the information content of the s&p 500 index and vix options on the dynamics of the s&p 500...

The authors are grateful to the National Science Council of Taiwan for the financial support provided for thisstudy.

*Correspondence author, Department of Finance, National Taiwan University, No. 85 Roosevelt Road,Section 4, Taipei 106, Taiwan. Tel.: 886-2-3366-1092, Fax: 886-2-8369-5581, e-mail: [email protected].

Received April 2011; Accepted April 2011

■ San-Lin Chung, Wei-Che Tsai, and Yaw-Huei Wang are at the Department of Finance,National Taiwan University, Taiwan.

■ Pei-Shih Weng is at the Department of Finance, National Central University, Jung-Li, Taiwan.

The Journal of Futures Markets, Vol. 31, No. 12, 1170–1201 (2011)© 2011 Wiley Periodicals, Inc.Published online May 27, 2011 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/fut.20532

THE INFORMATION CONTENT

OF THE S&P 500 INDEX AND

VIX OPTIONS ON THE

DYNAMICS OF THE S&P 500INDEX

SAN-LIN CHUNGWEI-CHE TSAIYAW-HUEI WANG*PEI-SHIH WENG

Given that both S&P 500 index and VIX options essentially contain informationabout the future dynamics of the S&P 500 index, in this study, we set out toempirically investigate the informational roles played by these two option marketswith regard to the prediction of returns, volatility, and density in the S&P 500index. Our results reveal that the information content implied from these twooption markets is not identical. In addition to the information extracted from theS&P 500 index options, all of the predictions for the S&P 500 index are signifi-cantly improved by the information recovered from the VIX options. Our findings

Information Content of the S&P 500 1171

Journal of Futures Markets DOI: 10.1002/fut

are robust to various measures of realized volatility and methods of density evalu-ation. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark 31:1170–1201, 2011

1. INTRODUCTION

The launch of VIX options took place on the Chicago Board Options Exchange(CBOE) on February 24, 2006, about two years after the introduction of VIXfutures. Since then, as a result of the increasing demand for practical marketrisk management, these options have become the most successful new productlaunch in the history of the CBOE.1 Thus, there is no doubt whatsoever thatVIX options play an extremely important role within the financial markets, par-ticularly in the aftermath of the sub-prime crisis.

VIX options are options that are written on the CBOE volatility index(VIX), an index that is compiled from the market prices of S&P 500 indexoptions as a means of approximating the expected aggregate volatility of theS&P 500 index during the subsequent 30 calendar-day period.2 The expecta-tions on the dynamics of the S&P 500 index are essentially contained not onlywithin S&P 500 index options, but also within VIX options, with the latteroffering the greater likelihood of providing traders with a more direct and effec-tive way of realizing their volatility information or hedging their volatility risk.

The information content of S&P 500 index options, which has been exten-sively explored in literally hundreds of studies, is found to be useful in deter-mining the future dynamics of the S&P 500 index. However, exploration intothe information content of VIX options with regard to the future dynamics ofthe S&P 500 index is still at an embryonic stage, despite the dramatic rise in theimportance of VIX options over recent years. We therefore contribute to the lit-erature by filling this gap through an examination of whether the informationembedded within the VIX options market can be used to improve the predictionof returns, volatility, and density in the S&P 500 index.3

1The trading volume of VIX options reached almost 4.5 million contracts during the first nine months of theirexistence. According to current data on trading activities in VIX options, during the first 10 months of 2009,the daily average trading volume had grown to more than 120,000 contracts.2When the VIX was first introduced in 1993, it was compiled from the implied volatility of eight S&P 100index options comprising near at-the-money, nearby and second nearby calls and puts; this was to reflect theimplied volatility of a 30 calendar day at-the-money option. Since 2003, however, the VIX has been calculat-ed from the prices of S&P 500 index options, with the calculation being undertaken based upon a model-freeformula with a wide range of strike prices, as opposed to only eight.3The present study is not, of course, the first to investigate the VIX options market; nevertheless, it is clearthat most, if not all, of the prior studies have tended to focus on the pricing of VIX options. Following the firstformula of Whaley (1993), derived from the concept of Black (1976), several studies (such as Carr & Lee,2009; Detemple & Osakwe, 2000; Grunbichler & Longstaff, 1996; Lin & Chang, 2009; Sepp, 2008a,b; )have subsequently gone on to explore the pricing of VIX options under different approaches, with Wang andDaigler (2010) empirically testing the pricing performance of alternative VIX option models.

1172 Chung et al.


Since several of the prior studies have provided support for the predictiveability of the VIX with regard to stock returns, the first channel for investigatingthe informational role of the VIX options market would clearly be to examinewhether the information implied in the VIX options market can furtherimprove the prediction of future returns on the S&P 500 index.4 In addition tothe prediction of returns on the S&P 500 index, a further channel for theexploration of the informational role of the VIX options market would involvean investigation into the incremental predictive ability of the informationimplied within the VIX options market for realized volatility on S&P 500 indexreturns, since the VIX is commonly found to be the best predictor; indeed, therelative success of option implied volatility has been documented in numerousstudies on volatility forecasting.5

In addition to investigating the forecasting of returns and volatility, anexploration of the informational role of VIX options with regard to density pre-diction in the S&P 500 index provides a further channel for the potentialenhancement of current knowledge on the information content implied withinthe VIX options market. This is an important task not only because density pre-diction of spot prices is useful in the risk management for many central banksand investment banks, as well as in policy decision-making, the pricing offinancial derivatives and the evaluation of the rationality of market prices, butalso because the density of the S&P 500 index has been satisfactorily predictedby the density estimated from the market prices of S&P 500 index options.6

Motivated by these studies proving the informativeness of the S&P 500index options with regard to future dynamics of stock prices, we first of all usepairs of VIX calls and puts to recover the VIX levels from put–call parity, andthen adopt regression models to examine whether this implied VIX containsany incremental information with regard to future returns and volatility on theS&P 500 index. Furthermore, we adopt the stochastic volatility (SV) model ofHeston (1993) to derive the pricing formulae for both the S&P 500 index and VIX options, to calibrate the parameters using the option market prices,

4Whaley (2000) proposed the VIX as an effective “fear” indicator. Giot (2005) found a strong negative corre-lation between contemporaneous changes, as compared with a positive relationship between the current lev-els of implied volatility indices and future market index returns. Guo and Whitelaw (2006) and Banerjee et al. (2007) also have similar findings.5See Xu and Taylor (1995), Fleming (1998), Blair, Poon, and Taylor (2001) and Jiang and Tian (2005). Fromtheir comprehensive review of studies on volatility forecasting, Poon and Granger (2003) concluded that theVIX is the best predictor of realized volatility, although it may be a biased one.6Based upon the theoretical results for complete markets derived by Breeden and Litzenberger (1978), exten-sive investigation has been carried out in many studies over recent years into density prediction with optionprices. A variety of methods are surveyed in Bahra (1997), Cont (1997), Jackwerth (1999), Jondeau andRockinger (2000), Bliss and Panigirtzoglou (2002), Liu et al. (2007) and Shackleton et al. (2010). It hasbeen demonstrated that most of these methods are likely to perform satisfactorily, providing that options aretraded with sufficient strike prices and that their range covers most of the distribution.



and then to compare the density predictions generated from these two optionsmarkets individually and jointly.7

Our empirical results reveal that the information content implied in theS&P 500 index and VIX option markets for the S&P 500 index are similar,although not identical. In addition to the information embedded in S&P 500index options, the information implied in VIX options can significantly improvethe predictive power on the returns, volatility, and density of the S&P 500index. Although the performance of VIX option implied information in volatili-ty forecasting is totally independent of market situations, its performance isparticularly effective in returns forecasting when the market is found to beextreme or volatile.

With regard to density prediction, we find that the volatility dynamic of theS&P 500 index recovered from VIX options is far more stable and reasonable.We also find that VIX options can significantly improve the density prediction ofthe S&P 500 index, particularly over longer horizons. In general, our results areconsistent with the argument of Becker, Clements, and McClelland (2009),that the VIX subsumes information to contribute to the jump component ofvolatility, since improvements made by VIX options can be achieved by includingjump components within the dynamics of price and/or volatility (Wang, 2009).

The remainder of this study is organized as follows. The option pricingmodel used in this study is described in Section 2, followed in Section 3 by adescription of the data adopted for our empirical analysis. Sections 4–6,respectively, provide details of the empirical methodologies adopted, the empir-ical results on the prediction of returns, and those on volatility and density.Tests for robustness are carried out and further discussed in Section 7. Finally,the conclusions drawn from this study are presented in Section 8.

