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© 2014 Nicolaus Copernicus University. All rights reserved.  
http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2014.001  Vol. 14 (2014) 5−28  

Submitted May 5, 2014  ISSN 
Accepted December 12, 2014 1234-3862 

 Juliusz Jabłecki*, Ryszard Kokoszczyński*,  
Paweł Sakowski, Robert Ślepaczuk*, Piotr Wójcik 

Does Historical VIX Term Structure Contain Valuable  
Information for Predicting VIX Futures? 

A b s t r a c t. We suggest that the term structure of VIX futures shows a clear pattern of 
dependence on the current level of VIX index. At the low levels of VIX (below 20), the term 
structure is highly upward sloping, while at the high VIX levels (over 30) it is strongly 
downward sloping. We use these features to predict future VIX futures prices more precisely. 
We begin by introducing some quantitative measures of volatility term structure (VTS) and 
volatility risk premium (VRP). We use them further to estimate the distance between the 
actual value and the fair (model) value of the VTS. We find that this distance has significant 
predictive power for volatility futures and index futures and we use this feature to design 
simple strategies to invest in VIX futures. 

K e y w o r d s: volatility term structure, volatility risk premium, VIX, VIX futures, volatility 
futures, realized volatility, implied volatility, investment strategies, returns forecasting, effi-
cient risk and return measures 

J E L Classification: G11, G14, G15, G23, C61, C22 

                                                 
* The views presented in this text are those of the authors and do not necessarily represent 

those of the National Bank of Poland or Union Investment TFI S.A. 
Correspondence to: Robert Ślepaczuk, University of Warsaw, Faculty of Economic Sci-

ences, ul. Długa 44/50, 00-241 Warsaw, Poland, tel: +48 22 55 49 174, fax: +48 22 831 28 
46, e-mail: rslepaczuk@wne.uw.edu.pl. 

 We gratefully acknowledge government financial support via grant no. UMO-2011/ 
03/B/HS4/02298. 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

6

Introduction  

 We observe that term structure of VIX futures shows a clear pattern of 
dependence on the current level of the VIX1. Figure 1 shows that the term 
structure is highly upward sloping when the level of VIX is relatively low 
(below 20) and it is significantly downward sloping when the level of VIX is 
high (over 30). We assume that this is mostly due to the market perception of 
risk in the short and the long-term. That perception reflects strong mean 
reversion effects and long memory – both visible in volatility time series.  

 
Figure 1. VIX and S&P500 index quotations from 01/01/2006 until 01/07/2013 

 In order to investigate the term structure of VIX futures phenomenon 
more deeply, we introduce two quantitative characteristics of volatility term 
structure derived directly from the levels of the consecutive VIX futures 
maturities. These measures are calculated independently for different quin-
tile groups2 of VIX and various times to expirations of VIX futures. Further, 
we use these measures in order to estimate the distance between the actual 
and the fair (or “theoretical”) value of volatility term structure for particular 
maturity of VIX futures. Subsequently, we include this information in the 

                                                 
1 VIX – volatility index quoted on CBOE, based on implied volatility of S&P500 index 

options. It reflects market’s expectation of stock market volatility in 30 days horizon. More 
information could be found in Whaley (1993), CBOE (2003) and CBOE (2009).  

2 As those dependence patterns are not homogenous in the whole sample, we divide our 
data into VIX levels quintile groups.  The rest of the study is conducted consistently on this 
basis. 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

7 

process of forecasting VIX levels and VIX futures and we find that this dis-
tance has significant predictive power. In the last section, we propose a sim-
ple investment strategy that uses VTS in order to predict VIX futures prices.  
 Volatility research and, in particular, volatility forecasting seems to be 
one of the most active and successful areas in financial econometrics in re-
cent decades (Andersen et al., 2005). The literature on the VIX and its deriv-
atives is growing very fast, but the number of studies testing the predictabil-
ity of VIX futures prices is – according to our best knowledge – still very 
low. Konstantinidi and Skiadopoulos (2011) show only weak evidence of 
statistically predictable patterns in the evolution of volatility futures prices. 
They also cannot find a trading strategy with economically significant  
profits. 
 Some papers suggest strongly that the use of the information content of 
the volatility term structure may improve this situation, but the number of 
studies attempting to examine directly the relationship between the term 
structure of VIX futures and their future returns or the underlying equity 
returns is still very limited (Fassas, 2012; Asensio, 2013; Huskaj and 
Nossman, 2013 and the references therein). In our previous research (Jabłec-
ki et al. 2013a) we found significant relationship between lagged term struc-
ture of VIX futures and current level of VIX, yet this didn’t translate into 
better predictions of VIX level. 
 The paper is thus organized as follows. The next section describes the 
data. Methodology of this research is presented in the second section. The 
third section presents the description of simple measures of volatility term 
structure (VTS). Next section describes the dependence between VIX and 
S&P500 index and measures of VTS. Forecasting properties of VTS are pre-
sented in the same section. The investment strategies based on these results 
are presented in the fifth section. The last section concludes and presents 
possible extensions of this research. 

