly different from those produced from a standard Black-Scholes parameterization. Given the empirical results which refute the hypothesis of normality, in particular, it is obvious that the market for S&P 500 Index options may be more realistically modeled by taking the higher-order risk-neutral moments into account.
Practically speaking, for liquid underlyings with observable option prices, inferring the risk-neutral density is feasible. However, most options are liquid at-the-money and for the near term expirations. In addition, strikes are set at discrete intervals and are a function of the underlying assets price. Thus, recovering the risk-neutral density may be difficult. Further, one must question the validity of this practice. That is, one area of future research is to examine the inferred versus actual realized distribution. As with numerous studies related to implied volatility versus actual realized, one may find that the inferred distribution is not an accurate (unbiased) predictor of the future. Quite frankly, as with volatility estimation, fusing information from historical and implied data may be a reasonable approach.
References
1. Abken, P. and S. Nandi. 1996. "Options and Volatility". Federal Reserve Bank of Atlanta - Economic Review, (December): 21-35.
2. Aggarwal, R. 1990. "Distribution of Spot and Forward Exchange Rates: Empirical Evidence and Investor Valuation of Skewness and Kurtosis". Decision Sciences, vol. 21. issue 3: 588-595.
3. Aggarwal, R. and P. Rao Ramesh. 1990. "Institutional Ownership and Distribution of Equity Returns". The Financial Review, 25: 211-29.
4. Aggarwal, R. and P. Rao and T. Hiraki. 1989. "Skewness and Kurtosis in Japanese Equity Returns: Empirical Evidence". The Financial Review, 12, no. 3 (Fall): 211-29.
5. Arrow, K.J. 1964 "The Role of Securities in the Optimal Allocation of Risk-Bearing". Review of Economic Studies, 31, no. 2 (April): 91-96.
6. Balakrishnan, N., Johnson, N., and Samuel Kotz. Continuous Univariate Distributions. Vol. 1, John Wiley & Sons, Inc., 1994.
7. Bhupinder, B. 1997 "Implied risk-neutral probability density functions from option prices: theory and application". Bank of England.
8. Black, Fischer and Myron Scholes. 1973. "The Pricing of Options and Corporate Liabilities". Journal of Political Economy, vol. 81, no. 3 (May/June): 637-54.
9. Breeden, D.T. and R.H. Litzenberger. 1978. "Prices of State-Contingent Claims Implicit in Option Prices". Journal of Business, vol. 51, no. 4: 621-51.
10. Campbell J.Y. and L. Hentschel. 1992. "No News Is Good News: An Asymmetric Model Of Changing Volatility In Stock Returns". Journal of Financial Economics, 31: 281-318.
11. Corrado, C.J. 2001. "Option Pricing Based on the Generalized Lambda Distribution". Journal of Futures Markets, (February).
12. Corrado, C.J. and T. Su. 1996. "Skewness and Kurtosis in S&P 500 Index Returns Implied by Option Prices". Journal of Financial Research, 19: 175-192.
13. Cox, J. and S. Ross. 1976. "The Valuation of Options for Alternative Stochastic Processes". Journal of Financial Economics, 3: 145-66.
14. Das, S.R. and R.K. Sundaram. 1999. "Of Smiles and Smirks: A Term Structure Perspective". Journal of Financial and Quantitative Analysis, 34(2): 211-239.
15. Debreu, G. 1959. The Theory of Value, Wiley, NY.
16. Derman, E. and I. Kani. 1994. "Riding on the Smile". Risk, 7 (February): 32-39.
17. Dupire, B. 1994. "Pricing with a Smile". Risk, 7 (January): 18-20.
18. Fama, Eugene F. 1965. "The Behavior ofStock Market Prices". Journal of Business, 38: 34-105.
19. Heston, S. 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options". Review of Financial Studies, 6: 327-43.
20. Hull, J. and A. White. 1987. "The Pricing of Options on Assets with Stochastic Volatilities". The Journal of Finance, vol. 42, no. 2 (June): 281-300.
21. Jackwerth, J.C. and M. Rubinstein. 1995. "Recovering Probability Distributions from Contemporaneous Security Prices". The Journal of Finance, December.
22. Jackwerth, J.C. and M. Rubinstein. 1996. "Recovering Probability Distributions from Option Prices". The Journal of Finance, December.
23. Jarrow, R. and A. Rudd. 1982. "Approximate Option Valuation for Arbitrary Stochastic Processes". Journal of Financial Economics, 10, no. 3 (November): 347-69.
24. Johnson, R.S., Pawlukiewicz, J.E., and J.M. Mehta. 1997. "Binomial Option Pricing with Skewed Asset Returns". Review of Quantitative Finance and Accounting, 9: 89-101.
25. Longstaff, F.A. 1995. "Option Pricing and the Martingale Restriction". The Review of Financial Studies, Vol. 8, Issue 4 (Winter): 1091-1124.
26. Macbeth, J. and L. Merville. 1979. "An Empirical Examination of the Black-Scholes Call Option Pricing Model". The Journal of Finance, vol. 34, no. 5 (December): 1173-86.
27. Mandelbrot, Benoit. 1963. "The Variation of Certain Speculative Prices". Journal of Business, 36: 394-419.
28. Merton, Robert C. 1976. "Option Pricing with Discontinuous Returns". Journal of Financial Economics, 3: 125-44.
29. Ritchey, R. 1990. "Call Option Valuation for Discrete Normal Mixtures". Journal of Financial Research, vol. 13, no. 4 (Winter): 285-96.
30. Rubinstein, M. 1994. "Implied Binomial Trees". Journal of Finance, 49, no. 3 (July): 771-818.
31. Rubinstein, M. 1998. "Edgeworth Binomial Trees". Journal of Derivatives, 5: 20-27.
Stamplfi, J. and Victor Goodman. The Mathematics of Finance: Modeling and Hedging. The Brooks/Cole Series in Advanced Mathematics, 2001.
32. Scott, L.O. 1987. "Option Pricing When the Variance Changes Randomly: Theory, Estimation, and Application". Journal of Financial and Quantitative Analysis, 22: 419-38.
33. Singleton, J.C. and John Wingender. 1986. "Skewness Persistence in Common Stock Returns". Journal of Financial and Quantitative Analysis, 21: 335-41.
34. Stein, E.M. and J.C. Stein. 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach". Review of Financial Studies, 4: 727-52.
35. Sterge, A.J. 1989. "On the Distribution of Financial Futures Price Changes". Financial Analysts Journal, May-June, 75-78.
36. Tian, Y.S. 1998. "A Trinomial Option Pricing Model Dependent on Skewness and Kurtosis". International Review of Economics and Finance, 7(3): 315-330.
37. Turner, A.L. and Eric J. Weigel. 1992. "Daily Stock Return Volatility: 1928-1989". Management Science, 38: 1586-1609.
38. Wiggins, J.B. 1987. "Option Values under Stochastic Volatilities". Journal of Financial Economics, 19: 351-72.
The article was delivered to the editor's office on 22.01.09.
Keywords: Equity returns, non-normality, options pricing, Edgeworth expansion.
