University of Gothenburg
Master Thesis
A Swedish Model-Free Implied Volatility
Index constructed from OMXS30
options
Author:
Eric O¨stro¨m
Supervisor:
Dr. Adam Farago
Abstract
In this paper I construct a model-free implied volatility index, SVIX, from OMXS30
options based on a variance replication technique, independent of any option pricing
model. The SVIX index exhibits several stylized properties of volatility indices such as
long memory components, mean reversion and volatility clustering. The relationship
between OMXS30 returns and SVIX is negative, with some indication of an asymmetric
component. There is some evidence that implied volatility, represented among other
by SVIX, is superior to historical volatility in predicting future volatility and there
is a contemporaneous volatility transmission between VIX and SVIX. In addition, I
construct another index, SSVIX, based on simple variance swap replication which can
be hedged and priced even if we allow for jumps in the underlying asset.
A thesis submitted in fulfillment of the requirements
for the degree of Master of Science in Finance
at
University of Gothenburg, School of Business, Economics Law
December 2015
”If you hear a ”prominent” economist using the word ’equilibrium,’ or ’normal distribu-
tion,’ do not argue with him; just ignore him, or try to put a rat down his shirt.”
- Nassim Nicholas Taleb, The Black Swan: The Impact of the Highly Improbable
Acknowledgements
I am grateful for the input and goodwill I have received from other people, without your
help it wouldn’t been possible to write this paper. First, I would like to thank Harry
Matilainen at the SIX-Group for helping me with out with the option data needed for
this thesis. Secondly, I would like to thank my sister Linn and my friend Daniel for
valuable discussions on coding. Last but not least, I would like to thank my supervisor,
Dr. Adam Farago, for all valuable comments, input and guidance along the way . . .
ii
Contents
Acknowledgements ii
Contents iii
1 Introduction 1
2 Previous research 3
2.1 Research on volatility and volatility indices . . . . . . . . . . . . . . . . . 3
2.1.1 Stylized properties of volatility . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Implied volatility and IV indices as a forecast of future volatility . 4
2.1.3 Relationship between returns and volatility indices . . . . . . . . . 5
2.1.4 Volatility spillover effects in Equity markets . . . . . . . . . . . . . 6
3 Theory and construction of SVIX 7
3.1 Theory behind SVIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 Introduction to variance swaps . . . . . . . . . . . . . . . . . . . . 8
3.1.2 Valuing variance swaps using replication technique . . . . . . . . . 9
3.1.3 Limitations and critique of variance swap replication . . . . . . . . 11
3.1.4 Valuing simple variance swaps using replication technique . . . . . 11
3.2 Practical construction of SVIX and SSVIX . . . . . . . . . . . . . . . . . 12
3.2.1 Step 1. Selection of options to include . . . . . . . . . . . . . . . . 12
3.2.2 Step 2a. Calculation of variance for SVIX . . . . . . . . . . . . . . 13
3.2.3 Step 2b. Calculation of variance for SSVIX . . . . . . . . . . . . . 14
3.2.4 Step 3. Inter- or extrapolation of variance . . . . . . . . . . . . . . 14
3.2.5 Final calculation of SVIX and SSVIX . . . . . . . . . . . . . . . . 15
4 Data 16
4.1 Input data for SVIX calculations . . . . . . . . . . . . . . . . . . . . . . . 16
4.1.1 Input price data: Bid-Ask or Close prices? . . . . . . . . . . . . . . 16
4.1.2 Data restrictions and missing observations . . . . . . . . . . . . . . 17
5 Results 18
5.1 The SVIX series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.1.1 Sub period 1 Jan-2005 - May 2006 - Low volatility regime . . . . . 18
5.1.2 Sub period 2 May 2006 - Aug 2008 - Mid volatility regime . . . . . 19
5.1.3 Sub period 3 Aug 2008 - Jul 2009 - High volatility regime . . . . . 19
5.1.4 Sub period 4 Jul 2009 - Jul 2011 - Mid volatility regime . . . . . . 20
iii
Contents iv
5.1.5 Sub period 5 Jul 2011 - Jun 2012 - High volatility regime . . . . . 20
5.1.6 Sub period 6 Jun 2012 - Jun 2015 - Low volatility regime . . . . . 20