1. Data Description 

For each trading day we gathered close prices for VIX, VIX futures and 
S&P500 index3. Initially, the data set included 24 expiration months (from 
January 2006 until July 2013), but we had to limit their number to 7 because 

                                                 
3 Data are from the following sources:  
 VIX - http://cfe.cboe.com/Products/historicalVIX.aspx 
 VIX futures - http://cfe.cboe.com/Products/historicalVIX.aspx 

 S&P500 - www.stooq.pl 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

8

of liquidity problems for longer maturities. Data preparation for VIX returns 
also included the process of gap correction in order to omit the problem of 
very high positive returns at the moment of series change. 
 Figure 1 and Table 1 present time series for VIX and S&P500. It is 
worth noticing that S&P500 returns are leptokurtic and negatively skewed 
while VIX returns are also leptokurtic and positively skewed. Additionally, 
both returns series are negatively correlated. 

Table 1. The descriptive statistics for daily returns of VIX and S&P500 index 

 VIX returns S&P500 returns 
Number of obs. 1872 1872 

Minimum –0.3505 –0.0946 
Maximum 0.4960 0.1095 

1st Quartile –0.0401 –0.0049 
3rd Quartile 0.0325 0.0061 

Mean 0.0001 0.0001 
Median –0.0052 0.0008 

SE Mean 0.0016 0.0003 
LCL Mean –0.0030 –0.0005 
UCL Mean 0.0034 0.0007 

Std. deviation 0.0719 0.0144 
Skewness 0.7022 –0.2994 
Kurtosis 4.1499 8.9920 

Correlation –0.7550 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013. 

2. Methodology 

 To answer research questions and verify hypotheses we decided to un-
dertake the following steps. As some earlier studies suggest (i.e. Giot, 2005; 
Simon and Wiggins, 2001), we investigate the relationship we are interested 
in having classified daily VIX closing prices into five quintile groups4. As 
we observe very low liquidity for longer VIX futures maturities, we with-
draw from the sample all futures prices for contracts with expiration longer 
than seven months5. Then, we use OLS to estimate quadratic regressions for 
closing prices of VIX and VIX futures, independently for each quintile 
group:  

, (1) 

                                                 
4 The selection of quintile groups is somewhat arbitrary, nevertheless it was dictated by 

very heterogeneous shapes of volatility term structure of VTS in different market conditions. 
It is important to add that it really does not matter if we choose quintile or quartile groups. 

5 This amounted to be less than 25% of initial sample size. 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

9 

where  – daily closing price of VIX or VIX futures,  – time to 
maturity of -th VIX futures ( 1,… , , for VIX 	0),  – size of 
the VIX -th quintile group ( 1,… ,5). 
 As a result, we are able to define the shape of relationship between pric-
es of VIX contracts and their times to expiry. We refer to this relationship as 
volatility term structure (VTS).  
 On Figure 2, we draw all observations and shapes of volatility term 
structure, separately for five VIX quintile groups. It confirms our initial pre-
sumption that volatility term structure is dependent on VIX level. It is up-
ward sloping for initial four quintile groups where  parameter is negative. 
On the other hand, it is downward sloping for the fifth VIX quintile group 
where  parameter has positive value. 

 
Figure 2.  Quadratic regressions for term structure of VIX futures in five different 

VIX quintile groups. All calculations were made on the data from 
01/01/2006 until 01/07/2013 on the basis of VIX futures with up to 7 
months to expiration. VIX futures are quoted on CFE 

 Based on this initial intuition that volatility term structure is dependent 
on VIX level and that the slope of VTS depends on the current level of VIX, 
we propose two different measures of VTS and three different measures of 
volatility risk premium (VRP) in order to quantify VTS and risk associated 
with it. We use the results of regression (1) to construct a reference (theoret-
ical) price of volatility futures as a function of the VIX level (quintile) and 
time to expiration. The distance between the actual price and the reference 
price allows us to estimate the VRP. Then, we show that future returns of 
VIX and S&P500 index are dependent on the actual level of VIX and the 
shape of volatility term structure. This is confirmed with simple regressions 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

10

trying to find some robust patterns which are used later on to construct in-
vestment strategy, where the investment algorithms are based on VTS. 

3. Measures of Volatility Term Structure 

 We propose two measures of VTS and two measures of VRP. The de-
tailed formulas are presented below. Slope1 is a sum of differences of VIX 
futures prices with consecutive maturities divided by actual VIX level, cal-
culated separately within each VIX quintile group: 

slope1 , (2) 

where  – close price of 7-th VIX futures within -th VIX quintile group. 

 Actual values of Slope1 confirm our observations from the previous 
section concerning volatility term structure. We see substantial positive dif-
ferences between last and first contract levels for first four quintile groups 
and significant negative differences for the last quintile group (Figure 3). 

