Does Skewness Matter? Evidence from the Index Options Market

Madhu Kalimipalli School of Business and Economics

Wilfrid Laurier University Waterloo, Ontario, Canada N2L 3C5

Tel: 519-884-0710 (ext. 2187) mkalimip@wlu.ca

Ranjini Sivakumar Centre for Advanced Studies in Finance

School of Accountancy University of Waterloo

Waterloo, Ontario, Canada N2L 3G1 Tel: 519-888-4567 (ext. 5703)

rsivakum@uwaterloo.ca

(preliminary version, April 2002)

Abstract We model the temporal properties of the first three moments of asset returns and examine if incorporating time varying skewness in underlying asset returns leads to profitable strategies using at-the-money S&P 500 index options. We devise trading rules that incorporate the skewness forecasts to trade in at-the-money delta-neutral strips, straps and straddles. We find that a simulated trading strategy using the GARCHS (skewness) model outperforms the GARCH model both before and after adjusting for transaction costs. The empirical evidence indicates that index option prices for at-the-money options do not reflect time varying skewness. Our results suggest that mispricing of options causes the negative skewness in the implicit risk-neutral distribution in option prices.

Key Words: conditional volatility and skewness, option pricing biases, at-the-money

delta-neutral strips, straps and straddles

JEL Classification: G10, G14

Does Skewness Matter? Evidence from the Index Options Market

1. Introduction:

Existing literature has documented significant time varying skewness in stock index

returns (Harvey and Siddique, 1999 and 2000 and Hansen, 1994). The natural

development of skewness persistence models is an extension of volatility persistence

models and a direct consequence of asset pricing equations that contain third central

return moments. Harvey and Siddique (1999) find strong evidence for time varying

variance and skewness in monthly and weekly stock index data. Inclusion of conditional

skewness is found to attenuate asymmetric variance and seasonality effects in conditional

moments and lead to lower persistence in the variance equation.

There is a significant empirical evidence (see e.g. Bates, 1996b for a summary) that

the Black-Scholes valuation model exhibits pricing biases across moneyness and

maturity. Bates (1991) shows that the out-of-the-money (OTM) puts became very

expensive relative to OTM money calls during the year preceding the stock market crash

in October 1987 as skewness premium implicit in OTM money options on S & P 500

futures became significantly negative. The negative skewness premium results in a

“smirk” pattern in index volatilities. In addition, Bates (2000) documents significant time

varying skewness in stock index option data.

The interesting question is how does the conditional skewness in the asset returns

affect the underlying risk neutral pricing distribution used in option valuation? Jackwerth

and Rubinstein (1996) document that in the pre 1987 period, both the risk-neutral

distribution (option implied distribution) and the actual distribution of S&P 500 returns

2

are about lognormal. However in the post 1987 period, while the actual distribution

looks about lognormal again, the risk-neutral distribution is left-skewed and leptokurtic.

Bates (2000) suggests three explanations for the negative skewness in the implicit risk-

neutral distribution. The first is that investors view the underlying stochastic process for

S&P 500 returns has changed, the second is a change in investor’s risk aversion and the

third reason being a mispricing of post-crash options. Bakshi, Cao and Chen (1997) and

Bates (2000) among others look at the first explanation and propose option valuation

models that incorporate the asymmetry in the risk neutral pricing distribution. Jackwerth

(2000) looks at the second explanation. He empirically derives risk aversion functions

implied by option prices and realized returns on the SP500 index for the period 1986-

1995. In the post 1987 period, he finds negative risk aversion functions that are

inconsistent with economic theory and concludes that the market misprices the options.

Bakshi et al. (1997) examine options on the S&P 500 index during the period 1988-

1991. Their empirical evidence suggests that overall a model with stochastic volatility

and random jumps is superior to the Black-Scholes model. Interestingly, they find that

for at-the-money (ATM) options, the Black Scholes model is superior to the more

complex models that include the stochastic volatility model with jumps (Bakshi et al.

1997 and Bates, 2000). Specifically, in the out-of-sample cross-sectional performance,

they find that ATM call options (moneyness between 0.97 and 1.03), valued using the BS

model do not show any maturity related bias.

In this paper, we investigate whether it is mispricing that causes the negative

skewness in the implicit risk-neutral distribution. We model the temporal properties of

the first three moments of asset returns following Hansen (1994) and Harvey and

3

Siddique (1999) and examine if incorporating time varying skewness in underlying asset

returns leads to profitable option based strategies. We examine S&P 500 index options

data during the period November 1998 to March 2000. Based on the Bakshi et al. (1997)

findings, we assume that the Black-Scholes model is the appropriate option valuation

model and ask whether embedding skewness in spot pricing models leads to profitable

strategies using ATM options.