2. A THEORETICAL PRICING MODEL FOR S&P500 INDEX AND VIX OPTIONS

The VIX is compiled from the market prices of S&P 500 index options for useas a measure of the volatility expectations of S&P 500 index returns. The feasi-bility of using the market prices of VIX options to extract information aboutfuture dynamics in the S&P 500 index is therefore largely dependent upon thetheoretical pricing model in which the S&P 500 index and VIX options aresimultaneously derived, and based upon the same assumptions for the dynam-ic process in the S&P 500 index.8

7In theory, if the prices of S&P 500 index and VIX options can be derived by assuming the same dynamicprocess for the S&P 500 index, the “risk-neutral density” (RND) and the corresponding “real-world density”(RWD) can be backed out from the market prices of either option.8See Detemple and Osakwe (2000), Sepp (2008a,b) and Lin and Chang (2009).

1174 Chung et al.


Considering the trade-off between model complexity and reasonableness,we adopt the SV model of Heston (1993) for the introduction of SV, even with-out jumps or other additional features, as the most crucial step during thedevelopment of option pricing models. The price dynamics of the S&P 500index in a risk-neutral measure are represented by

(1)

where St is the underlying S&P 500 index at time t, sv represents the volatilityof volatility, r is the risk-free rate, q is the continuous dividend yield, k is themean-reversion speed, u is the long-run mean level of variance, Vt is the vari-ance rate at time t, and ris the correlation coefficient between the two stochas-tic processes, and .

The theoretical price for a European-style call option on the S&P 500index under the SV model of Heston (1993) is given as

(2)

where K is the strike price and tis the time-to-maturity of the contract. Detailsof the two risk-adjusted probabilities, P1 and P2, are provided in Appendix A.

The next step is to derive the theoretical VIX option pricing formula basedon the underlying assumptions in Equation (1). Following the framework ofLin and Chang (2009), we obtain the option pricing formula of a VIX call asfollows:

(3)

where is the current VIX futures price, X is the strike price, and t is thetime-to-maturity. The risk-adjusted probabilities, �1 and �2, are

summarized in Appendix B.9 Similarly, one can derive the theoretical VIX putoption pricing formula as follows:

(4)PVIX0 (X, t) � e�rt[X(1 � �2) � VIXF

0(t)(1 � �1)].

VIXF0(t)

CVIX0 (X, t) � e�rt[VIXF

0(t)�1 � X�2]

C0(K, t) � S0e�qtP1 � Ke�rtP2

ZQvZQ

s

dVt � k(u � Vt)dt � sv2VtdZQv

dSt � (r � q)Stdt � 2VtStdZQs

9Here, we select VIX futures as the underlying asset of VIX options; however, in practice, VIX options traderswho do not fully understand the relationship between VIX and VIX option prices are often frustrated whenthe prices of options do not seem to follow the movement of the VIX. After a spike in the VIX level, VIXoptions often appear to be trading at a discount. The reasons for such behavior may be attributable to the factthat VIX options are European-style options, and also that the VIX is a mean-reverting index. Specifically,since VIX options are European-style options, they can only be exercised on their expiration date, and thus,their valuation is based on the expected (or forward) value of the VIX on the expiration date, as opposed tothe current or “spot” VIX value.



According to the framework of Zhang and Zhu (2006), the theoretical price ofVIX futures under the SV model with maturity t can be derived as

(5)

where and with t0 � 30/365.Following the derivation of Cox, Ingersoll, and Ross (1985) for the square rootprocess of instantaneous short rate, the transition probability density, fQ(Vt|V0),is as follows:

(6)

where , , , , andIq(�) is the modified Bessel function of the first kind, of order q.Given the market prices of a VIX call and the corresponding VIX put, we canapply the put–call parity to infer the VIX level from the VIX option market asfollows:10

(7)

where and denote the market prices of the VIX call and put,respectively. This VIX recovered from the put–call parity essentially reflects theconsensus of the participants within the VIX option market.

3. DATA DESCRIPTION

The primary data adopted for the present study comprise the end-of-day settle-ment prices of S&P 500 index and VIX options, both of which are European-style options traded on the CBOE. The daily levels of the S&P 500 index andVIX are also provided within the data set, which is obtained from the MarketData Express of CBOE and covers a sample period running from March 2006to June 2008.11

Filtering of the option prices was undertaken based upon the following cri-teria. First, we excluded those option settlement prices that violated any arbi-trage-free bounds, as well as those observations with either a bid or ask pricestated in points that were less than US$ 0.5 for S&P 500 index, and less thanUS$ 0.1 for VIX options, as these options prices could be found to be insensi-tive to the information contained within.

MPVIX0MCVIX

0

implied VIX � MCVIX0 (X, t) � Xe�rt � MPVIX

0 (X, t)

q � (2ku�s2v ) � 1v � cVtu � cV0e

�ktc � 2k�(s2v (1 � e�kt) )

f Q(Vt ƒ V0) � ce�u�va vub

q�2

Iq(22uv)

at0� (1 � e�kt0)�kt0bt0

� u(1 � ((1 � e�kt0)�kt0))

VIXF0(t) � EQ

0 (VIXt) � EQ0 (2bt0

� at0Vt) � �

�

0

2bt0� at0

Vt fQ(Vt ƒ V0) dVt

10This put–call parity differs from that for options on traded assts and is shown in Grünbichler and Longstaff(1996).11The market prices of VIX futures used to convert ITM VIX puts into OTM VIX calls for density predictioncan be freely downloaded from the CBOE website.

1176 Chung et al.


Second, in order to avoid the issue of liquidity being imposed upon shortermaturity (longer maturity) options, we excluded those options with a time-to-maturity of less than one week (greater than one year). Our sample ultimatelycomprised a total of 330,176 S&P 500 index options and 77,515 VIX options,with the option premiums of the former being found to be much higher.

The summary statistics for the S&P 500 index options are presented inTable I, whereas those for the VIX options are presented in Table II. As expect-ed, for both options markets, the call (put) price is found to have a positiveassociation with time-to-maturity and a negative (positive) association withmoneyness.12 In terms of time-to-maturity, for both the S&P 500 index and VIXoptions markets, more than 50% of the contracts are in the 30–180 category. Asfor moneyness, the proportion of near-the-money contracts for VIX options isfound to be much lower, whereas the proportion of deep moneyness contractsfor VIX options is found to be much higher than for S&P 500 index options. Inparticular, deep moneyness VIX calls (puts) account for 85.83% (80.87%) ofthe total contracts in their categories. This phenomenon may reveal thatinvestors trade VIX options more aggressively, while also bearing more risk.

Since OTM options are usually traded more actively due to the demand forhedging, we compare the number of contracts for OTM calls and puts, and findthat the OTM puts are traded much more than that of the OTM calls in theS&P 500 index option market. By contrast, OTM calls account for a higherpercentage as compared with the number of OTM puts in the VIX options mar-ket. The ratio of OTM calls over puts is 0.74 for the S&P 500 index options and1.85 for VIX options. Assuming that investors have greater demand for OTMS&P 500 index puts to hedge their spot risk, it seems reasonable to find moreOTM call contracts in the VIX option market.13

In addition to the prices of the options, the corresponding risk-free ratesused in this study are interpolated from the zero curve surfaces available in theOptionMetrics database, while the intraday S&P 500 index levels used to cal-culate realized volatility levels are obtained from Olsen Data AG. The summarystatistics of the VIX and the returns and realized volatility of the S&P 500 indexare presented in Table III.14 Consistent with the prior studies, the distributionof returns is negatively skewed and fat tailed, and the significance of the first-order autocorrelation is non-negligible.

12The moneyness is defined as the ratio of a strike price over the current index level.13The reason may be due to the fact that the market makers have net short positions on OTM S&P 500 indexputs and thus require net long positions on OTM VIX calls to hedge their volatility risk.14As proposed in Andersen et al. (2001), the daily realized volatility is calculated as the square root of the sumof the five-minute squared returns within a trading day. In order to minimize the estimation variance, Arealand Taylor (2002) proposed an alternative measure within which intraday squared returns were weightedaccording to the intraday pattern of realized volatility. We also adopt this measure within our robustnessanalysis as an alternative proxy for realized volatility.