 

Figure 3.  Boxplot for Slope1 with respect to VIX quintile groups. All calculations 
were made on the data from 01/01/2006 until 01/07/2013 on the basis of 
VIX futures with up to 7 months to expiration. Diamonds denote group 
mean values, while black dots denote outliers which are outside 150% of 
interquartile range 

 The second VTS measure, Slope2, is the slope coefficient of the simple 
linear regression, estimated using OLS separately for every quintile group of 
VIX levels: 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

11

β β ,	 3 		

where  – daily closing price of VIX and VIX futures,  – time to 
maturity of -th VIX futures (for VIX  = 0, 1,… , ),  – size of the 
VIX -th quintile group ( 1,… ,5).  

Slope2 results give support to very similar conclusions as those inferred 
from Slope1 values (Table 2). 

Table 2. The descriptive statistics of Slope2 for VIX quintile groups  

VIX quintile 
group 

size 
parameters 

Min Max Avg Med SD 
VIX levels 

1 374 (0;13.14] 0.9 20.3 5.7 5.4 3.1 
2 374 (13.14;16.03] –5.4 20.1 9.4 10.6 5.6 
3 375 (16.03;19.55] –2.2 19.0 8.0 8.7 4.9 
4 373 (19.55:25.41] –10.5 16.1 4.3 4.6 5.1 
5 374 (25.41;80.86] –71.8 8.4 –9.9 –5.6 13.9 

ALL 1870 (0:80.86] –71.8 20.3 3.5 4.8 10.3 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. 

 As the next step we calculate individual volatility risk premium (VRP) by 
comparing actual VIX futures prices with their „theoretical” values given by 

formula (1), for each quintile group separately. ,  is defined as 

a percentage deviation of -th futures current price from its „theoretical” 
price: 

,
, , , (4) 

where:  – the number of the quintile group ( ∈ 1,2,3,4,5 ,  – reference 
to consecutive VIX futures contracts with ascending time to maturity 
( ∈ 1,2,3,4,5,6,7 ),  – size of the VIX -th quintile group ( 1,… ,5), 

,  – closing price of -th VIX futures contract within -th quintile group, 

,  – theoretical value of -th VIX futures index future, calculated from eq. 

(1), within VIX -th quintile group. 
  shows that inside first four VIX quintile groups we observe on 
average quite substantial departures from „theoretical” volatility term struc-
ture. They range between –20% and 20%. What is more, these deviations are 
neither skewed towards positive nor negative direction, which means that on 
average they are equal to zero. The fifth VIX quintile group shows much 
different picture. Deviations here are much more volatile and they are heavi-
ly skewed towards positive values (Figure 4). At the same time, much more 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

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than half of observations from this group have values below zero (median is 
on the level of 10%). 

 

Figure 4.  Boxplot for  with respect to VIX quintile groups. All calculations 
were made on the data from 01/01/2006 until 01/07/2013 on the basis of 
VIX futures with up to 7 months to expiration. Diamonds denote group 
mean values, while black dots denote outliers which are outside 150% of 
interquartile range 

 In the next step, we estimate the aggregated volatility risk premium for 
all maturities and for each quintile group. This measure is proposed in two 
versions in order to correctly present the direction and the magnitude of de-
viations from the “theoretical” shape: 
 aggregated volatility risk premium is the sum of individual volatility risk 

premiums for all VIX futures maturities, separately for each VIX quintile 
group: 

, ∑ , , (5) 

 absolute aggregated volatility risk premium is the sum of absolute indi-
vidual volatility risk premium for all VIX futures maturities separately 
for each VIX quintile group: 

,
| | ∑ , , (6) 

where:	  – the number of the quintile group ( ∈ 1,2,3,4,5 ),  – reference 
to consecutive VIX futures contracts with ascending time to maturity 
( ∈ 1,2,3,4,5,6,7 ). 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

13

 , contains information about the magnitude of departures for all 

maturities taken together. This could be an important factor for estimating 

the value of overall shift of VTS. On the other hand, ,
| | contains in-

formation about the direction of departures for all maturities together. Simi-
larly, this could helpful to estimate the direction of overall shift of  VTS. 

Figure 5 shows the direction of deviation ( , ), while Figure 6 pre-

sents their aggregated values ( ,
| |). 

 

Figure 5.  Boxplot for ,  with respect to VIX quintile groups. All calcula-
tions were made on the data from 01/01/2006 until 01/07/2013 on the ba-
sis of VIX futures with up to 7 months to expiration. Diamonds denote 
group mean values, while black dots denote outliers which are outside 
150% of interquartile range 

 Figure 5 allows for similar conclusions as Figure 4 but the range of fluc-
tuations inside each quintile group is now much wider – the reason for that 
being quite trivial because this is the sum of individual volatility risk premi-
ums. Moreover, we observe a large number of outliers inside the first quin-
tile group. They signal quite substantial deviations of the VIX futures from 
their theoretical shape, most often for maturities from the second one up – 
the result thereof is the highly upward sloping VTS for days with very high 

values of , or ,
| |. Furthermore, we observe a positive skew-

ness for the fifth VIX quintile group, as is the case for .  



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

14

 The situation is quite similar when we analyze Figure 6. Once again we 
observe much wider fluctuations in each quintile group than in the case of 
Figure 4. Moreover, we identify much more outliers in each quintile group 
than in Figure 5. Observations in each quintile group are more or less posi-
tively skewed with the highest skewness in the fifth quintile group. Addi-
tionally, median and mean in fifth quintile group are much higher than in 
other four quintile groups what is partly the result of the highest fluctuations 
in case of this group on Figure 5. 