We use a framework proposed by Noh, Engle and Kane (1994) to estimate the profits

from the options trading strategies. Noh et al. (1994) show that simple GARCH models

(that incorporate time varying volatility) outperform implied volatility models for

investors trading in at-the-money straddles, after accounting for transaction costs. We use

the GARCHS (GARCH with conditional skewness) model as in Harvey and Siddique

(1999) and obtain the latent volatility and skewness from spot data. The GARCHS

trading strategy leads to trading in a strip or a strap. When conditional skewness is indeed

constant, the GARCHS reduces to a GARCH model and both models should yield similar

returns.

We find that a simulated trading strategy using the GARCHS (skewness) model

outperforms the GARCH model both before and after adjusting for transaction costs. The

empirical evidence suggests that index option prices for ATM options do not reflect time

varying skewness. Our results suggest that mispricing of options causes the negative

skewness in the implicit risk-neutral distribution.

This paper is organized as follows. In section 2, we provide a brief literature review.

In section 3, we describe the data and provide the sample description statistics. In section

4

4, we discuss the empirical methodology and present the results on the volatility models.

In the next section we present the results on the trading strategies. Section 6 concludes.

2. Background and Literature Review:

What causes skewness or asymmetry in returns? There are at least four possible

explanations in the literature. The first explanation is the “leverage effect” whereby a

drop in stock price leads to higher operating and financial leverage and hence high

volatility in subsequent returns (Black, 1976). The second is based on the “volatility

feedback mechanism” whereby the direct effect of a positive shock on volatility is

mitigated by an increase in risk premium, while in the presence of negative shock both

direct and indirect effects work to increase the risk premium (Campbell and Hentschel,

1992). The third explanation is based on a possible bursting of a “bubble”, a low

probability scenario with large negative consequences (Blanchard and Watson, 1982).

Finally investor heterogeneity and short sale constraints of investors explain skewness

(Hong and Stein, 1999).

Hansen (1994) provides a model of skewness evolution in the context of conditional

density estimation using a skewed Student-t distribution. He proposes a model of

skewness that evolves much like a GARCH process in squares of cubed residuals and

applies the approach to the estimation of US Treasury securities and the US dollar/Swiss

Franc exchange rate. He finds evidence of skewness persistence. Harvey and Siddique

(1999) adapt Hansen's approach to a wide number of daily and monthly equity return

series. Harvey and Siddique (2000) introduce skewness in the CAPM framework by

expressing the stochastic discount factor or inter-temporal marginal rate of substitution as

5

a quadratic function of the market return. They find that the coskewness factor (defined

as that part of an asset’s skewness that is related to market portfolio’s skewness) has

value in cross-sectional CAPM regressions across assets. This is in addition to size and

book-to-market factors that were proposed by Fama and French (1992). The momentum

effect in portfolios is found to be related to the systematic skewness factor. The question

that follows is what does a negatively skewed empirical distribution imply for the

implicit risk-neutral distribution in option prices. We next review some of the options

related literature that looks at this issue.

Bates (1991) shows that the out-of the money puts became very expensive during the

latter half of 1986, remained so until early 1987 and again during August of 1987 as

skewness premium implicit in out-of- the money options on S & P 500 futures became

significantly negative. No such effects were found during the months immediately

preceding the October 1987 crash. Following the 1987 crash, the negative skewness

premium continued to be significant till the end of 1987. Citing the specification of the

underlying stochastic process as a possible reason for the skewness premium, the paper

introduces a diffusion model with systemic jump risk to capture the time varying

skewness in the data.

Using a jump-diffusion model, Bates (1996a) finds a significant positive implicit

skewness in currency options on Deutsche mark during the period 1984-87, but not from

1988-91. The paper shows that a stochastic volatility (SV) model with jumps can explain

high kurtosis and skewness across different option maturities. Bakshi et al. (1997)

propose an option pricing model with stochastic volatility, stochastic interest rates and

random jumps. Their empirical evidence suggests that a model with stochastic volatility

6

and random jumps is superior to the Black-Scholes model. Bates (2000) again considers

a SV model now with time varying jumps to explain the skewness implicit in the S & P

500 futures option markets. The paper shows that models with SV or a negative

correlation between returns and volatility alone are not sufficient to generate the high

negative skewness or high volatility of volatility in the data.

In related research on the underlying stochastic process, Heston and Nandi (2000)

point out that a GARCH option valuation model that captures the negative correlation of

spot returns with volatility and the historical information in volatility model results in

reduced moneyness and maturity biases in option valuation. They also show that the

GARCH option valuation model is superior to an ad-hoc (smoothed) Black-Scholes

model proposed by Dumas, Fleming and Whaley (1998).

Chen, Hong and Stein (1999) using a panel data of U.S firms, find that negative

skewness is most pronounced in stocks with high past trading volume and returns and for

larger sized stocks. Bakshi, Kapadia and Madan (2000) show that risk-neutral

distributions for individual stocks differ from that of the market index by being far less

negatively skewed and substantially more volatile.