5.2 Descriptive statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3 Tests of statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Forecast quality of the SVIX index . . . . . . . . . . . . . . . . . . . . . . 24
5.5 Relationship between OMXS30 returns and SVIX . . . . . . . . . . . . . . 27
5.6 Spillover effects in volatility indices . . . . . . . . . . . . . . . . . . . . . . 29
5.7 The relationship between SVIX and SSVIX . . . . . . . . . . . . . . . . . 30
6 Conclusion 32
6.1 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
A Appendix 34
A.1 Normality tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
A.2 Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
A.3 Spillover effects in volatility indices - plot of VIX, VSTOXX and SVIX . . 35
A.4 Comparison of SVIX and SIXVX . . . . . . . . . . . . . . . . . . . . . . . 36
A.5 SVIX - OMXS30 relationship . . . . . . . . . . . . . . . . . . . . . . . . . 37
A.6 Option data used in the SVIX/SSVIX calculation . . . . . . . . . . . . . . 38
Bibliography 39
Chapter 1
Introduction
A central concept within asset pricing is the uncertainty of asset returns. The most
common way to determine the uncertainty of an asset is to estimate it’s volatility. Within
a financial setting, volatility is most often defined as the standard deviation of asset
returns:
σˆ = 1T−1
√
T∑
t=1
(Rt − R¯)2
Although volatility can not be translated directly into risk, it can serve as a risk measure.
Since σˆ (the sample standard deviation) is distribution free, Poon and Granger (2003)
argues that it is useless to use σ as a risk measure unless it is connected to a distribution
or pricing dynamic. Most often, investors assume a normal distribution for returns when
σ is used as a risk measure (Poon and Granger (2003)). The concept of risk and return
is very central within asset pricing and portfolio theory. This has led to comprehensive
research on the subject, with many ways to estimate uncertainty, or volatility, of financial
assets. Henceforth, this uncertainty will be referred to as volatility.
Many methods have been proposed to properly estimate volatility, including stochastic
and deterministic time-series methods such as Realized and Implied volatility models.
Implied volatility (IV) is derived from an option price and thus shows what the market
”implies” about future volatility of the underlying asset. In the Black-Scholes1 model,
volatility is one of six inputs, but the only which is not observable in the market it-
self (Black and Scholes (1973)). Thus, the Black-Scholes implied volatility is what the
volatility of the underlying asset ”must-be”, given all the other observable variables.
Models based on Black-Scholes options pricing formula have been used to extract the
1I discuss more on the BS assumptions in A.4 Appendix together with a description of the BS model
dependent IV index SIXVX which I use in this paper.
1
Chapter 1. Introduction 2
implied volatility from market prices, while in later years, model-free implied volatility
(MFIV) have grown in popularity. One drawback of Black-Scholes implied volatility is
that it depends on all underlying assumptions of the Black-Scholes model. What distin-
guishes Model-free IV from Black-Scholes IV, is that it relies on a replication technique
of a variance swap and does not rely on the same strict assumptions as the Black-Scholes
model. MFIV indices have been constructed on a historical as well as live-updated ba-
sis, providing investors with expected volatility in a coming period, derived from market
option prices. The most known index of this kind is represented by the VIX, provided
by CBOE since 1993. VIX measures the 30-day expected market volatility implied by
S&P 500 index options and relies on a variance replication technique to capture the
price of variance. There has been a lot of research on properties of MFIV indices, how
well they predict future realized volatility and how they compare to other volatility
models for forecasting. Several stylized facts about volatility have been established, e.g.
mean-reversion, long-memory components, stationarity and non-normality.