 
Figure 6.  Boxplot for ,

| | with respect to VIX quintile groups. All calcula-
tions were made on the data from 01/01/2006 until 01/07/2013 on the ba-
sis of VIX futures with up to 7 months to expiration. Diamonds denote 
group mean values, while black dots denote outliers which are outside 
150% of interquartile range 

4. Forecasting Properties of Volatility Term Structure 

 In order to check predictive power of proposed measures of VTS and 
VRP we present below tables with median 1-month returns of S&P500 index 
and VIX conditional on five quintile groups of VIX and: 
 five quintile groups for Slope1 (Table 3 and Table 4), 
 five quintile groups for Slope2 (Table 5 and Table 6), 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

15

 four quintile groups6 for  (Table 7 and Table 8), 

 four quintile groups for ,  (Table 9 and Table 10), 

 four quintile groups for ,
| | (Table 11 and Table 12). 

Table 3. Median 1-month S&P500 returns (in %) conditional on VIX and Slope1 
quintile groups 

VIX 
quintile 
group 

Slope1 
I 

quintile 
group 

II  
quintile 
group 

III  
quintile 
group 

IV  
quintile 
group 

V  
quintile 
group 

ALL 

VIX level 
[–0.53: 
 –0.02] 

(–0.02: 
0.1] 

(0.1: 
0.25] 

(0.25: 
0.39] 

(0.39: 
0.92] 

[–0.53: 
0.92] 

1 [9.89:14.18] NA 3.03 1.22 1.74 1.14 1.27 
2 (14.18:17.74] 5.46 0.81 2.17 –0.18 0.72 0.86 
3 (17.74:21.68] 0.31 –3.13 0.11 2.34 3.14 1.19 
4 (21.68:26.85] 1.98 1.39 2.84 3.2 4.12 2.48 
5 (26.85:80.86] 0.52 2.94 8.00 NA NA 1.23 

ALL [9.89:80.86] 0.91 0.87 1.55 1.74 1.4 1.33 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. 

Table 4. Median 1-month VIX returns (in %) conditional on VIX and Slope1 quin-
tile groups 

VIX 
quintile 
group 

Slope1 
I 

quintile 
group 

II  
quintile 
group 

III  
quintile 
group 

IV  
quintile 
group 

V  
quintile 
group 

ALL 

VIX level 
[–0.53: 
 –0.02] 

(–0.02: 
0.1] 

(0.1: 0.25] 
(0.25: 
0.39] 

(0.39: 
0.92] 

[–0.53: 
0.92] 

1 [9.89:14.18] NA –0.04 0.91 1.43 5.01 2.52 
2 (14.18:17.74] –27.7 –2.88 –8.46 9.84 6.54 5.36 
3 (17.74:21.68] 1.27 14.95 –0.92 –7.27 –8.53 –3.39 
4 (21.68:26.85] –14.99 –1.34 –7.70 –8.50 –11.94 –7.62 
5 (26.85:80.86] –11.44 –12.78 –16.01 NA NA –12.06 

ALL [9.89:80.86] –11.95 –1.9 –3.07 –1.12 2.92 –2.68 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. Bold fonts denote values significantly different from zero. 

 We try further to find with various VTS and VRP measures any patterns 
of relationship between S&P500 index and VIX returns. Table 3.  shows that 
we cannot find any clear dependence between S&P500 return and VIX quin-
tile groups and this observation does not change for all other tables describ-

                                                 
6 We divide VRP measures into quartile groups because their fluctuations are not homog-

enous. We choose quartile groups for VRP instead of quintile groups selected for VIX to 
make results and then definitions of investment strategies more transparent. 



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DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

16

ing S&P500 returns (Table 3, Table 5, Table 7, Table 9, Table 11). Addi-
tionally, we can see that S&P500 returns do not depend on Slope1 levels. 

Table 5. Median 1-month S&P500 returns (in %) conditional on VIX and Slope2 
quintile groups 

VIX 
quintile 
group 

Slope2 
I 

quintile 
group 

II  
quintile 
group 

III 
 quintile 
group 

IV  
quintile 
group 

V  
quintile 
group 

ALL 

VIX level 
[–71.83: 
–0.72] 

(–0.72: 
2.87] 

(2.87: 
6.88] 

(6.88: 
10.54] 

(10.54: 
20.36] 

[–71.83: 
20.36] 

1 [9.89:14.18] NA 1.07 1.34 1.23 2.61 1.27 
2 (14.18:17.74] 5.46 0.81 1.76 1.67 0.07 0.86 
3 (17.74:21.68] 1.04 –2.57 –2.29 1.63 2.99 1.19 
4 (21.68:26.85] 1.98 –2.72 2.79 3.27 3.21 2.48 
5 (26.85:80.86] 0.59 2.88 1.97 9.10 NA 1.23 

ALL [9.89:80.86] 0.98 0.55 1.44 1.75 1.83 1.33 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. 