Jackwerth (2000) rules out changes in investor risk aversion as a reason for the

negative skewness and suggests mispricing as a possible reason. We explore this

explanation in this paper.

3. Data and Sampling Procedure:

In this study, we use S&P 500 daily options data and daily index levels from October

1998 to March 2000. We examine the S&P500 index options data because these options

7

are widely traded. For each day, we use the closing option price and the closing index

level as reported in the Datastream International database. We assume that the S&P 500

daily dividend yield interpolated to match the maturity of the option contract is a

reasonable proxy for the dividends paid on each option contract. We use the six-month

Treasury-bill rate as a proxy for the risk-free rate in the Black-Scholes valuation model.

Only options with moneyness (strike price/ index level) in the range 0.80 to 1.20 are

included. Options with maturity less than fifteen days and greater than 180 days are

excluded. Only options with daily volumes greater than 100 are retained. For a given

exercise price and maturity, only options that have both put and call prices are retained.

Options that violate the put-call parity relationship are excluded. Since the option market

closes after the stock market, the option holder has a wildcard option. As in Noh et al.

(1994), we ignore the wildcard option, understating the profits from the trading rules.

Based on these criteria, our sample consists of 1,742 call-put options pairs in 271 trading

days.

Figure 1 presents the weekly S & P 500 index data for the period 1970-2001. We see

that the index surged from mid 90’s onwards and peaked in the year 2000 followed by a

decline .

Table 1 presents the summary statistics of the weekly S & P 500 index data for the

period 1970-2001. In general we see that volatility, skewness and kurtosis have been

varying over time and have been high during the periods of oil shocks in the 70s, the

1987 crash period and more recently during 2001. The sample period for our index

options (Nov 1998-Mar 2001) seems to be characterized by particularly high volatility

compared to the historical average.

8

Table 2 presents the summary statistics of the daily S & P 500 index data for the

period 1990-2001. In general we see that volatility, skewness and kurtosis vary over the

week and are usually high on Mondays compared to the rest of the week.

Panel A in Table 3 presents the Augmented Dickey-Fuller unit root tests for the daily

and weekly time series data. We cannot reject the unit root null hypothesis for the index

data. The first differencing however seems to gives us the stationary return series. Panel

B presents the Ljung Box statistics for the squared AR(1) return residuals. They indicate

high auto-correlations in the daily and weekly data that imply time dependence in higher

order moments such as GARCH effects.

Figures 2 and 3 present the density functions of the weekly and daily time series

index data. We see large negative skewness and fat tails in the data.

4. Results from Conditional Volatility Models:

In this section we describe the conditional volatility models and their results based on

the time series index data. We use the GARCH(1,1)-M with leverage conditional

skewness and degrees of freedom –referred to as GARCHS(1,1) model- as the omnibus

specification. Hansen (1994) obtains a density function for a random variable driven by

its skewness and degrees of freedom in addition to the first two moments (details in the

Appendix).This specification is very general and it reduces to several known

distributions as special cases. The GARCHS(1,1) specification is described below.

9

The above specification is the GARCH (1,1)–in mean model with leverage effect and

time varying conditional skewness and degrees of freedom (df). We refer to this model as

Model 6 in our tables.

Model 1 has the usual GARCH(1,1) specification and is obtained by setting df in

Model 6 to a high number above 30 and by constraining skewness, leverage and lagged

variance effect in the mean to 0. Model 2 is the GARCH(1,1)-M with leverage effect

and is obtained by setting df in Model 6 to a high number above 30 and by constraining

skewness to 0. Model 3 is the EGARCH(1,1)-M model.

Model 4 is obtained by constraining the conditional df equation in Model 6 to

have intercept only and its skewness to zero. Model 5 is obtained by constraining the

conditional df and skewness equations in Model 6 to only have intercepts.

Table 4 presents the results for weekly index data for the period 1970-2001.

From the Panel A in table 4 we see that there is a high persistence in variance equation

and a strong evidence for leverage and skewness in the data. Fat tails are driven by large

(predominantly) negative shocks to the returns as evidenced by significant coefficient on

lagged squared residual in the df equation. The evidence for the risk premium in the

mean equation (the GARCH –in-mean effect) is rather weak. Panel B tells us that the

0 10 0

),|(~)|( ,

ˆˆ6 Model

1

11

212110

212110

2113

21210

1

12110

<≥

=

++=

++=

+++=

Ω=

+++=

−

−−

−−

−−

−−−

−

−−

t

tt

ttt

ttt

ttttt

ttttt

tttt

uifuif

d

uSkudf

u duhh

Zghu

uhrr

δεδδγεγγ

ββββ

ληεε

ααα

10

Model 6 outperforms others in terms of highest likelihood, AIC and SBC values. Model 2

and 3 come out as winners in terms of Jarque- Bera metric. The likelihood ratio metric

for nested specifications confirms that Model 6 is a definite improvement over models

1,2 and 4. However there is not much improvement over Model 5 .