In this paper, I construct a MFIV index, SVIX, representing the expected 30-day volatil-
ity implied by Swedish index option prices. The index is constructed on a daily basis
from January 2005 to June 2015 using observations on OMXS30 index options. The
theory and methodology is based on the same variance replication technique that is
used for the VIX. After constructing the SVIX, I investigate its statistical properties,
how it relates to the Swedish equity market, its information content on future realized
volatility and how these findings can be interpreted. In addition, I construct a second
index, SSVIX, which relies on a replication technique of simple variance swaps (inspired
by Martin (2013)) and has been proposed to more properly capture the variation in
index movements and to allow for jumps in the underlying asset’s price.
Similar papers have been conducted on the Swedish market by Reuterhall (2005) and
Dahlman and Wallmark (2007), but on earlier time periods. With this paper, I aim to
first provide an updated version of a MFIV index on the Swedish stock market ranging
from 2005-2015, including the latest financial crisis. I provide a broad overview of many
different aspects of this index (e.g. statistical properties, relationship with underlying
index and predictive power). Secondly, there still does not (to my knowledge) exist
any official MFIV index on the Swedish stock market and as mentioned by Dahlman
and Wallmark (2007), research on smaller equity market volatility is scarce. Thirdly,
there has not been any research covering an MFIV index based on simple variance swap
replication (my SSVIX index), nor any constructed index of such kind on the Swedish
market.
Chapter 2
Previous research
2.1 Research on volatility and volatility indices
This part will outline previous academic research on the subject, relevant to volatility
and volatility indices. The first part will cover research on general statistical properties
of volatility and volatility indices. The second part will cover implied volatility and how
well it predicts future volatility. The third part covers the relationship between volatility
indices and index returns. The last part covers volatility transmission in international
equity markets.
2.1.1 Stylized properties of volatility
There are several well documented characteristics of volatility of financial time series,
including non-normality, excess kurtosis, volatility clustering, mean reversion and au-
tocorrelation. Andersen and Bondarenko (2007) find that the sample autocorrelation
pattern is very slowly decaying, supporting the hypothesis that volatility processes con-
tain long memory components. The same result is found by Ahoniemi (2006) on the
logarithmic level series of the VIX together with negative autocorrelation in the log
VIX first differenced series which point towards mean reversion. Early papers by Perry
(1982) and Pagan and Schwert (1990) conclude that volatility time series have a unit
root, while newer papers (e.g. Ahoniemi (2006), Dahlman and Wallmark (2007) Ski-
adopoulos (2004)) conclude volatility series to be stationary. Granger et al. (2000) find
volatility of intra-day returns to experience a long memory with autocorrelations that
are significantly above zero, for up to a thousand lags or more. In a study of the Greek
derivative market, Skiadopoulos (2004) finds evidence for autocorrelation and mean re-
version for the Greek implied Volatility Index (GVIX).
3
Chapter 2. Previous research 4
2.1.2 Implied volatility and IV indices as a forecast of future volatility
An implied volatility index is constituted of observations on implied volatility, extracted
from historical and current option prices. Thus, real-time implied volatility can be
compared to historical implied volatility. CBOE VIX is probably the most well known
index representing 30 day market expected volatility. But can implied volatility indices
really be considered forward looking or should they be taken as a mere indicator of
fear? If volatility indices are good estimates of future realized volatility, they should
provide unbiased estimates of future realized volatility. Whaley (2000) concludes, over
a fourteen year period, that VIX/VXO has acted reliably as a fear gauge and that high
levels of the VIX has been followed by market turmoil, however it should be noted that
Whaley’s analysis is made on the old VIX/VXO construction. Simon (2003) writes that
if implied volatility indices rather represents ’investor fear gauge’ they will incorporate
investors emotions. Thus, after a large market drop such indices would reflect investors
increased demand to buy put options. He finds evidence that the VXN index (rep-
resenting the implied 30-day volatility of Nasdaq 100 options) averages around 7-1/2
percentage points higher than subsequent realized volatility. Shaikh and Padhi (2015)
performs a similar analysis on the India VIX and reach the conclusion that the India
VIX represents both the ’investor fear gauge’ as well as being the best unbiased estimate
of future stock market volatility. Gonza´lez and Novales (2009) estimates a daily volatil-
ity index for the Spanish market using the Eurex method, used to estimate the German
and Swiss volatility indices VDAX-NEW and VSMI. They reach the conclusion that
volatility indices capture investors current attitude towards risk, but are not very useful
to predict future behavior of realized volatility over longer periods. Similar results are
found by Skiadopoulos (2004) who constructs an implied volatility index, GVIX, for the
Greek derivative market. He reaches the conclusion that GVIX cannot forecast future
FTSE/ASE-20 returns and cannot be treated as a leading indicator for the stock market.