Table 6. Median 1-month VIX returns (in %) conditional on VIX and Slope2 quin-
tile groups 

VIX 
quintile 
group 

Slope2 
I 

quintile 
group 

II  
quintile 
group 

III  
quintile 
group 

IV  
quintile 
group 

V  
quintile 
group 

ALL 

VIX level 
[–71.83: 
–0.72] 

(–0.72: 
2.87] 

(2.87: 
6.88] 

(6.88: 
10.54] 

(10.54: 
20.36] 

[–71.83: 
20.36] 

1 [9.89:14.18] NA 8.63 0.85 3.04 –0.84 2.52 
2 (14.18:17.74] –27.7 –2.88 –11.78 2.53 10.79 5.36 
3 (17.74:21.68] –14.21 13.89 13.99 –5.05 –8.47 –3.39 
4 (21.68:26.85] –14.99 0.63 –3.38 –11.93 –8.33 –7.62 
5 (26.85:80.86] –11.91 –11.24 –14.69 –16.10 NA –12.06 

ALL [9.89:80.86] –12.64 0.87 –1.71 –0.99 –0.57 –2.68 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. Bold fonts denote values significantly different from zero. 

 On the other hand, we can observe – in all relevant tables describing 
VIX returns (Table 4, Table 6, Table 8, Table 10 and Table 12) – that VIX 
returns decrease almost monotonically, while VIX level moves from the first 
to the fifth quintile group. At the same table, we can see that VIX returns 
increase almost monotonically while Slope1 moves from the first to the fifth 
quintile group. Generally, Table 4 informs us about very high positive VIX 
returns for the first and the second VIX quintile groups and for the fourth 
and the fifth Slope1 quintile groups. On the other hand, we observe very 
high negative VIX returns for the fourth and the fifth VIX quintile groups 
and for the first and for the second Slope1 quintile groups. 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

17

 Slope2 discloses additional information. Table 5 shows that S&P500 
index returns increase monotonically while Slope2 moves from the first to 
the fifth quintile group. Table 6, describing the dependence of VIX returns 
on the Slope2 values, allows for almost the same conclusions as in the case 
of Table 4. The general conclusion concerning the part of the table with high 
positive and high negative returns is once again the same as in the case of 
Table 4. 

Table 7. Median 1-month S&P500 returns (in %) conditional on VIX quintile groups 
and  quartile groups 

VIX 
quintile 
Group 

 
I 

quartile 
group 

II  
quartile 
group 

III  
quartile 
group 

IV  
quartile 
group 

ALL 

VIX level 
[–0.34: 
–0.08] 

(–0.08: 
–0.01] 

(–0.01: 
0.06] 

(0.06: 
0.79] 

[–0.34:  
0.79] 

1 [9.89:14.18] 0.68 0.95 1.83 1.56 1.27 
2 (14.18:17.74] 1.77 0.44 –1.74 2.18 0.86 
3 (17.74:21.68] 1.84 1.19 0.19 1.78 1.19 
4 (21.68:26.85] 0.56 1.23 2.6 4.78 2.48 
5 (26.85:80.86] 1.08 2.99 5.68 –1.60 1.23 

ALL [9.89:80.86] 1.15 1.1 1.28 1.92 1.33 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. 

Table 8. Median 1-month VIX returns (in %) conditional on VIX quintile groups 
and 	quartile groups 

VIX  
quintile 
group 

 
I 

quartile 
group 

II 
quartile 
group 

III  
quartile 
group 

IV  
quartile 
group 

ALL 

VIX level 
[–0.34:  
–0.08] 

(–0.08:  
–0.01] 

(–0.01: 
0.06] 

(0.06: 
0.79] 

[–0.34:  
0.79] 

1 [9.89:14.18] 5.74 0.26 0.49 5.43 2.52 
2 (14.18:17.74] –6.32 2.10 15.27 1.69 5.36 
3 (17.74:21.68] –12.00 0.31 3.52 –6.55 –3.39 
4 (21.68:26.85] –3.67 –8.01 –8.59 –6.31 –7.62 
5 (26.85:80.86] –11.79 –15.9 –16.55 –9.76 –12.06 

ALL [9.89:80.86] –5.37 –2.28 0.15 –3.03 –2.68 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. Bold fonts denote values significantly different from zero. 

 Table 7 shows that S&P500 returns increase monotonically with the 
increase of  values. On the other hand, we cannot see any clear de-

pendence between VIX returns and  values (Table 8). Contrary to the 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

18

latter, the general conclusion concerning the part of the table with high posi-
tive and high negative returns still holds. 
 In Table 9 we find the most ideal dependence of S&P500 returns. They 

increase monotonically with the increase of , values. There is no 

clear dependence between VIX returns and , values (Table 10). 

Once again, the general conclusion concerning the part of the table with high 
positive and high negative returns still holds. 