Table 5 presents the results for daily index data for the period 1970-22-01. From

the Panel A in table 5 we see that there is a high persistence in variance equation and a

strong evidence for leverage and skewness in the data. The evidence for the risk premium

in the mean equation (the GARCH –in-mean effect) is rather weak. Panel B tells us that

the Model 6 outperforms others in terms of highest likelihood, AIC and SBC values.

Model 2 and 3 come out as winners in terms of Jarque- Bera metric. The likelihood ratio

metric for nested specifications confirms that Model 6 is a definite improvement over

models 1,2 4 and 5 .

Figure 4 plots returns, latent conditional volatility, skewness and degrees of

freedom from the conditional skewness model –Model 6- for the S & P 500 index weekly

series 1970-2001 and figure 5 has a similar plot for the daily index data for the period

1994-2001. In general we find that periods of high volatility are also periods of high

negative skewness and fatness in the return distributions.

5. Results for Option Trading Strategies:

Table 6 presents the summary statistics of S & P 500 index options data for the period

Nov 1998-Mar 2000. In general puts are cheaper relative to calls and trade more heavily.

At-the-money options (ATMs) also seem to have a shorter maturity (about 1.3 months)

compared to out-of the money options - OTMs (about 2-2.5 months).

11

Table 7 presents the results for delta-neutral straddles based on competing models for

S & P 500 index options data for the period Nov 1998-Mar 2000. Panel A shows us that

the put prices are much higher relative to the call prices. Model 1 (GARCH (1,1)–M with

normal distribution for the error term) comes closest to the actual market prices of calls

and put ands straddles, while the Model 4 (GARCH (1,1)–M with unconditional

skewness for the error term) gives us the lower bounds. In general the model prices are

much lower compared to the option prices implying that options are over priced. Panel B

(table 7) gives us the number of buys and sells of the delta- neutral straddles for

competing models. We find that in general straddles are sold in 74% of the trades and

purchased in the remaining 26%. Models 3 and 4 (both with t distributions for the error

terms) involve larger short positions in straddles than other models.

Panel C (table 7) presents percentage returns on trading in the delta-neutral straddles

for competing models. We find that Model 1 beats the simple unconditional volatility

model. Moreover the conditional skewness model (Model 6) outperforms all other

models both before and after 0.25% transaction costs.

Panels D and E (table 7) replicate Panels B and C results using a $ 0.50-filter rule for

stock price changes. The filter represents the trading costs per straddle. Trading takes

place only if the absolute price deviation is greater than $0.50. We find that the numbers

of trades are now lower because of attrition due to the filter rule; the straddles are still

sold more often than they are bought. Model 1 now outperforms all others before and

after 0.25% transaction costs.

Figure 7 shows the % returns from straddle based on unconditional volatility plotted

over each day during the sample period . We find that returns are more or less stationary

12

around zero except for a few (four or five) extreme positive outliers that would induce

positive skewness in returns.

Next we turn to delta-neutral strips, straps and straddles. Figure 6 shows us the

differences between the straddles only strategy and that based on all strips, straps and

straddles.

Table 8 presents the results for delta-neutral strips and straps and straddles based on

competing models for S & P 500 index options data. Panel A shows us that Model 1

comes closest to the actual market prices of all the three option strategies straddles, while

the Model 4 gives us the lower bounds. In general the model prices are much lower

compared to the option prices implying that options are over priced. Panel B (table 8)

gives us the number of buys and sells of the delta- neutral strategies for competing

models. There is attrition in the actual number of trades from 269- this corresponds to

those trades that do not satisfy the trading decisions laid out in figure 6. In general we

find that strips, straps and straddles are sold in 74% of the trades and purchased in the

remaining 26%. The buys and sells are now spread over strips, straps and straddles

unlike straddles only in table 7.

Panel C (table 8) presents percentage returns on trading in the delta-neutral strategies

for the competing models. We find that returns from both conditional skewness models,-

Models 5 and 6, outperform all the results reported in table 7 both before and after 0.25%

transaction costs. The t-statistics indicate that the returns from the strategy are

significantly different from zero.

Panels D and E (table 8) replicate Panels B and C results using a $ 0.50-filter rule for

stock price changes. We find that the numbers of trades are now lower because of

13

attrition due to the filter rule; the number of sells still overwhelms the number of buys.

Returns from both skewness models still outperform all others reported in table 7 before

and after 0.25% transaction costs.