However, Skiadopoulos (2004) finds results in line with Whaley (2000), Giot (2002) and
Simon (2003), that the index can be seen as ’investor fear gauge’.
Implied volatility extracted from option prices will give the market expectation of fu-
ture volatility. Christensen and Prabhala (1998) concludes that ”If option markets are
efficient, implied volatility should be an efficient forecast of future volatility, i.e., implied
volatility should subsume the information contained in all other variables in the market
information set in explaining future volatility”. Numerous studies have been conducted
on whether implied volatility is a good predictor of future volatility with rather mixed
conclusions. Poon and Granger (2003) find that option implied volatility contains most
information about future volatility and is superior to historical, GARCH and stochastic
volatility models. By using a data sample of S&P 100 index options from November 1983
Chapter 2. Previous research 5
to May 1995, Christensen and Prabhala (1998) also find implied volatility to be superior
to historical volatility in forecast quality. They also reach the conclusion that implied
volatility contains incremental information beyond that of historical volatility. Results
derived by Jiang and Tian (2005) on S&P 500 index options suggests that model-free
implied volatility is a more efficient predictor for future realized volatility and incorpo-
rate all information contained in the BS implied volatility and past realized volatility.
Carr and Wu (2006) find that VIX has predictive power in future realized variance and
that GARCH models do not provide additional information when VIX is included as a
regressor in the model.
On the other hand, contradicting results are found by Canina and Figlewski (1993),
Lamoureux and Lastrapes (1993) and Day and Lewis (1992). Canina and Figlewski
(1993) find that implied volatility is a poor forecast of realized volatility and that implied
volatility does not incorporate information contained in recent observed volatility. Day
and Lewis (1992) find mixed results when comparing forecast quality of implied volatility
with conditional volatility from GARCH and EGARCH models and state that neither
of the models completely characterize conditional stock market volatility.
2.1.3 Relationship between returns and volatility indices
Already in 1976, Fischer Black documented a relationship between stock returns and
volatility changes. Black (1976) determines that a drop in firm value will cause returns
of its stock to be negative and most often increase the leverage of the firm. If the debt-
equity ratio rises (assuming fixed debt), Black concludes that an effect of this will be
a rise in volatility of the stock. This effect has been referred to by academics as ”the
leverage effect”. Figlewski and Wang (2000) study this effect further and find evidence
that the suggested ”leverage effect” is more of a ”down market effect” that may have not
as much connection to firm leverage as believed. They find a strong ”leverage effect”
connected to falling stock prices but also many deviations that question the leverage
changes as the explanation. Further they state that those effects are much smaller or
even nonexistent when positive stock returns reduce leverage. This suggests that the
relationship is asymmetric, something pointed out by Whaley (2000). As mentioned
above, he argues that VIX can be used as a barometer of investors fear of downside risk,
and as investors greed in markets trending upwards. If there should be an asymmetric
relationship, negative returns should cause larger changes in implied volatility indices
than equal sized positive returns. Other empirical work that support this asymmetric
relationship is conducted by Bollerslev and Zhou (2006), Fleming et al. (1995), Simon
(2003) and Giot (2005) but results of no asymmetry is found by Giot (2002) and Dahlman
and Wallmark (2007). The negative relationship between volatility indices and returns
Chapter 2. Previous research 6
has been widely documented. However, results on whether there exists an asymmetric
component are not as clear cut and seem to differ between markets and time periods.