According to the definition of VIX provided by the CBOE, VIX is a portfo-lio of S&P 500 index calls and puts and calculated as

VIX � B2T ai

¢Ki

K2i

eRTQ(Ki) �1Tc FK0

�1d 2

TABLE I

Summary Statistics of the S&P500 Index Options Data Set

Moneyness

Deep ITM ITM 0.9 � Near the Money OTM 1.03 � Deep OTMMaturity m 0.9 m 0.97 0.97 � m 1.03 m 1.1 m 1.1 Subtotals

A: Call options� 30 DaysNo. of contracts 16,199 8,243 8,067 5,111 1,173 38,793Share of total (%) 9.71 4.94 4.83 3.06 0.70 23.25Average price (US$) 280.17 89.07 21.66 1.82 0.37 140.67

30–180 DaysNo. of contracts 34,897 15,703 18,656 15,637 5,558 90,451Share of total (%) 20.91 9.41 11.18 9.37 3.33 54.21Average price (US$) 320.64 104.73 43.04 10.45 2.83 152.75

180 DaysNo. of contracts 16,329 4,648 4,652 5,251 6,733 37,613Share of total (%) 9.79 2.79 2.79 3.15 4.04 22.54Average price (US$) 383.81 150.91 94.74 46.99 12.67 205.82

SubtotalsNo. of contracts 67,425 28,594 31,375 25,999 13,464 166,857Share of total (%) 40.41 17.14 18.80 15.58 8.07 100.00Average price (US$) 326.21 107.72 45.21 16.14 7.53 161.90

B: Put options� 30 Days

No. of contracts 7,495 8,194 8,073 8,007 3,938 35,707Share of total (%) 4.59 5.02 4.94 4.90 2.41 21.86Average price (US$) 0.87 3.61 20.97 90.52 232.49 51.69


180 DaysNo. of contracts 15,440 4,649 4,649 5,251 7,320 37,309Share of total (%) 9.45 2.85 2.85 3.22 4.48 22.84Average price (US$) 13.10 44.22 69.37 115.84 272.52 89.35


This table presents details on the number of available contracts, the percentage of the total number, and the average price (in US$)of S&P 500 index options across various categories of moneyness for call (Panel A) and put (Panel B) options, where moneyness isdefined as the ratio of a strike price to the current index level. The sample period runs from March 2006 to June 2008.

1178 Chung et al.


where T is the time-to-maturity, R is the risk-free rate to expiration, F is the for-ward index derived from index option prices, K0 is the first strike below the forward index level, Ki is the strike price of the ith out-of-money option, andQ(Ki) is the midpoint of the bid–ask spread for each option with strike Ki.

TABLE II

Summary Statistics of the VIX Options Data Set

Moneyness

Deep ITM ITM 0.9 � Near the Money OTM 1.03 � Deep OTMMaturity m 0.9 m 0.97 0.97 � m 1.03 m 1.1 m 1.1 Subtotals

A: Call options� 30 Days

No. of contracts 2,418 336 297 387 1,888 5,326Share of total (%) 6.00 0.83 0.74 1.04 5.07 14.31Average price (US$) 6.71 1.92 1.35 0.99 0.42 3.46


180 DaysNo. of contracts 3,842 684 592 717 4,943 10,778Share of total (%) 9.54 1.70 1.47 1.93 13.28 28.95Average price (US$) 5.80 3.21 2.73 2.40 1.16 3.11


B: Put options� 30 Days

No. of contracts 622 329 297 388 3,667 5,303Share of total (%) 1.67 0.88% 0.80 1.04 9.85 14.25Average price (US$) 0.53 0.84 1.36 2.15 10.57 7.66


180 DaysNo. of contracts 2,635 684 591 716 5,110 9,736Share of total (%) 7.08 1.84 1.59 1.92 10.05 26.16Average price (US$) 0.99 2.15 2.78 3.60 10.05 6.13


This table presents details on the number of available contracts, the percentage of the total number, and the average price (in US$)of VIX options across various categories of moneyness for call (Panel A) and put (Panel B) options, where moneyness is defined asthe ratio of a strike price to the current VIX level. The sample period runs from March 2006 to June 2008.



In addition to being a portfolio of calls and puts, VIX is also well known as avolatility measure. During our sample period, realized volatility and VIX arehighly correlated (correlation: 0.76) and both of them exhibit a highly clusteredpattern (first-order autocorrelation: 0.74 and 0.97, respectively); furthermore,as can be clearly seen from Figure 1, the latter is found to be consistently high-er than the former. Figure 1 also reveals that the sample period can be roughlydivided into two sub-periods based upon mid-2007, with the earlier period rep-resenting a bull market (with lower and more stable volatility), while the laterperiod represents a bear market (with higher and more unstable volatility). Itshould, therefore, be of some interest to compare the informational roles ofVIX options under different market statuses.

4. RETURNS FORECASTING

4.1 Empirical Methodology

Following on from the contemporaneous and lead–lag relationships betweenimplied volatility levels and returns identified by Giot (2005), Banerjee, Doran,and Peterson (2007) subsequently went on to provide both theoretical andempirical support for the predictability of the VIX for future returns. We first of

TABLE III

Summary Statistics of the VIX, and S&P 500 Index Returns and Realized Volatility

Variables

Statistics Returns Realized Volatility VIX

Mean �8.77E�07 0.155 0.171Median 6.80E�04 0.131 0.155Maximum 0.040 0.694 0.322Minimum �0.034 0.047 0.100S.D. 0.010 0.085 0.056Skewness �0.218 1.891 0.586Kurtosis 5.051 8.894 2.134

AutocorrelationLag 1 �0.103 0.742 0.967Lag 2 �0.009 0.668 0.947Lag 3 �0.019 0.635 0.931Lag 4 �0.045 0.614 0.915Lag 5 �0.023 0.580 0.899

This table present the summary statistics of the VIX and the returns and realized volatility on the S&P 500 index for the sample peri-od running from March 2006 to June 2008, giving a total of 588 daily observations. The returns are calculated as the differencebetween two successive daily closing logarithmic index levels, whilst realized volatility is estimated by annualizing the square root ofthe sum of the five-minute squared returns within a day, including the overnight interval.

1180 Chung et al.


all use pairs of market prices of nearest-month ATM VIX calls and puts torecover the implied VIX levels from the put–call parity of VIX options, as shownin Equation (7), and then regress the 30 calendar day log return of the S&P500 index on VIX (VIX2) and implied VIX (VIX2) in order to examine their indi-vidual contribution to the determination of future returns. To further investi-gate whether the information implied in the VIX options market can provideany incremental contribution to the prediction of the S&P 500 index returns,we also run the following two alternative regression models:

(8)

(9)

where rt30 is the 30 calendar day log return of the S&P 500 index.15

Since VIX and VIX2 are highly correlated with their respective impliedmeasures, in order to avoid the problem of collinearity, we first regress theimplied VIX-related variables on their corresponding measures and then use

r30t � a2 � b2VIX2

t � c2eimpv2t � e2,t

r30t � a1 � b1VIXt � c1e

impvt � e1,t

1200Mar May Jul Sep Nov Jan Mar May Jul Sep Nov Jan Mar May

Apr Jun Aug

2006 2007 2008

Oct Dec Feb Apr Jun Aug Oct Dec Feb Apr Jun

1250

1300

1350

S&

P 5

00 In

dex

Pric

e

1400

1450

1500

S&P 500 IndexVIXRealized Volatility

1550

1600

0.0

0.1

0.2

0.3

0.4

0.5

VIX

Rea

lized

Vol

atili

ty

0.6

0.7

0.8

0.9

1.0

FIGURE 1Time series of the S&P 500 index, VIX and realized volatility of index returns. This figure plots the timeseries of the S&P 500 index, the VIX and the realized volatility of the index returns. The sample period

runs from March 2006 to June 2008 with daily frequency. Realized volatility is estimated by annualizingthe square root of the sum of the five-minute squared returns within a trading day.

15As VIX is regarded as a fear indicator or a signal to time the market (Giot, 2005; Whaley, 2000), it is natureto expect that there exists a time-series relation between VIX and future market returns. Although it is alsopossible, based on the return-risk trade-off in assets pricing theories, to examine the cross-sectional relationamong stocks, this study inevitably focuses only on the time-series relation because we investigate thedynamics of an equity index rather than individual stocks.



the residuals and to investigate the incremental contribution of theimplied VIX. As reported in the prior studies, b1 and b2 are expected to be posi-tively significant. If c1 and c2 are found to be significant, we can conclude thatthe implied VIX provides incremental information for the determination offuture S&P 500 index returns.