6. Summary and Conclusions:

We investigate whether it is mispricing that causes the negative skewness in the

implicit risk-neutral distribution in S&P 500 index option prices. We model the temporal

properties of the first three moments of asset returns following Hansen (1994) and

Harvey and Siddique (1999) and examine if incorporating time varying skewness in

underlying asset returns leads to profitable strategies using at-the-money options. We

find that a simulated trading strategy using the GARCHS (skewness) model outperforms

the GARCH model both before and after adjusting for transaction costs. The empirical

evidence suggests that index option prices for ATM options do not reflect time varying

skewness. Our results suggest that mispricing of options causes the negative skewness in

the implicit risk-neutral distribution.

14

Appendix: Conditional Skewness Model:

The GARCHS (1,1) specification for the conditional mean, conditional variance and conditional skewness, where the error term in the mean has a skewed conditional student t distribution with changing degrees of freedom, is as follows: Conditional mean: tttt uhrr +++= −− 12110 ˆˆˆ ααα

where, ttt hu ε= and ( )ληε ,|~| 1 zgtt −Ω

where g ( ) is as described below.

Conditional variance: 2113

212110 −−−− +++= ttttt uduhh ββββ

where,

<≥

=−

−− 01

00

1

11

t

tt uif

uifd

Conditional skewness 212110 −− ++= ttt uSk δεδδ

Degrees of freedom: 212110 −− ++= ttt udf γεγγ where ∞<< df2

The likelihood function for the skewed t distribution (Hansen 1994) is:

−≥

++

−+×

−<

−+

−+×

= +−

+−

bazabzcb

bazabzcb

zg2

12

21

2

1211

1211

),|( η

η

λη

ληλη

where the eta stands for the degrees of freedom and is bounded as ∞<<η2 and lambda

is the skewness parameter and is bounded as 11 <<− λ . Further the constants a, b and c

are as defined below.

15

Γ−

+Γ

=

−+=

−−=

2)2(

21

31

124

222

ηηπ

ηληηλ

c

ab

ca

Hansen (1994) show that this density function has a zero mean and unit variance.

Setting lambda to zero gives us a regular t-distribution and setting eta to a high number

over 30 and lambda to zero gives us a regular standard normal distribution.

16

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18

Table 1

Summary statistics based on weekly S & P 500 index returns 1970-2001

Year Number of obs.

% mean weekly returns

% median weekly returns

% annualized standard deviation

skewness kurtosis

1970 52 0.002 0.250 17.458 -0.275 2.363 1971 51 0.190 0.454 12.507 0.325 3.595 1972 51 0.239 0.063 9.845 -0.167 2.333 1973 51 -0.405 -0.372 16.831 0.095 1.874 1974 51 -0.768 -0.518 29.611 0.399 2.817 1975 51 0.475 0.324 17.062 0.073 2.179 1976 52 0.327 0.278 11.766 0.403 2.761 1977 51 -0.198 -0.279 10.589 0.430 3.205 1978 51 0.073 0.073 13.627 -0.393 3.112 1979 51 0.178 0.353 12.216 -0.503 3.442 1980 51 0.501 0.807 18.044 -0.560 3.240 1981 52 -0.197 -0.207 13.784 0.008 3.162 1982 51 0.324 0.032 20.625 0.733 3.042 1983 51 0.248 -0.007 12.038 0.450 2.455 1984 51 -0.036 -0.365 14.108 0.355 3.404 1985 51 0.456 0.646 10.049 -0.231 2.768 1986 51 0.320 0.882 15.264 -1.104 5.027 1987 52 0.039 0.576 27.444 -2.238 11.226 1988 51 0.133 -0.073 14.608 -0.382 3.210 1989 51 0.441 0.737 10.638 -0.433 3.055 1990 51 -0.157 -0.018 17.283 -0.644 3.845 1991 51 0.449 0.318 13.790 0.169 3.185 1992 52 0.083 0.069 8.329 0.598 3.852 1993 51 0.165 0.162 8.460 -0.517 4.040 1994 51 -0.025 0.190 10.272 -0.568 3.718 1995 51 0.565 0.621 7.816 -0.315 2.748 1996 51 0.396 0.917 13.072 -0.490 2.592 1997 51 0.462 0.438 15.813 -0.013 2.393 1998 52 0.455 1.017 19.286 -0.128 4.000 1999 51 0.280 0.289 16.645 -0.094 2.220 2000 51 -0.099 -0.124 19.936 -0.195 2.735 2001 51 -0.278 0.278 21.945 -0.869 4.193

Whole period 1990-2001

1682 0.150 0.217 15.890 -0.552 7.019

Time series data 1994-2001

417 0.219 0.363 16.306 -0.506 4.596

Options data Nov 1998-Mar

2001

71 0.331 0.419 17.372 -0.119 2.089

19

Table 2 Summary statistics based on daily S & P 500 index returns 1990-2001

Number of obs.