Table 9. Median 1-month S&P500 returns (in %) conditional on VIX quintile groups 

and , 	quartile groups 

VIX 
quintile 
Group 

.  
I 

quartile 
group  

II  
quartile 
group 

III  
quartile  
group 

IV  
quartile 
group 

ALL 

VIX level 
[–2.19:  
–0.55] 

(–0.55:  
–0.11] 

(–0.11: 
0.58] 

(0.58:  
3.68] 

[–2.19: 
3.68] 

1 [9.89:14.18] 0.77 0.82 2.11 2.02 1.27 
2 (14.18:17.74] 1.41 1.40 –1.93 2.16 0.86 
3 (17.74:21.68] 0.28 –0.40 –0.17 2.75 1.19 
4 (21.68:26.85] –0.56 1.88 2.60 4.21 2.48 
5 (26.85:80.86] 0.42 2.90 5.50 0.28 1.23 

ALL [9.89:80.86] 0.6 1.05 1.4 2.58 1.33 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. 

Table 10. Median 1-month VIX returns (in %) conditional on VIX quintile groups 

and , 	quartile groups 

VIX 
quintile 
group 

.  
I 

quartile 
group 

II  
quartile 
group 

III 
quartile 
group 

IV 
quartile 
group 

ALL 

VIX level 
[–2.19:  
–0.55] 

(–0.55:  
–0.11] 

(–0.11: 
0.58] 

(0.58: 
3.68] 

[–2.19: 
3.68] 

1 [9.89:14.18] 7.59 3.01 –2.64 2.96 2.52 
2 (14.18:17.74] –7.72 0.06 12.13 2.17 5.36 
3 (17.74:21.68] –10.23 5.36 –1.48 –9.96 –3.39 
4 (21.68:26.85] –5.87 –12.34 –5.21 –7.42 –7.62 
5 (26.85:80.86] –8.72 –16.39 –18.52 –11.68 –12.06 

ALL [9.89:80.86] –5.13 –0.59 –0.56 –4.63 –2.68 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. Red font denotes significantly negative returns while green font 
denotes significantly positive returns. 

 Table 11 and Table 12 do not add any new information. There is no any 

clear dependence between ,
| | and S&P500 return. The same conclu-



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

19

sion we can drawn in the case of VIX returns and ,  values (Ta-

ble 12). The general conclusion concerning the part of the table with high 
positive and high negative VIX returns holds only in case of the negative 
returns. 

Table 11. Median 1-month S&P500 returns (in %) conditional on VIX quintile 

groups and ,
| |	quartile groups 

VIX  
quintile 
group 

.
| | 

I 
quartile 
group 

II  
quartile 
group 

III  
quartile 
group 

IV  
quartile 
group 

ALL 

VIX level [0.03: 0.33] 
(0.33: 
0.58] 

(0.58: 
0.87] 

(0.87: 
3.68] 

[0.03: 
3.68] 

1 [9.89:14.18] 1.69 0.75 1.31 2.24 1.27 
2 (14.18:17.74] –2.28 0.87 1.63 2.16 0.86 
3 (17.74:21.68] 0.95 –0.76 1.66 2.01 1.19 
4 (21.68:26.85] 2.74 1.12 2.22 3.83 2.48 
5 (26.85:80.86] 4.25 3.27 0.38 0.22 1.23 

ALL [9.89:80.86] 1.47 0.86 1.36 1.82 1.33 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. 

Table 12. Median 1-month VIX returns (in %) conditional on VIX quintile groups 

and ,
| |	quartile groups 

VIX 
quintile 
group 

.
| | 

I 
quartile 
group 

II  
quartile 
group 

III 
quartile 
group 

IV 
quartile 
group 

ALL 

VIX level 
[0.03: 
0.33] 

(0.33: 
0.58] 

(0.58: 
0.87] 

(0.87: 
3.68] 

[0.03: 
3.68] 

1 [9.89:14.18] –2.49 7.54 6.39 –0.44 2.52 
2 (14.18:17.74] 13.68 5.89 –3.32 –0.57 5.36 
3 (17.74:21.68] 0.35 1.56 –10.17 –8.34 –3.39 
4 (21.68:26.85] –13.8 –3.5 –3.67 –9.24 –7.62 
5 (26.85:80.86] –26.61 –15.39 –8.24 –10.27 –12.06 

ALL [9.89:80.86] –3.24 1.45 –2.73 –6.42 –2.68 
Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. Bold fonts denote values significantly different from zero. 

 Having found strong dependence patterns we will try to use them to de-
sign a simple investment strategy which is supposed to beat the market 
(S&P500 buy&hold strategy). 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