4.2 Empirical Results

The results of the alternative regression models on returns forecasting for thefull sample and extreme scenario are reported in Table IV. The annualized 30-day returns are regressed on the VIX (implied VIX) levels in Model 1 (Model 2)and on the VIX levels and the orthogonal implied VIX residuals in Model 3. Thecoefficient estimates and standard errors of the three models are reported inPanel A, while those of the models with VIX2-related variables as regressors arereported in Panel B.

As indicated by the regression coefficients, VIX does not significantly pre-dict the subsequent 30-day return, whereas implied VIX does, and orthogonalimplied VIX also provides a significant incremental contribution to the deter-mination of future returns at the 1% significance level. However, in contrast tothe findings of Banerjee et al. (2007), the individual predictive signs of impliedVIX are negative in the full sample. As shown in the VIX whitepaper issued bythe CBOE, VIX is contemporaneously, negatively related to market movement.Since the underlying assets of VIX options are VIX futures contracts ratherthan VIX itself, the implied VIX derived from VIX options essentially representsthe expectation of VIX at the option expiration. Therefore, the negative regres-sion coefficient between the 30 calendar day log return and the implied VIXmay just reflect the contemporaneous relation between future VIX and marketmovement.

To further explore this inconsistency, we rerun all of the regression modelswith alternative sub-samples. As noted by Giot (2005), the VIX can be used asa market timing indicator, but only when it reaches a certain extreme level; wetherefore form a sub-sample in which the VIX is beyond � one standard devia-tion from the long-term mean. As shown in Figure 1, our sample period can beclearly separated into two sections with different market status, based uponmid-2007. We therefore go on to form two additional sub-samples, separatedby July 1, 2007, with the earlier period representing a stable bull market, andthe later period, a volatile bear market.

The estimation results for the extreme sub-sample are shown in the lowersections of Panels A and B in Table IV. The results on the signs and significanceof the regression coefficients are consistent with the findings of both Giot(2005) and Banerjee et al. (2007); thus, both VIX and implied VIX can serve as

eimpv2teimpv

t

1182 Chung et al.


predictors of market returns, although they are clearly more reliable whenreaching a certain extreme level. Furthermore, in addition to the informationprovided by the VIX, that contained within the implied VIX can also provide asignificant incremental contribution to the determination of future returns atthe 1% significance level.

The estimation results of the two sub-samples with different market sta-tus are shown in Table V. Similar to the findings on the extreme sub-sample

TABLE IV

Estimation Results of Returns Forecasting Models for Full Sample and Extreme Scenario

Model (1) Model (2) Model (3)

Independent Variables Coeff. S.E. Coeff. S.E. Coeff. S.E.

A: Alternative 1Full sample (n � 588)

Intercept 0.073 0.059 0.199*** 0.063 0.073 0.054VIX �0.501 0.326 – – �0.501 0.321Implied VIX – – �1.213*** 0.343 – –VIX residual – – – – 15.275*** 1.500Adj. R2(%) 0.23 1.92 15.12

Extreme scenario (n � 223)Intercept �0.083 0.073 �0.058 0.081 �0.083 0.071VIX 0.772** 0.351 – – 0.772** 0.341Implied VIX – – 0.635* 0.382 – –VIX residual – – – – �9.830*** 2.569Adj. R2(%) 1.70 0.73 7.41

B: Alternative 2Full sample (n � 588)

Intercept �0.001 0.034 0.073** 0.035 �0.001 0.030VIX2 �0.374 0.864 – – �0.374 0.786Implied VIX2 – – �2.553*** 0.910 – –VIX 2 residual – – – – �39.591*** 3.560Adj. R2(%) �0.14 1.16 17.20

Extreme scenario (n � 223)Intercept �0.039 0.049 �0.013 0.052 �0.039 0.046VIX2 2.436*** 0.928 – – 2.436*** 0.896Implied VIX 2 – – 1.861* 1.037 – –VIX 2 residual – – – – �21.148*** 5.143Adj. R2(%) 2.59 0.99 9.13

This table presents the estimation results of the regression models used to investigate returns forecasting in the S&P 500 index forthe full sample and the extreme scenario (the extreme scenario is defined as beyond � one standard deviation from the long-termmean of the VIX level). In addition to regressing the 30 calendar day log return on the VIX (VIX2) or the implied VIX (VIX2) so as toinvestigate their individual information content on future returns, we also investigate whether the information implied within the VIXoptions market can provide any incremental contribution to the return prediction by running the following two alternative regressionmodels: and , where are the 30 calendar day log returns of

the S&P 500 index, commencing at time t � 1, and ( ) are the residuals from regressing the implied VIX (VIX2) on their cor-responding measures so as to avoid the problem of collinearity. The sample period runs from March 2006 to June 2008. * indicatessignificance at the 10% level; ** indicates significance at the 5% level; and *** indicates significance at the 1% level.

eimpv2teimpv

t

r 30tr 30

t � a2 � b2VIX 2t � c2e

impv 2t � e2,tr 30

t � a1 � b1VIXt � c1eimpvt � e1,t



period, both the VIX and the implied VIX are found to be significantly andpositively related to future returns, with the orthogonal implied VIX residualalso providing a significant incremental contribution to the determination offuture returns (independent of market status) at the 5% significance level.However, the explanatory power in the volatile bear market (sub-sample 2) isclearly much higher, with the adjusted R2 being found to be as high as36.47%.

TABLE V

Estimation Results of Returns Forecasting Models for Alternative Samples



A: Alternative 1Sub-sample 1 (n � 336)

Intercept �0.178** 0.083 �0.396*** 0.119 �0.178** 0.083VIX 2.324*** 0.632 – – 2.324*** 0.632Implied VIX – – 3.881*** 0.882 – –VIX residual – – – – 5.761** 2.275Adj. R2(%) 3.61 5.10 5.14

Sub-sample 2 (n � 252)Intercept �2.109*** 0.172 �2.151*** 0.230 �2.109*** 0.168VIX 8.482*** 0.751 – – 8.482*** 0.734Implied VIX – – 8.522*** 0.993 – –VIX residual – – – – �8.863*** 2.488Adj. R2(%) 33.51 22.46 36.47

B: Alternative 2Sub-sample 1 (n � 336)

Intercept �0.003 0.041 �0.103* 0.058 �0.003 0.041VIX2 7.306*** 2.183 – – 7.306*** 2.168Implied VIX2 – – 12.467*** 3.085 – –VIX2 residual – – – – 18.568** 7.783Adj. R2(%) 2.95 4.38 4.30

Sub-sample 2 (n � 252)Intercept �1.172*** 0.091 �1.238*** 0.122 �1.172*** 0.090VIX 2 18.658*** 1.645 – – 18.658*** 1.622Implied VIX 2 – – 19.437*** 2.208 – –VIX 2 residual – – – – �15.490*** 5.417Adj. R2(%) 33.71 23.36 35.56

This table presents the estimation results of the regression models used to investigate returns forecasting in the S&P 500 index forthe alternative samples. In addition to regressing the 30 calendar day log return on the VIX (VIX2) or the implied VIX (VIX2) so as toinvestigate their individual information content on future returns, we also investigate whether the information implied within the VIXoptions market can provide any incremental contribution to the return prediction by running the following two alternative regressionmodels: and , where are the 30 calendar day log returns of the

S&P 500 index, commencing at time t � 1, and are the residuals from regressing the implied VIX (VIX 2) on their corre-sponding measures so as to avoid the problem of collinearity. The sample period runs from March 2006 to June 2008 with two sub-samples separated by 1 July 2007. * indicates significance at the 10% level; ** indicates significance at the 5% level; and *** indicatessignificance at the 1% level.

(eimpv2t )eimpv

t

r30tr30

t � a2 � b2VIX2t � c2e

impv2t � e2,tr 30

t � a1 � b1VIXt � c1eimpvt � e1,t

1184 Chung et al.


In summary, our results indicate that both the VIX and implied VIX canpredict future returns, with the predictive power being found to be more signif-icant in a volatile bear market. As regards serving as a predictor of marketreturns, they appear to be much more reliable when the volatility level isextreme, or when the market status is identifiable. Furthermore, as comparedwith the VIX compiled from S&P 500 option prices, the VIX recovered fromVIX options prices consistently provides a significant incremental contributionto the forecasting of returns.

5. VOLATILITY FORECASTING

5.1. Empirical Methodology

As noted in many of the prior studies,16 the volatility forecasting power of a par-ticular information set can be evaluated by a volatility model (such as aGARCH model) in which the information serves as an additional conditionalvariable or an encompassing regression within which the information is speci-fied as an additional regressor. By definition, the VIX is the market expectationwith regard to the aggregate volatility for the subsequent 30 calendar days. Wetherefore surmise that it would be more intuitive or straightforward to followthe idea of an encompassing regression.