% mean daily

returns

% median daily returns

% annualized standard deviation

skewness kurtosis

Monday 626 0.092 0.033 15.879 -1.438 13.074 Tuesday 626 0.054 0.000 15.987 0.372 6.046

Wednesday 626 0.028 0.035 14.058 0.284 5.764 Thursday 626 0.006 0.000 15.338 0.266 5.769

Friday 626 0.008 0.034 15.723 -0.572 6.025 Whole period:

1990-2001 1682 0.038 0.011 15.412 -0.254 7.530

20

Table 3 Unit root and GARCH tests based on the weekly S & P 500 index data 1970-2001 and daily S & P 500 index data 1990-2001

Panel A: ADF tests based on regressions with intercept

weekly data daily data Index 0.7743 -0.6908 Returns -23.2768 -21.9143 We report the ADF test statistics for the gamma coefficient for the following regression. The null of unit root is represented as γ=0. The critical value is –2.86 at 95% confidence level. Panel B: Ljung-Box test statistic value for the squared AR(1) residuals from return series

Lag Ljung-Box Statistic

Weekly data 1970-2001

Ljung-Box Statistic

Daily data 1994-2001

χ2(lag) statistic

( 95% confidence level)

6 23.6137 58.6133 12.60 7 22.0310 54.5005 14.07 8 21.4673 50.9705 15.51 9 19.5214 49.0633 16.92 10 18.8460 46.6977 18.31 12 17.4505 43.5050 21.00 18 12.3322 35.3900 28.90

We report the Ljung-Box statistic for the squared residuals from the AR(1) return process at different lags. The Ljung-Box statistic for squared residuals is significant for daily data and weekly data up to lag 10.

ti

ititt yyy εβγα +∆++=∆ ∑=

+−−

8

2110

21

Table 4 Estimates of competing conditional volatility and skewness models based on weekly S & P 500 index data 1970-2001

Panel A: Model estimates Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Mean- Equation: intercept 0.211

(0.046) 0.099

(0.088) 0.030

(0.074) 0.101

(0.084) 0.107

(0.076) 0.119

(0.077) rt-1 -0.043

(0.026) -0.020 (0.026)

-0.011 (0.021)

-0.019 (0.039)

-0.035 (0.026)

-0.029 (0.025)

ht-1 0.014 (0.02)

0.026 (0.019)

0.021 (0.019)

0.013 (0.017)

0.010 (0.016)

Variance Equation:

intercept 0.191 (0.078)

0.312 (0.073)

0.095 (0.022)

0.261 (0.079)

0.226 (0.078)

0.236 (0.075)

ht-1 0.843 (1.161)

0.812 (0.25)

0.935 (0.015)

0.838 (0.361)

0.850 (0.39)

0.838 (0.364)

Ut-12 0.122

(0.401) 0.024 (0.15)

0.029 (0.17)

0.033 (0.165)

0.038 (0.169)

It-1*Ut-12 0.193

(0.039) 0.145

(0.04) 0.131

(0.037) 0.146

(0.042) |εt-1|-sqrt(2/π) 0.199

(0.027)

εt-1 -0.126 (0.021)

Degrees of freedom Equation:

intercept -0.386 (0.657)

-0.163 (0.477)

0.431 (0.516)

Ut-1 -0.055

(0.052) Ut-1

2 -0.047 (0.005)

DF: 13.392 17.18 Skewness Equation:

intercept -0.389 (0.086)

-0.327 (0.092)

Ut-1 0.021

(0.012) Ut-1

2 -0.012 (0.006)

SK 0.173 # of

parameters 5 7 7 8 9 13

Log likelihood

-3592.500 -3571.982 -3565.945 -3561.540 -3551.099 -3547.920

T : 1680. Standard errors in parentheses

22

Panel B: Model Comparisons

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 skewness -0.468 -0.374 -0.397 -0.396 -0.408 -0.398 kurtosis 4.408 3.815 3.845 3.946 3.994 3.917

JB statistic 200.167 85.519 94.120 106.449 115.830 103.268 AIC -3597.500 -3578.982 -3572.945 -3574.540 -3564.099 -3560.920 SBC -3597.018 -3576.668 -3570.631 -3566.535 -3556.095 -3552.915

LR statistic 89.16*** 48.12*** - 27.24*** 6.36 - *** all significant at 1% significance level

23

Table 5 Estimates of competing conditional volatility and skewness models based on daily S & P 500 index data 1994-2001

Panel A: Model estimates Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Mean- Equation: intercept 0.066

(0.018) 0.025

(0.029) 0.003

(0.032) 0.044

(0.031) 0.034

(0.029) 0.032

(0.023) rt-1 0.043

(0.023) 0.070

(0.025) 0.068

(0.024) 0.032

(0.024) 0.030

(0.021) 0.064

(0.023) ht-1 0.016

(0.036) 0.041

(0.034) 0.020

(0.044) 0.015

(0.033) 0.022

(0.024)