20

5. The Investment Model 

 The main objective of this section is to design investment strategies 
which will implement the dependence of VIX and VIX futures returns on 
various VTS and VRP measures. This idea has been directly motivated by our 
previous study (Jabłecki et. al. 2013a), where we found evidence of signifi-
cant relationship between lagged slope of VIX futures term structure and 
current level of VIX. This finding however, couldn’t help us to get signifi-
cantly lower average prediction errors of futures VIX level, when compared 
with naïve forecasts. We decided therefore to try to translate apparent con-
nection between term structure of VIX futures and VIX level into a profita-
ble investment strategy. 
 Our investment algorithms use the idea of mean reversion characteristics 
of VIX fluctuations and additionally, the information hidden in term struc-
ture of VIX futures shape and VRP values. Tables and figures presented 
earlier confirmed our initial intuition that VIX returns (and partly S&P500 
returns7) depend on the current level of VIX, VTS shape and VRP values. 
Higher slope of VTS (or high VRP value) together with lower VIX quintile 
group generally implicates high VIX returns, while lower slope of VTS (or 
low VRP value) together with higher VIX quintile group generally impli-
cates very low VIX returns. Additionally, we observed that average S&P500 
index returns rise almost monotonically with the increase of  both VRP val-
ues and Slope2, revealing strong dependence between index future returns 
and the current level of risk perceived by market participants. 
 We propose five simple strategies which invests in VIX futures con-
tracts. For comparison purposes we use S&P500 buy&hold strategy results.  
 The general assumptions for all strategies are as follows:  
 transaction costs = 0.1%, 
 data gathering window: minimum one year, 
 first signals: 2008-01-01, then  the current closing price is used to gener-

ate a new signal for each consecutive day, 
 trade price: closing price of VIX futures with nearest maturity, 
 switch to the 2nd contract on the last trading day (rolling yields inc-

luded), 
 leverage: 100%, 
 margin: we do not receive any additional interests from cash above  

margin. 

                                                 
7 We admit that dependence in case of S&P500 returns is much weaker than in case of 

VIX returns. 



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DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

21

 The detailed assumptions for each strategy and theirs results are present-
ed below. 

5.1. Strategy I 

 Positions are determined by following conditions: 
 buy: if  is in 1st quartile group, 

 sell: if  in 4th quartile group, 

 close: if in 2nd quartile group after sell signal or in 3rd after buy 
signal, 

 hold: if in 2nd quartile group after buy signal or in 3rd after sell  
signal. 

 Strategy I utilizes the assumption that VRP is mean reverting process 
and every departure from „theoretical” volatility term structure should be 
reverted and should go back to zero. The problem is that while it is true in-
side VIX quintile groups, the logic of the signal can be weakened by switch-
es between VIX quintile groups. This could be the reason of rather poor 
results of this strategy, which are presented in detail on Figure 7 and in Ta-
ble 13. 

5.2. Strategy II 

 Positions are determined by following conditions: 
 buy: if `in 1st quartile group, 

 sell: if in 4th quartile group, 

 hold: if in 2nd or 3rd quartile group. 
 Strategy II is very similar to Strategy I but it has less strict rules concern-
ing the moment when we close the position. The problem with switches be-
tween VIX quintile groups still exists but due to less frequent close signals 
the results of this strategy are most striking among all strategies (Figure 8 
and Table 13). 

5.3. Strategy III 

 Positions are determined by following conditions: 
 buy: if in 1st or 2nd quartile group, 

 sell: if in 3rd or 4th quartile group. 
 Strategy III is characterized by most frequent switches between short and 
long positions. Nevertheless, this modification does not enhance the results 
(Figure 9 and Table 13). 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

22

 

 

Figure 7.  Equity line and signal of Strategy I for VIX futures and S&P500 index. 
All calculations were made on the data from 01/01/2006 until 01/07/2013 
on the basis of VIX futures with up to 7 months to expiration 

 

Figure 8.  Equity line and signal of Strategy II for VIX futures and S&P500 index. 
All calculations were made on the data from 01/01/2006 until 01/07/2013 
on the basis of VIX futures with up to 7 months to expiration 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

23

 

Figure 9.  Equity line and signal of Strategy III for VIX futures and S&P500 index. 
All calculations were made on the data from 01/01/2006 until 01/07/2013 
on the basis of VIX futures with up to 7 months to expiration 

 

Figure 10. Equity line and signal of strategy IV for VIX futures and S&P500 index. 
All calculations were made on the data from 01/01/2006 until 01/07/2013 
on the basis of VIX futures with up to 7 months to expiration 

5.4. Strategy IV 

 Positions are determined by following conditions: 
 buy: if VIX in 1st or 2nd quintile group, 
 



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

24

 sell: if VIX in 4th or 5th quintile group, 
 hold: otherwise. 

Strategy IV signals depend only on VIX level. It uses extreme quintiles 
of VIX in order to generate buy and sell signals for VIX futures. The posi-
tion is hold until the opposite signal is generated. The results of this ap-
proach are rather poor (Figure 10 and Table 13). 

 

Figure 11. Equity line and signal of strategy V for VIX futures and S&P500 index. 
All calculations were made on the data from 01/01/2006 until 01/07/2013 
on the basis of VIX futures with up to 7 months to expiration 

5.5. Strategy V 

 Positions are determined by following conditions: 
 buy: if VIX in 1st or 2nd quintile group, 
 sell: if VIX in 4th or 5th quintile group, 
 close: otherwise. 

Strategy V uses the same logic as Strategy IV but closes its positions more 
often what in fact is once again not the best option (Figure 11 and Table 13). 