In particular, we first of all regress the aggregate realized volatility (vari-ance) of the subsequent 30 calendar-day period on the VIX (VIX2) or theimplied VIX (VIX2) separately in order to examine their individual contributionto the determination of future volatility, and then apply the following two alter-native regression models to investigate the incremental contribution of theimplied VIX to future volatility

(10)

(11)

where is the annualized aggregate realized volatility of the S&P 500 indexreturns for the subsequent 30 calendar-day period.17

We noted earlier that VIX and VIX2 are found to be highly correlated withtheir corresponding implied measures; thus, in order to avoid the problem ofcollinearity, we take the residuals and from the regressions of theeimpv2

teimpvt

RV30t

(RV30t )2 � a2 � b2VIX2

t � g2eimpv2t � e2,t

RV30t � a1 � b1VIXt � g1e

impvt � e1,t

16Examples include Blair et al. (2001), Jian and Tian (2005), Becker, Clements, and White (2007), Taylor,Yadav, and Zhang (2008) and Goyal and Saretto (2009)17Since the distribution of realized volatility is approximately lognormal (Andersen et al., 2001), we also runthe regression models with the logarithmic forms of the examined variables. The results are similar and avail-able upon request.



implied VIX-related variables on their corresponding measures to investigatethe incremental contribution of the implied measures.

As pointed out in several prior studies, b1 and b2 are expected to be posi-tively significant. If g1 and g2 are found to be significant, then we can concludethat the implied VIX provides incremental information for the forecasting ofrealized volatility in S&P 500 index returns.

5.2 Empirical Results

Following the analysis framework for returns forecasting, we run regressionmodels for the full sample and sub-samples comprising the extreme scenario, astable bull market, and a volatile bear market. The estimation results for thefull sample and the extreme sub-sample are reported in Table VI, whereasthose for the two sub-samples with different market status are presented inTable VII.

The annualized 30-day realized volatility levels are regressed on the VIXlevel in Model (1), on the implied VIX level in Model (2), and on both the VIXlevel and the orthogonal implied VIX residuals in Model (3). The coefficientestimates and standard errors of the three models are reported in Panel A,whereas those of the corresponding three models with realized variance as theregressand and VIX2-related variables as regressors are reported in Panel B.

In contrast to the earlier findings on the forecasting of returns, the resultsfor volatility forecasting are found to be consistent across all samples andvolatility proxies. Both the VIX and implied VIX are found to have significantlypositive correlations with future realized volatility. Furthermore, in addition tothe VIX, the implied VIX consistently provides a significant incremental contri-bution to volatility forecasting at the 5% significance level.

A finding which is of particular interest is that the slope coefficient andexplanatory power for both the full sample and the extreme scenario are muchhigher than those for the two sub-samples with different market statuses. Theadjusted R2 for both the full sample and the extreme scenario is also found tobe considerably higher (about 50%) than that for the two sub-samples (usuallysmaller than 10%). We conjecture that the lower slop coefficient and explana-tory power for the two sub-samples are due to the unobvious trend in these twosections of volatility process.18

In summary, for returns and volatility forecasting, although both the VIXand implied VIX provide significant information, there is no clear evidence to

18As shown in Figure 1, compared with the volatility evolution for the whole sample period, the volatilityevolves with a less clear trend in the two sub-sample periods. The consequence of the less clear trend in real-ized volatility process is that the fitted regression line is close to the mean level of the dependent variable(realized volatility) and thus the explained sum of squares and adjusted R2 are lower.

1186 Chung et al.


enable us to distinguish their individual predictive power. Nevertheless, inaddition to the information implied in the S&P 500 index options market, theinformation implied in the VIX options market can also provide a significantcontribution to the prediction of both returns and volatility in the S&P 500index. The performance in volatility forecasting is independent of market situ-ations; however, the performance in returns forecasting is found to be particu-larly effective when the market is extreme or volatile.

TABLE VI

Estimation Results of Volatility Forecasting Models for the Full Sample and Extreme Scenario



A: Alternative 1Full sample (n � 588)

Intercept 0.000 0.005 �0.017*** 0.005 0.000 0.005VIX 0.810*** 0.029 – – 0.810*** 0.028Implied VIX – – 0.887*** 0.029 – –VIX residual – – – – 1.020*** 1.137Adj. R2(%) 57.44 61.05 61.05

Extreme Scenario (n � 223)Intercept 0.017** 0.008 �0.004 0.009 0.017** 0.008VIX 0.722*** 0.039 – – 0.722*** 0.037Implied VIX – – 0.820*** 0.041 – –VIX residual – – – – 1.247*** 0.280Adj. R2(%) 61.04 63.87 64.10

B: Alternative 2Full sample (n � 588)

Intercept 0.002* 0.001 �0.001 0.001 0.002* 0.001VIX2 0.644*** 0.029 – – 0.644*** 0.027Implied VIX2 – – 0.724*** 0.029 – –VIX2 residual – – – – 1.091*** 0.124Adj. R2 (%) 45.37 50.97 51.65

Extreme Scenario (n � 223)Intercept 0.005** 0.002 0.001 0.002 0.005** 0.002VIX2 0.572*** 0.043 – – 0.572*** 0.040Implied VIX2 – – 0.669*** 0.045 – –VIX2 residual – – – – 1.273*** 0.229Adj. R2 (%) 44.72 49.93 51.30

This table presents the estimation results of the regression models used to investigate volatility forecasting on the S&P 500 indexreturns for the full sample and the extreme scenario (the extreme scenario is defined as beyond � one standard deviation from thelong-term mean of the VIX level). In addition to regressing the 30 calendar day log return on volatility (variance) on the VIX (VIX2) orthe implied VIX (VIX2) so as to investigate their individual information content on future volatility, we also investigate whether the infor-mation implied within the VIX options market can provide any incremental contribution to the volatility prediction by running the follow-ing two alternative regression models: and , where is the annualized aggregate realized volatility of the S&P 500 index returns for the subsequent 30 calendar days, commencing at timet � 1, and and are the residuals from regressing implied VIX and VIX 2 on their corresponding measures so as to avoid theproblem of collinearity. The sample period runs from March 2006 to June 2008. * indicates significance at the 10% level; ** indicatessignificance at the 5% level; and *** indicates significance at the 1% level.

eimpv2teimpv

t

RV 30t(RV 30

t )2 � a2 � b2VIX 2t � g2e

impv2t � e2,tRV 30

t � a1 � b1VIXt � g1eimpvt � e1,t



6. DENSITY PREDICTION

6.1. Empirical Methodology

Based upon the analogy of the pricing formula for S&P 500 index options tothe Black–Sholes formula, P2 in Equation (2) is the probability of ST Kunder the risk-neutral measure. Therefore, 1 � P2 is the risk-neutral distribu-tion function, F(x), of the S&P 500 index at expiration, and the RND can be

TABLE VII

Estimation Results of Volatility Forecasting for Alternative Samples



A: Alternative 1Sub-sample 1 (n � 336)

Intercept 0.046*** 0.006 0.041*** 0.010 0.046*** 0.006VIX 0.393*** 0.048 – – 0.393*** 0.047Implied VIX – – 0.417*** 0.071 – –VIX residual – – – – �0.620*** 0.172Adj. R2(%) 16.32 9.13 19.21

Sub-sample 2 (n � 252)Intercept 0.135*** 0.018 0.114*** 0.022 0.135*** 0.018VIX 0.262*** 0.079 – – 0.262*** 0.079Implied VIX – – 0.350*** 0.097 – –VIX residual – – – – 0.389*** 0.168Adj. R2(%) 3.81 4.61 4.63

B: Alternative 2Sub-sample 1 (n � 336)

Intercept 0.005*** 0.001 0.005*** 0.001 0.005*** 0.001VIX 2 0.258*** 0.036 – – 0.258*** 0.035Implied VIX2 – – 0.265*** 0.052 – –VIX 2 residual – – – – �0.456*** 0.125Adj. R2(%) 13.38 6.84 16.46

Sub-sample 2 (n � 252)Intercept 0.029*** 0.004 0.023*** 0.005 0.029*** 0.004VIX 2 0.210*** 0.073 – – 0.210*** 0.073Implied VIX2 – – 0.319*** 0.091 – –VIX 2 residual – – – – 0.547*** 0.243Adj. R2(%) 2.77 4.31 4.32