2.1.4 Volatility spillover effects in Equity markets
There are several studies that find results of volatility transmission between different
equity markets. Factors that have lead to increasing spillover effects between markets
have often been that markets have become more integrated and that trade flows are
multidirectional. Baele (2005) states that ”Increased trade integration, equity market
development, and low inflation are shown to have contributed to the increase in EU shock
spillover intensity”. He finds evidence for volatility transmission from the US market to
a number of European markets during high world volatility regimes. The same result,
that volatility spillover increases in high volatility regimes, have been concluded by King
and Wadhwani (1990) as well. Results derived by Ciffarelli and Paladino (2012) state
that the Latin American and European markets are highly sensitive to news in the US
markets, while Hong Kong was the only market that experienced volatility spillover from
the US market. From Granger causality tests, they conclude that there is no clear cut
hierarchy in causality of volatility transmission among the worlds equity markets, even
if US markets are often placed on top of the causal pyramid. 1
Badshah (2009) finds significant spillover effects between VIX, VXN, VDAX and VS-
TOXX with bi-directional causality which covers the implied volatility transmission. He
finds evidence that markets within the same region are of high integration, but even
markest in different regions show signs of integration in implied volatility. However,
Aboura (2003) finds evidence that the French and German implied volatility indices
show less correlation with each other than with the US index which is in contrast to the
previous studies above. Skiadopoulos (2004) concludes that there is a contemporaneous
spillover of changes in implied volatility between VIX and GVIX but no lead effects.
1For the interested reader I refer to Ciffarelli and Paladino (2012) page 52 to read more about results
on ranking of causality.
Chapter 3
Theory and construction of SVIX
3.1 Theory behind SVIX
The theory behind the construction of a MFIV index, is based on Demeterfi et al. (1999)
who covers valuation of variance swaps and how a variance swap can be replicated by
a hedged portfolio of options with suitable strikes. The CBOE VIX index relies on the
same theory with some practical adaptions. First I will briefly cover the theory behind
variance swaps and then move on to how one can use a variance replicating portfolio
to value these instruments. The theoretical discussion will end with some limitations
which arise when relying on the framework of Demeterfi et al. (1999) and how those
may affect this paper. After outlining the theoretical part, I will move on to explain the
method used to construct the SVIX.
Just as investors trade stocks for the exposure of stock prices movements, investors may
trade volatility or variance because they believe to have some insight of future volatility.
However, investors who want a pure exposure to volatility cannot make use of single
options on an index or stock due to the fact that such options provide exposure to
more variables than the volatility. By examining the option greeks in Black-Scholes
(Hull (2000)), we find that options provide exposure to stock price movements (Delta),
volatility, (Vega), time (Theta), interest rates (Rho) as well as several higher order
partial derivatives of the Black-Scholes PDE which can be computed. Demeterfi et al.
(1999) concludes that delta-hedging is at best inaccurate due to the violation of many of
the Black-Scholes assumptions; volatility cannot be accurately estimated, stocks cannot
be traded on a continuously basis, liquidity problems and so on.
A MFIV index aims to capture market expected implied variance, or generally return
variation over the coming 30 day period under a so called risk-neutral measure. However,
7
Chapter 3. Theory and construction of SVIX 8
volatility is stochastic, which makes a MFIV index to typically differ from an expected
return variation under an actual or objective measure. Thus, a MFIV index will not
provide a pure forecast of volatility for the underlying asset, but rather bundles a forecast
with market pricing of the uncertainty surrounding the forecast. An effect of this, is
that implied volatilities will include premiums, compensating investors for systematic
risk due to equity-index volatility (Andersen and Bondarenko (2007)).