Variance Equation: intercept 0.007

(0.363) 0.018

(0.255) 0.000

(0.004) 0.013

(0.018) 0.014

(0.016) 0.014

(0.016) ht-1 0.927

(.2.521) 0.904

(0.241) 0.970

(0.007) 0.921

(0.309) 0.920

(0.276) 0.914

(0.264) Ut-1

2 0.067 (0.651)

0.000 (0.109)

0.000 (0.032)

0.000 (0.022)

0.000 (0.013)

It-1*Ut-12 0.161

(0.024) 0.131

(0.022) 0.134

(0.022) 0.146

(0.024) |εt-1|-sqrt(2/π) 0.130

(0.017)

εt-1 -0.132 (0.016)

Degrees of freedom Equation:

intercept -1.644 (0.254)

-1.534 (0.27)

-1.476 (0.29)

Ut-1 -0.132

(0.138) Ut-1

2 -0.021 (0.029)

DF: 6.616 7.049 Skewness Equation:

intercept -0.159 (0.07)

-0.174 (0.074)

Ut-1 0.304

(0.065 Ut-1

2 0.045 (0.022)

SK -0.07 # of

parameters 5 7 7 8 9 13

Log likelihood

-2828.796 -2787.217 -2777.152 -2742.340 -2739.618 -2728.160

T : 2083. Standard errors in parentheses

24

Panel B: Model Comparisons

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 skewness -0.531 -0.492 -0.420 -0.518 -0.518 -0.504 kurtosis 5.371 4.889 4.596 5.059 5.053 4.976

JB statistic 585.560 393.869 282.178 461.228 459.213 426.885 AIC -2833.796 -2794.217 -2784.152 -2755.340 -2752.618 -2741.160 SBC -2833.422 -2792.011 -2781.946 -2747.443 -2744.721 -2733.263

LR statistic 201.28*** 118.12*** - 28.36*** 22.92*** -

*** all significant at 1% significance level

25

Table 6 Summary statistics of S & P 500 index option data Nov 1998-Mar 2000

Panel A: Averages

average maturity (days)

average prices $

average moneyness (X/S)

average volume

ATM calls 40.9 40.71 1.0004 1858.64 ATM puts 40.9 34.82 1.0004 2320.20 OTM calls 60.1 32.03 1.0318 1372.58 OTM puts 72.7 28.70 0.9481 1601.58

Panel B: Medians

median maturity (days)

median prices $

median moneyness (X/S)

median volume

ATM calls 35.5 38.00 1.001 1296.50 ATM puts 35.5 33.44 1.001 1420.00 OTM calls 49.5 29.00 1.0232 874.50 OTM puts 57.5 27.00 0.9581 931.50

26

Table 7 Delta-neutral straddles based on competing models for S & P 500 index option data Nov 1998-Mar 2000

Panel A: Average prices of the delta-neutral straddles for competing models

call put straddle market price 29.38 42.89 72.26 Unconditional volatility 25.34 39.80 65.14

Model 1 GARCH(1,1)-M 25.88 40.34 66.21 Model 3 EGARCH(1,1)-M 25.24 39.70 64.95 Model 4 GARCH(1,1)-M+ df 25.06 39.52 64.58 Model 6 GARCH(1,1)-M +cdf+cskew 25.61 40.07 65.68

Number of observations 270. Average moneyness and maturity of the delta-neutral straddles are 1.0144, and 36.07 days respectively

Panel B: Number of buys and sells of delta-neutral straddles for the competing models

total trades buys sells Unconditional volatility 269 70 199

Model 1 GARCH(1,1)-M 269 69 200 Model 3 EGARCH(1,1)-M 269 50 219 Model 4 GARCH(1,1)-M (t-distn) 269 54 215 Model 6 GARCH(1,1)-M +cdf+cskew 269 68 201

Panel C: % Returns on trading in the delta-neutral straddles for competing models

Before transaction costs After transaction costs of 0.25%

% daily return % daily return # of

obs mean median std.

dev t-stat mean median std. dev t-stat

Unconditional volatility 269 3.77 1.23 26.10 2.37 3.41 0.88 26.06 2.15 Model 1: GARCH(1,1)-M

269 3.98 1.23 26.29 2.48 3.62 0.96 26.26 2.26

Model 3: EGARCH(1,1)-M

269 2.36 1.12 26.57 1.46 2.00 0.74 26.53 1.24

Model 4: GARCH(1,1)-M+df

269 3.12 1.46 26.11 1.96 2.76 1.06 26.07 1.74

Model 6: GARCH(1,1)-M +cdf+ cskew

269 4.09 1.46 27.81 2.41 3.72 1.06 27.77 2.20

27

Panel D: Number of buys and sells of delta-neutral straddles for the competing models with $0.50 filter for stock prices

total trades buys sells Unconditional volatility 255 63 192

Model 1 GARCH(1,1)-M 253 59 194 Model 3 EGARCH(1,1)-M 249 40 209 Model 4 GARCH(1,1)-M (t-distn) 252 47 205 Model 6 GARCH(1,1)-M +cdf+cskew 256 61 195