This table presents the estimation results of the regression models used to investigate volatility forecasting on the S&P 500 indexreturns for the alternative samples. In addition to regressing the 30 calendar day realized volatility (variance) on the VIX (VIX2) or theimplied VIX (VIX2) so as to investigate their individual information content on future volatility, we also investigate whether the informa-tion implied within the VIX options market can provide any incremental contribution to the volatility prediction by running the followingtwo alternative regression models: and , where isthe annualized aggregate realized volatility of the S&P 500 index returns for the subsequent 30 calendar days, commencing at timet � 1, and and are the residuals from regressing implied VIX and VIX2 on their corresponding measures so as to avoid theproblem of collinearity. The sample period runs from March 2006 to June 2008 with two sub-sample separated by 1 July 2007. * indi-cates significance at the 10% level; ** indicates significance at the 5% level; and *** indicates significance at the 1% level.

eimpv2teimpv

t

RV 30t(RV 30

t )2 � a2 � b2VIX 2t � g2e

impv2t � e2,tRV 30

t � a1 � b1VIXt � g1eimpvt � e1,t

1188 Chung et al.


generated by taking the first derivative of 1 � P2 with respect to the strike priceK evaluated at x and given by

(12)

where Re [�] denotes the real part of a complex number and is the condi-tional characteristic function of the Heston (1993) model.19

The transformation from RNDs to RWDs is reliant upon the assumptionsof a risk premium function, a representative agent model, or the calibrationtheory. Since the computational load for the SV model under the first twoapproaches is extremely heavy, in this study, we adopt a method of transforma-tion based upon statistical calibration. As noted in Shackleton, Taylor, andPeng (2010) and Wang (2009), a non-parametric calibration performs betterthan certain parametric functions, such as the b-distribution function; thus,we utilize this to implement the transformation.20 The RWD and its correspon-ding distribution function are respectively given as

(13)

where h(y) and H(y) are the respective non-parametric kernel “probability den-sity function” (PDF) and “cumulative distribution function” (CDF), defined bya normal kernel with bandwidth B; these are respectively written as

and with f(�)

denoting the PDF and � (�) denoting the CDF, for standard normal distribu-tion. Given a sample of n observations, we define ui � F(xi) (i �1, 2, . . . n) andtransform ui into a new series, yi � ��1(ui).

21

According to the pricing formulae of S&P 500 index and VIX options, theprices of both options depend on the same parameters specifying the SVdynamic process.22 Therefore, the RNDs and RWDs of the S&P 500 index canbe estimated from either of the options markets. We estimate the dynamicparameters by minimizing the following loss functions:

(14)

where MC0 denotes the market prices and C0 denotes the theoret-ical prices of S&P 500 index (VIX) calls, M(N) is the number of contracts for the

(CVIX0 )(MCVIX

0 )

SSE � aN

i�1(MCVIX

0 (Xi, Ti) � CVIX0 (Xi, Ti))

2

SSE � aM

i�1(MC0(Ki, Ti) � C0(Ki, Ti))

2

H(y) � (1�n)an

i�1�[(y � yi)�B]h(y) � (1�nB)a

n

i�1f[(y � yi)�B]

g(x) �h(y) f(x)

f(y) and G(x) � H(��1(F(x)))

g~2(c)

f(x) �1px �

�

0

Re[e� ic ln xg~2(c)]dc

19The details can be found in Appendix A.20We also carry out transformation with the b-distribution function for our robustness analysis.21The details of the transformation procedure can be found in Shackleton et al. (2010).22Except that the correlation coefficient, r, does not appear in the VIX option pricing formulae.



S&P 500 index (VIX) options. In the loss functions, the market prices of ITM callsare converted from the market prices of their corresponding OTM puts. Essentiallycall and put option prices for the same strike and maturity should essentially con-tain the same information. However, given that the OTM options are usually moreheavily traded, we discard the ITM options for density prediction.

To further examine whether VIX options, as compared with S&P 500 indexoptions, can provide any incremental contribution to density prediction, wealso estimate the risk-neutral dynamic parameters of the S&P 500 index byminimizing the sum of the squared errors between the market and theoreticalprices of both S&P 500 index and VIX options. Since the premiums of S&P500 index options are generally much higher than those of VIX options, and thenumber of available contracts for S&P index options is usually much largerthan that for VIX options, in order to give both options markets equivalentweights in the loss function, we adjust the loss function in Equation (14) as

(15)

where the ratio of M over N represents a weighting factor. The percentage pric-ing errors adjust for the difference in magnitude between the option premiums,while the weighting factor adjusts for the difference in the total number of con-tracts.23

Density prediction can be evaluated by a variety of methods, including sev-eral methods surveyed by Tay and Wallis (2000). In addition to examining theKolmogorov–Smirnov (K-S) statistics for all pairs of density levels estimatedfrom the S&P 500 index or the VIX option markets, we assess both the individ-ual and joint density predictive power of the two markets using likelihood crite-ria24 and the Berkowitz (2001) test.25

For a particular forecast horizon, we have n density forecasts and the log-likelihood of n realized asset prices for method m is denoted as Lm. When comparing

� aN

j�1aM

N

MCVIX0 (Xi, Ti) � CVIX

0 (Xi, Ti)

MCVIX0 (Xi, Ti)

b2

SSE � aM

i�1aMC0(Ki, Ti) � C0(Ki, Ti)

MC0(Ki, Ti)b2

23A few different loss functions are used within the extant literature to back out the parameters from theoption prices; the primary reason for our adoption of one specified in Equation (13) is that it guarantees the same magnitude of pricing errors for the OTM puts from which the synthetic ITM calls are converted.However, due to the different general price levels and number of contracts for S&P 500 index and VIXoptions, in order to ensure equivalent weights for the two options markets, it is inevitable that we must adopta loss function with percentage pricing errors, such as Equation (14).24The same criteria are used in Bao, Lee, and Saltoglu (2007), Liu et al. (2007) and Shackleton et al. (2010).25Since most of the statistic diagnostic tests based on the probability integral transformations, such as the K-S and Anderson–Darling tests, are not sufficiently powerful for small samples, Berkowitz (2001) proposeda method for jointly testing uniformity and independence. The results of the Anderson–Darling tests are usedto check the robustness of our analysis.

1190 Chung et al.


the performance of two methods, we prefer the method with a higher value ofLm and the null hypothesis that two methods have equal log-likelihood can betested by assessing the statistical significance of the sample mean of the differ-ences in n pairs of log-likelihood values, dt.

26 Alternatively, we can assess thestatistical significance of the sample median of the log-likelihood differenceswith a non-parametric method, such as the Wilcoxon-signed rank test.

Based upon the transformed variable, ,

where fi,t(x) and , respectively, denote the true and the estimated densi-ties for the forecast horizon t and Si,t denotes realizations, the Berkowitz(2001) test assesses whether or not the values of z are consistent with the nullhypothesis of i.i.d. observations from a standard normal distribution.

The alternative hypothesis is of a stationary AR(1) process with Gaussianresiduals; that is, a process within which no restrictions are placed on themean, variance and AR parameters. If the log-likelihood functions for the nullhypothesis and the alternative hypothesis are, respectively, denoted as L0 andL1, then the statistic for the likelihood ratio test will be LR3 � 2(L1 � L0),which follows a x2(3) distribution under the null hypothesis.27

6.2. Empirical Results

6.2.1. Parameter estimation

We first estimate the risk-neutral dynamic parameters of the S&P 500 index byminimizing the sum of the squared errors between the market and the theoret-ical prices of the S&P 500 index or the VIX options. Table VIII presents thesummary statistics of the two sets of risk-neutral parameters in Panels A and B,with Panel C reporting the test results of mean (median) equality based uponthe t-test (Wilcoxon-signed rank test).

Since the correlation coefficient, r, between two Wiener processes (priceand volatility) does not come into play in the option pricing formula for VIXoptions, we have no correlation estimates within the parameter set of VIXoptions; thus, we use the correlation estimates recovered from the S&P 500index options to evaluate the density predicted by the VIX options.