3.1.1 Introduction to variance swaps
To provide a pure exposure to volatility, investors can make use of volatility swaps
which are forward contracts on future realized volatility, or variance swaps which are
similar forward contracts on the square of future volatility. Demeterfi et al. (1999)
further states that investors mainly talk about volatility, ”but it is variance that has
more fundamental theoretical significance”. This is due to the fact that the correct
way to value a swap contract, is to value it’s replicating portfolio. Variance swaps can
more reliably be replicated, by using portfolios of options at varying strike prices. The
variance rate between time 0 and time T can be replicated using a portfolio of put and
call options (Hull (2000)). The replication technique of variance swaps will make ground
for the construction of SVIX where, due to non-arbitrage conditions, the fair value of a
variance swap is determined by the cost of the replicating portfolio of options (Demeterfi
et al. (1999)). The SVIX index will then be calculated as the square root of the price of
variance.
A swap contract on variance provides investors with a clean exposure to variance, and
as with all swap contracts, the initial risk-neutral value is zero. Variance swap contracts
are forward contracts on annualized variance with a payoff at expiration equal to (σ2R−
Kvar) × N where σ2R is realized variance (expressed in Equation (3.1)) of the underlying
asset over the contract life, Kvar is the delivery price for variance and N is a notional
amount in dollars per annualized variance point. Thus, the payoff to the holder of the
contract receive N times the points of which the realized variance sigma has differed
from the delivery price Kvar.
So the variance swap will be a contract to exchange1
σ2R =
(
log
S1
S0
)2
+
(
log
S2
S1
)2
+ ...+
(
log
ST
ST−1
)2
(3.1)
1The realized variance during a time period can be expressed more formally as the sum in Equation
(5.1).
Chapter 3. Theory and construction of SVIX 9
for some fixed ”strike” V˜ at time T , where the market convention is to set V˜ so that no
money is exchanged by the initiation of the contract. The expectation of Equation 3.1
in the ∆→ 0 limit converges to2
V˜ = E∗
[∫ T
0
(d log St)
2
]
(3.2)
Following the derivation in Martin (2013), the strike on a variance swap is determined
by the price of a notional contract paying the log of the asset’s simple return at time T.
V˜ = 2rT − 2E∗ log
ST
S0
(3.3)
Demeterfi et al. (1999) derives the equation to price this contract in terms of European
call and put options as
e−rT E∗ log
ST
S0
= rTe−rT−
∫ F0,T
0
1
K2
put0,T (K) dK −
∫ ∞
F0,T
1
K2
call0,T (K) dK (3.4)
By substituting (3.4) back into (3.3) we will then end up with the general result under
assumptions 1-53 as
V˜ = 2erT
{∫ F0,T
0
1
K2
put0,T (K) dK +
∫ ∞
F0,T
1
K2
call0,T (K) dK
}
(3.5)
3.1.2 Valuing variance swaps using replication technique
Consider a portfolio of index options. This portfolio will provide exposure to both Delta,
as the index level moves, as well as Vega, measuring the variance of the index return. To
achieve a portfolio which depends only on variance, the Delta must first be neutralized
which can be done using an index forward contract with matching time to maturity.
Once the portfolio has been Delta-hedged, its value will solely be dependent on variance
movements (under Black-Scholes assumptions, Demeterfi et al. (1999) shows that the
effect of a negative Theta is offset by the Gamma). The sensitivity to variance of such a
portfolio will not be constant as I show in my example in Figure 3.1, the Vega is highest
2Martin (2013)
3Assumption 1-5: 1. European puts and calls can be traded at arbitrary strikes. 2. The underlying
asset does not pay dividends. 3. The continuously compounded interest rate is constant. 4. The
underlying asset and the risk-free bond can be traded continuously in time. 5. The underlying asset
price follow an Ito process dSt = rStdt+ σtSt dZt under the risk-neutral measure.