Panel E: % Returns on trading in the delta-neutral straddles for competing models with $0.50 filter for stock prices

Before transaction costs After transaction costs of 0.25%

% daily return % daily return # of

obs mean median std.

dev t-stat mean median std. dev t-stat

Unconditional volatility 255 3.62 0.53 25.55 2.32 3.28 0.19 25.51 2.11 Model 1: GARCH(1,1)-M

253 3.66 0.53 25.22 2.38 3.32 0.19 25.18 2.16

Model 3: EGARCH(1,1)-M

249 1.84 0.19 25.26 1.19 1.51 0.00 25.22 0.98

Model 4: GARCH(1,1)-M+df

252 3.36 1.10 25.69 2.14 3.02 0.70 25.65 1.93

Model 6: GARCH(1,1)-M +cdf+ cskew

256 3.30 0.75 25.99 2.08 2.95 0.38 25.95 1.87

28

Table 8 Delta- neutral strips and straps based on the conditional skewness model for S & P 500 index option data Nov 1998-Mar 2000

Panel A: Average prices of the delta-neutral strips, straps and straddles for competing models

call put straddle market price 72.26 101.64 115.15 Unconditional volatility 65.14 90.48 104.94

Model 1 GARCH(1,1)-M 66.21 92.09 106.55 Model 3 EGARCH(1,1)-M 64.95 90.19 104.65 Model 4 GARCH(1,1)-M+ df 64.58 89.64 104.10 Model 5 GARCH(1,1)-M+ df+ skew 64.97 90.23 104.69 Model 6 GARCH(1,1)-M +cdf+cskew 65.68 91.30 105.76

Number of observations 270. Average moneyness and maturity of the delta-neutral straddles are 1.0144, and 36.07 days respectively

Panel B: Number of buys and sells of delta-neutral strips, straps and straddles for the competing models

trading in straddles trading in strips trading in straps total trades buys (%) sells(%) buys (%) sells(%) buys (%) sells(%)

Model 5: GARCH(1,1)-M +df+skew

268 3 0 28 97 33 107

Model 6: GARCH(1,1)-M +cdf+cskew

265 2 0 23 107 45 88

Panel C: % Returns on trading in the delta-neutral strips, straps and straddles for competing models

Before transaction costs After transaction costs of 0.25%

% daily return % daily return # of

obs mean median std.

dev t-stat mean median std. dev t-stat

Model 5: GARCH(1,1)-M+df + skew

268 4.95 2.23 28.67 2.83 4.92 1.91 29.20 2.79

Model 6: GARCH(1,1)-M +cdf+ cskew

265 4.27 1.85 28.93 2.42 4.25 1.53 29.46 2.39

29

Panel D: Number of buys and sells of delta-neutral strips, straps and straddles for the competing models with $0.50 filter for stock prices

trading in straddles trading in strips trading in straps total trades buys (%) sells(%) buys (%) sells(%) buys (%) sells(%)

Model 5: GARCH(1,1)-M +df+skew

256 0 0 24 97 29 106

Model 6: GARCH(1,1)-M

+cdf+cskew

256 0 0 20 107 43 86

Panel E: % Returns on trading in the delta-neutral strips, straps and straddles for competing models with $0.50 filter for stock prices

Before transaction costs After transaction costs of 0.25%

% daily return % daily return # of

obs mean median std.

dev t-stat mean median std. dev t-stat

Model 5: GARCH(1,1)-M+df + skew

256 4.43 1.47 28.09 2.59 4.41 1.10 28.63 2.56

Model 6: GARCH(1,1)-M +cdf+ cskew

256 4.34 1.63 28.69 2.48 4.31 1.33 29.22 2.45

30

Figure 1

S & P 500 index weekly series 1970-2001

31

Figure 2 Density function for the S & P 500 index weekly series 1970-2001

Figure 3 Density function for the S & P 500 index daily series 1990-2001

32

Figure 4 Plots of returns, latent conditional volatility, skewness and degrees of freedom from the conditional skewness model for the S & P 500 index weekly series 1970-2001

33

Figure 5 Plots of returns, latent conditional volatility, skewness and degrees of freedom from the conditional skewness model for the S & P 500 index daily series 1994-2001

34

Figure 6 Trading strategies involving options

35

Figure 7 Plots of the S & P 500 index ATM straddle prices, returns and maturity for the period Nov 1998-Mar 2000