Estimations have been carried out on the dynamic parameters of the SVmodel from S&P 500 index levels and/or option prices in several of the prior

f̂i,t(x)

zi,t��1(ui,t)��1a�Si,t

��

f̂i,t(x) dxb

26The details of the theory behind this test can be found in Amisano and Giacomini (2007), Bao et al. (2007),Liu et al. (2007) and Shackleton et al. (2010).27If the data are found to be serially correlated, then the rejection of the LR3 test may actually arise from theautocorrelation of the data series. Berkowitz (2001) therefore proposed the LR1 test to separately examinetheir independence; however, since all of the predictions in the present study are non-overlapping, the LR1test should prove to be unnecessary.



studies, with a typical finding being that the estimation outcomes are ratherinconsistent and heavily dependent upon the market, sample periods and esti-mation approaches.28 As compared with the risk-neutral volatility dynamic ofthe S&P 500 index generated from the S&P 500 index options, the dynamicprocess obtained from the VIX options is found to be more stable (a lower sv)with a lower long-term average (a lower u) and a slow mean-reversion speed (a lower k).

Although a higher value for the volatility of volatility (sv) implies fattertails for the index-level distribution, Bates (1996, 2000) and Bakshi, Cao, andChen (1997) found that the volatility level of the SV model backed out fromthe S&P 500 index options appeared to be too high; and indeed, we have a sim-ilar finding, since the average sv estimates are as high as 0.64. Nevertheless,the mean level of sv estimated from the VIX options is only 0.47, which ismuch more reasonable, and quite close to that estimated by Wang (2009) withthe inclusion of jump components in the price and/or volatility dynamic(s).This finding may indicate that the tail distribution of the S&P 500 indeximplied in VIX options is more realistic.

Panel C of Table VIII indicates that the only parameter that is found tohave no significant difference across the markets is the initial SV (V0). The sig-nificant difference for most of the parameters implies that the informationalroles of the S&P 500 index and VIX options markets on the S&P 500 index maynot be identical. Finally, consistent with the prior studies, the correlation coef-ficient between price and volatility shocks (r) is found to be highly negative.

The summary statistics of the parameters jointly estimated from both ofthe options markets are shown in Panel D of Table VIII. Basically, the parame-ters backed out using both S&P 500 index and VIX options are bounded bythose recovered individually from the two markets. As compared with the find-ings of good or reasonable estimates in the prior studies, the volatility of volatility(sv) implied in the S&P 500 options is found to be too high, while the long-term level of volatility (u) implied in the VIX options is found to be too low.Therefore, the parameters recovered jointly from both data sources appear tobe much more reasonable.

These findings indicate that VIX options can improve the estimation of the dynamic parameters of the S&P 500 index, particularly with regard to thevolatility dynamic, an issue that has been achieved in the prior studies throughthe inclusion of jump components in the price and/or volatility dynamicprocesses of the S&P 500 index. These findings are also consistent with theargument of Becker et al. (2009) that VIX subsumes information to contributeto the jump component of volatility.

28The estimates of many studies are summarized in Shackleton et al. (2010).

1192 Chung et al.


6.2.2. Density evaluation

Although the parameter estimates differ somewhat between the S&P 500 indexand VIX options, we still do not have any clear idea to what extent the differ-ences affect the price density. Therefore, in this sub-section we separately eval-uate the implied densities in the S&P 500 index and VIX options, as well as inboth of the options markets. Based upon the estimation of the densities onceper week, we are provided with a total of 122 observations.

We evaluate both RNDs and RWDs transformed by the non-parametriccalibration method for one-, two-, three-, and four-week prediction horizons,beginning with the use of the K-S test to examine the null hypothesis that thedensities of the S&P 500 index implied in both the S&P 500 index and VIXoptions are drawn from the same distribution. As shown in Table IX, we fail toreject the null hypothesis for all of the one-week RNDs, and reject it for only 2of the two-week-RNDs, 8 of the three-week-RNDs, and 22 of the four-week-RNDs. In contrast, the null hypothesis is rejected for all RWDs for all predic-tion horizons. These findings indicate that the risk preference implied within

TABLE VIII

Summary Statistics of the SV Model Parameter Estimates for Alternative Data Sources

Parameters k u sv r V0

A: S&P 500 optionsMean 9.650 0.035 0.644 �0.818 0.027Median 8.358 0.023 0.567 �0.815 0.018S.D. 5.003 0.024 0.355 0.101 0.023

B: VIX optionsMean 5.707 0.011 0.469 – 0.030Median 5.301 0.007 0.406 – 0.021S.D. 1.753 0.010 0.155 – 0.023

C: Tests for differencesMean 8.215*** 9.844*** 4.987*** – �1.010Median 7.406*** 9.536*** 3.531*** – �1.155

D: Combination of S&P 500 and VIX optionsMean 8.661 0.022 0.528 �0.676 0.026Median 7.862 0.016 0.521 �0.715 0.019S.D. 3.307 0.014 0.190 0.152 0.019

This table presents the summary statistics of the SV model parameter estimates obtained from S&P 500 index options, VIX optionsand a combination of both options, with the sample period running from March 2006 to June 2008. The results on the S&P 500options, VIX options and combination of both options are respectively reported in Panels A, B and D, whilst Panel C reports the sta-tistics of the tests for the mean and median differences between the parameters estimates on the S&P 500 index options and thoseon the VIX options. The parameters are estimated by minimizing the loss function defined as the sum of the squared differencesbetween the market and theoretical option prices. The mean (median) tests are carried out using t-test (Wilcoxon signed rank test).* indicates significance at the 10% level; ** indicates significance at the 5% level; and *** indicates significance at the 1% level.



the S&P 500 index and VIX option markets may be quite different, althoughthey clearly share similar risk-neutral information with regard to density pre-diction.

We further employ both the Berkowitz (2001) test and likelihood criteriato explore the general predictive power of the densities obtained from threealternative data sources, the S&P 500 index options (SXO), the VIX options(VXO), and a combination of both (MIX). In order to avoid the existence ofautocorrelation in our prediction series, we select predictions only once perprediction horizon; thus, the total number of observations is reduced to 61 forthe two-week densities, 40 for the three-week densities, and 30 for the four-week densities.

The results of the Berkowitz (2001) test are shown in Panel A of Table X.For RNDs, as indicated by the LR3 statistics and their corresponding P-values,with the exception of the one-week predictions and the two-week SXO predic-tion, all of the remaining density predictions are found to be satisfactory at the5% significance level. Although there is no obvious difference between theP-values of the SXO and VXO predictions, the P-values of all of the MIX pre-dictions, with the exception of the one-week predictions, are clearly higherthan those of the predictions from the two individual options markets. Thesefindings suggest that the information implied in the VIX options market canimprove the predictive power of RNDs.

TABLE IX

Evaluation of the Consistency of the Density Predictions from Alternative Markets

K-S TestNo. of Tests Rejecting

Prediction Horizon Average P-Value the Null Hypothesis

A: Risk-neutral density (RND)1 week 0.014 0.948 0/1222 weeks 0.019 0.861 2/1223 weeks 0.025 0.779 8/1224 weeks 0.041 0.659 22/122

B: Real-world density (RWD)1 week 0.340 0.000 122/1222 weeks 0.179 0.000 122/1223 weeks 0.177 0.000 122/1224 weeks 0.231 0.000 122/122

This table reports the results of the Kolmogorov–Smirnov (KS) test to evaluate the null hypotheses of consistent density estimationsfrom the S&P 500 index and VIX options. The sample period runs from March 2006 to June 2008, giving a total of 122 weeks, with thetest being undertaken once a week for one-, two-, three-, and four-week-ahead predictions. The average of the KS statistics, the P-values and the number of tests rejecting the null hypothesis are shown for the evaluation of both “risk-neutral density” (RND) and“real-world density” (RWD) transformed by the non-parametric statistical calibration.

TA

BL

E X

Eva

luat

ion

of D

ensi

ty P

redi

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ns B

ased

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e B

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Hor

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1 W

eek

2 W

eeks

3 W

eeks

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Tes

tsS

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IXS

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IXS

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A: B

erko

witz

(20

01)

test

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(R

ND

)LR

3 st

atis

tic15

.44

10.7

012

.63

8.79

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alue

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Rea

l-wor

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RW

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LR3

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6.54

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7.40

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1.74

1.15

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1.34

1.13

1.18

1.06

P-V

alue

0.08

0.06

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0.53

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0.79

B: L

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(R

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)Lo

g Li

kelih

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�54

5.09

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6.00

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ce—

Mea

n (m

edia

n)V

XO

-SX

O�

0.03

9* (

�0.

023*

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3 (�

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7)0.

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27)

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-SX

O�

0.03

0* (

�0.

020*

)�

0.01

2 (0

.009

)0.

013

(0.0

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l-wor

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RW

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7)0.