5.6. Comparison of All Strategies 

 Results are summarized in Table 13. To compare the strategies under 
consideration, we use several popular risk-return measures.  
 After detailed analysis of presented results we can notice that only Strat-
egy II can beat the market represented by S&P500 buy&hold (Strategy VI). 
Return (ARC) and return-risk statistics (information ratio, Sharpe and Trey-
nor coefficients) are much better for this strategy. On the other hand, risk 



Does Historical Volatility Term Structure Contain Valuable Information… 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

25

statistics show that strategy II is characterized by relatively high beta coeffi-
cient, and what is more, by much higher level of standard deviation (ASD) 
and maximum drawdown (MaxD) when compared with the benchmark  
strategy. 

Table 13. Return and risk statistics for all investment strategies 

 
Strategy 

I 
Strategy 

II 
Strategy 

III 
Strategy 

IV 
Strategy 

V 

Strategy 
VI 

buy&hold 
ARC (%) 6.56 30.81 –36.84 –17.80 –29.24 2.28 
ASD (%) 71.48 81.02 81.17 54.97 81.09 25.43 

IR 0.09 0.38 –0.45 –0.32 –0.36 0.09 
Sharpe 0.08 0.37 –0.46 –0.34 –0.37 0.06 
Treynor 0.03 0.18 –0.21 –0.14 –0.20 0.02 

Beta 1.67 1.67 1.80 1.32 1.47 1.00 
MaxD (%) –69.58 –71.57 –95.04 –89.55 –90.82 –53.25 

Note: All calculations were made on the data from 01/01/2006 until 01/07/2013 on the basis of VIX 
futures with up to 7 months to expiration. ARC – annualized return compounded, ASD, annualized 
standard deviation, IR – information ratio, Sharpe – Sharpe ration, Treynor – Treynor ratio, Beta – slope 
of regression of a given strategy on buy&hold strategy, MaxD – maximum drawdown. 

Conclusions 

 Based on the presented results we can conclude that: 
1. Volatility time structure (VTS) shape and volatility risk premium (VRP) 

values are important in order to predict VIX and partly S&P500 index 
futures. 

2. We observe very high positive VIX returns for the first and the second 
VIX quintile groups and for the fourth and the fifth Slope1 (Slope2, 

, , , ,
| |  quintile groups. On the other hand, we ob-

serve very high negative VIX returns for the fourth and the fifth VIX 
quintile groups. 

3. It is possible to use information from term structure of VIX futures to 
construct profitable strategies (Strategy II) which enhance our return-
risk ratio when compared with S&P500 buy&hold Strategy VI. 

It seems that it would be important to extend these conclusions of this re-
search on other volatility and equity index futures (e.g. VSTOXX and Euro-
Stoxx50, VNKY and Nikkei 225 and other volatility futures quoted on 
CBOE/CFE). Additionally, we would like to test investment strategies where 
each characteristics will be calculated on rolling two years window instead 
of anchored window used in this study. Further research on various defini-
tion of volatility term structure are even more important because more ade-



J. Jabłecki, R. Kokoszczyński, P. Sakowski, R. Ślepaczuk, P. Wójcik 

DYNAMIC ECONOMETRIC MODELS 14 (2014) 5–28 

26

quate reference to so called “normal” or equilibrium level of VTS is the cru-
cial point in defining diverse volatility arbitrage strategies. 

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Czy struktura terminowa zmienności zawiera istotne informacje 
w celu prognozowania zachowania się kontraktów terminowych 

na zmienność i indeksy giełdowe? 

Z a r y s  t r e ś c i. Badanie pozwoliło nam stwierdzić, że struktura terminowa kontraktów na 
zmienność (indeks VIX), a konkretniej jej nachylenie, jest zależna od aktualnego poziomu 
indeksu VIX. W momencie, w którym indeks VIX jest na niskim poziomie (poniżej 20) struk-
tura terminowa ma wysokie dodatnie nachylenie, natomiast w momencie, w którym indeks 
VIX jest na wysokim poziomie (powyżej 30) to wtedy struktura ma ujemne nachylenie. Wy-
korzystujemy te obserwacje, aby lepiej przewidywać zachowanie się stóp zwrotu kontraktów 
terminowych na zmienność. Na początek wprowadzamy miary ilościowe struktury termino-
wej zmienności (VTS) oraz premii za ryzyko zmienności (VRP). VRP pozwala nam określić 
stopień odchylenia obecnej struktury terminowej od tzw. modelowej dla danego poziomu 
indeksu VIX. Zauważamy, że wielkość odchylenia ma istotne własności predykcyjne i dlate-
go w końcowej części artykułu proponujemy strategie inwestycyjne wykorzystujące tę kon-
cepcję przy budowie algorytmów inwestycyjnych generujących sygnały na rynku kontraktów 
terminowych na indeks VIX. 

S ł o w a  k l u c z o w e: struktura terminowa zmienności, premia za ryzyko zmienności, 
kontrakty terminowe na zmienność i indeksy giełdowe, zmienność zrealizowana, zmienność 
implikowana, strategia inwestycyjna, prognozowanie stóp zwrotu, efektywne miary ryzyka 
i stóp zwrotu 