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2001

) an

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erko

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(20

01)

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, w

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the

mea

n (m

edia

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f th

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lues

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, VIX

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and

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kets

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Ds

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-wee

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the

5% le

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***

indi

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canc

e at

the

1% le

vel.



In line with the findings of several of the prior studies,29 the RWD predic-tions in the present study are found to be consistently satisfactory at the 5%significance level, and indeed, there is no clear evidence in favor of any partic-ular data source. The log-likelihood values evaluated with realized observationsof the S&P 500 index are reported in Panel B of Table X, along with the meanand median equality tests of the individual log-likelihood values for the threealternative data sources.

By and large, we can see that the results of the likelihood criteria providegeneral confirmation of the findings of the Berkowitz tests, albeit with greaterinsights. When comparing the log-likelihood values of the SXO and VXO pre-dictions, whether for RNDs or RWDs, the SXO predictions generate highervalues for shorter horizons (one- and two-week horizons), whereas the VXOpredictions generate higher values for longer horizons (three- and four-weekhorizons). However, as indicated by the mean and median equality tests ofVXO–SXO, these patterns are more significant for RNDs.

Of particular interest is the finding that if we use both S&P 500 index andVIX options to estimate the density levels, the log-likelihood values for almostall of the longer horizon predictions are further raised, a phenomenon that isindependent of the density type. As is generally indicated by the results of themean and median equality tests of the MIX-SXO, for longer (shorter) horizons,the advantage (disadvantage) of MIX over SXO predictions become more (less)significant than that of VXO over SXO predictions.

In summary, the results of the Berkowitz (2001) tests and the likelihoodcriteria jointly suggest that, in addition to S&P 500 options, VIX options cansignificantly improve the predictive power of option-implied densities overlonger prediction horizons.

7. ROBUSTNESS ANALYSIS AND FURTHER DISCUSSION

For the robustness analyses undertaken in this study, we refer to the suggestionof Areal and Taylor (2002) that the intraday periodic pattern of volatility shouldbe taken into consideration when estimating realized volatility; thus, we employtheir optimally weighted formula to replace the equally weighted formula ofAndersen, Bollerslev, Diebold, and Ebens (2001) for the estimation of realizedvolatility, and find similar volatility forecasting results for these two alternativerealized volatility proxies.

29Examples include Bliss and Panigirtzoglou (2004), Liu et al. (2007), Shackleton et al. (2010) and Wang(2009).

1196 Chung et al.


As for the density prediction, we use the b-distribution function to trans-form RNDs to RWDs,30 while also employing the Anderson–Darling test toevaluate the density prediction, and find that the results remain generallyunchanged. In order to imply a risk-averse utility function, the necessary andsufficient condition for the b-distribution calibrated density, as derived by Liu,Shackleton, Taylor, and Xu et al.(2007), is a 1 b (with a � b). The rela-tionship between the RWD, g(x), and RND, f(x), is detailed as

where F(x) is the risk-neutral cumulative distribution function and B(a, b) isthe b function. If either a 1 or b � 1, then the representative agent is a risk-seeker for an interval of wealth, with respect to which, the second derivative ofthe utility function is positive.

Consistent with the findings reported in numerous prior studies,31 all ofour estimates of a are found to be greater than 1, indicating that the risk pref-erence implied in both the S&P 500 index and VIX options markets differsfrom the rational assumption imposed in most of the asset pricing models.However, when including VIX options in the estimation of density, we find thatthe estimated values of a are still greater than 1, but are closer to 1. In otherwords, the participants in the VIX options market may tend to provide a muchcloser fit to the rational assumption. This finding supports the economic valueof VIX options and leaves some questions for future studies, including whetheroptions investors are really risk-lovers, and if so, why they behave so differentlyin the two markets.

8. CONCLUDING REMARKS

Since both S&P 500 index and VIX options essentially contain informationabout the future dynamics of the S&P 500 index, we empirically investigate theinformational roles of these two option markets on returns, volatility, and den-sity predictions in the S&P 500 index, and find that the information contentsimplied in the two options markets are similar, but not identical. In addition tothe information extracted from S&P 500 index options, the information recov-ered from VIX options significantly improves all of the predictions on the S&P500 index. Our findings are robust to various measures of realized volatility andmethods of density evaluation.

g(x) �F(x)a�1(1� F(x))b�1

B(a, b) f(x)

30As proposed by Fackler and King (1990) and applied in Liu et al. (2007), Shackleton et al. (2010) andWang (2009).31See Carr, Geman, Madan, and Yor (2002), Liu et al. (2007), Shackleton et al. (2010), Constantinides,Jackwerth, and Perrakis (2009) and Wang (2009).



Although the performance of VIX options implied information in volatilityforecasting is independent of market situations, the forecasting performanceon returns is particularly effective when the market is extreme or volatile. VIXoptions can also significantly improve density predictions in the S&P 500 indexover longer horizons. Our results are generally consistent with the argument ofBecker et al. (2009) that the VIX subsumes information to contribute to thejump component of volatility, since it has been shown that improvements madeby VIX options can be achieved by including jump components in the dynamicsof price and/or volatility.

There are several possible extensions for future research. First, it may befeasible to extract useful information from the trading volume of VIX options,since the informational role of options volume in the equity options market isalready a central theme in many of the prior studies.32 In addition to equityoptions, VIX options and variance swaps, Brenner, Ou, and Zhang (2006) alsointroduce a volatility tool (an option on a straddle) to hedge volatility risk;hence, the second extension would be to investigate which market may be moreconducive to traders realizing their expected profits or hedging the volatilityrisk. Finally, another potential dimension would be to focus on an asset alloca-tion perspective if investors can span the mean-variance frontier by adding VIXoptions.33

APPENDIX A

P2 is the risk-neutral probability that the option expires in-the-money, and P1 isthe probability of the same event when a different measure is applied. The con-ditional characteristic function for P2 in the Heston model is

(A1)

where

and , with k � h/(h � 2d),and h � k � rsvfi � d.

Similarly, the conditional characteristic functions for probability P1 can bewritten as follows:

(A2)g~1(c) � eC�DV0� ic lnS0

d �2(rsvci�k)2�s2v (�ci�c2)

D � [h(1 � edt)]�[s2v(1 � kedt)]

C � (r � q)tci � kus�2v cht � 2 lna1 � kedt

1 � kb d

g~2(c) � eC�DV0� ic ln S0

32Including Easley, O’Hara, and Srinivas (1998), Cao, Chen, and Griffin (2005), Pan and Poteshman (2006)and Ni, Pan, and Poteshman (2008).33The spanning test using VIX futures has been carried out by Chen, Chung, and Ho (2011).

1198 Chung et al.


where

, withk�h/(h�2d), and h � k �

rsv � rsvyi � d.The characteristic functions can then be inverted to obtain the desired

probabilities P1 and P2. A standard inversion formula gives the conditional risk-neutral density, denoted by f (x):

(A3)

APPENDIX B

Two probabilities, �1 and �2, are recovered by inverting the respective charac-teristic functions of ln(VIX2)

(B1)

(B2)

Re[�] denotes the real part of a complex number. The characteristic function,f2(0, t; if), is given as follows:

(B3)

where

�k bt0

at0

d a at0

t0(VIXF0(t))2b a 1

ln(VIXF0(t) )2b

B � cku �12

s2v

at0

t0(VIXF0(t) )2 at0(VIXF

0(t) )2

at0

�bt0

at0

b

A �s2

v

2 at0(VIXF

0(t) )2

at0

�bt0

at0

ba at0

t0(VIXF0 (t) )2b

2a 1ln(VIXF

0(t) )2b and

D2(t) ��BA

� eAB

� caif �BAb�1

�ABd eBtf�1

C2(t) �BAkt�

k

AeBt� ln eA

B� caif�

B

Ab�1

�A

BdeBtf� ln caif�

B

Ab�1d f

f2(0, t; if) � exp[C2(t) � D2(t)ln(VIXF0 (t))2]

�2 �12

�1p �

�

0

Re c e� iwlnK2

f2(0, t; if)

ifddf

�1 �12

�1p �

�

0

Re c e� iwlnK2

f2(0, t; if � 0.5)

iff2(0, t; 0.5)ddf

f(x) �1px �

�

0

Re[e�ic ln xg~2(c)]dc

d�2(rsvci�k�rsv)2�s2

v(ci�c2)D � [h(1�edt)]�[s2v(1�kedt)

C � (r � q)tci � kus�2v cht � 2 lna1 � kedt

1 � kb d



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the information content of the s&p 500 index and vix options on the dynamics of the s&p 500...

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