Chapter 3. Theory and construction of SVIX 10
for at-the-money options and drops dramatically as the options go in-the-money or out-
of-the-money. To achieve a constant sensitivity to variance movements, a wide range of
options with different strike prices must be weighted inversely proportional to the square
of the strike price K2. An intuitive explanation to this is that as the index level moves
to higher values, each additional option of higher strike in the portfolio will provide
an additional contribution to the variance sensitivity proportional to that strike. An
option with higher strike will therefore produce a variance sensitivity contribution that
increases with the index level. In addition, the contributions of all options overlap at any
definite index level. Therefore, to offset this accumulation of index level dependence, one
needs diminishing amounts of higher-strike options, with weights inversely proportional
to K2 (Demeterfi et al. (1999)). The idealized version, in continuous time, of the SVIX
index looks the following from the derivation above
SV IX2 =
2erT
T
{∫ F0,T
0
1
K2
put0,T (K) dK +
∫ ∞
F0,T
1
K2
call0,T (K) dK
}
(3.6)
40 60 80 100 120 140 160 1800
5
10
15 Strikes 80, 100, 120
40 60 80 100 120 140 160 180
×10 -3
0
0.5
1
1.5 Weighted inversely proportional to strike2
40 60 80 100 120 140 160 1800
5
10
15
20 Strikes 60 to 140, spaced 20 apart
40 60 80 100 120 140 160 180
×10 -3
0
0.5
1
1.5
2 Weighted inversely proportional to strike2
40 60 80 100 120 140 160 1800
5
10
15
20 Strikes 60 to 140, spaced 10 apart
40 60 80 100 120 140 160 180
×10 -3
0
0.5
1
1.5
2
2.5 Weighted inversely proportional to strike2
0 50 100 150 2000
5
10
15
20
25 Strikes 20 to 180, spaced 2 apart
0 50 100 150 2000
0.005
0.01
0.015 Weighted inversely proportional to strike2
Figure 3.1: Vega exposure - Left figures show call option Vega as a function of stock
price. Right figures show the sum of variance exposure for a portfolio of the same
options, weighted inversely proportional to the square of their individual strike price
Chapter 3. Theory and construction of SVIX 11
3.1.3 Limitations and critique of variance swap replication
To perfectly replicate a variance swap, it takes an infinite number of correctly weighted
strikes, which in practice is impossible. There is only a finite and discrete number of
options even in the most liquid market, which imposes some limitations to applying the
replication technique of variance swaps in theory. As seen in Figure 3.1, when more
strikes are added and the spaces between each strike are fewer, the Vega straightens out
and becomes almost constant. However, in the tails of available strikes and below/above
those, the replication will be imperfect and make the portfolio of options less responsive
than a true variance swap. Another issue is the fact that indices, and to a higher extent,
stocks undergo jumps, which is discussed by Carr and Wu (2006) and Martin (2013).
The VIX index squared has a very concrete economic interpretation and can be seen as
either the price of a portfolio of options, or as an approximation of a variance swap rate
up to the discretization error discussed above and the error induced by jumps (Carr and
Wu (2006)). Martin (2013) states that in presence of jumps, the relationship between
VIX and the quadratic variation breaks down and hence, VIX2 does not correspond to a
fair strike of a variance swap and the replicating portfolio does not replicate a variance
swap payoff. To investigate the importance of this effect, I also calculate the SSVIX
index following the derivation found in Martin (2013). The SSVIX is introduced in the
next section.
3.1.4 Valuing simple variance swaps using replication technique
Variance swaps cannot be hedged when we allow for jumps in the series and market
participants have had to impose caps on payoffs. Martin (2013) writes ”These caps
which have become since 2008, the market convention in index variance swaps as well
as single name variance swaps, limit the maximum possible payoff on a variance swap,
but further complicate the pricing and interpretation of the contract. A fundamental
problem with the definition of a conventional variance swap can be seen very easily: if
the underlying asset, and individual stock goes bankrupt, so that St hits zero at some
point before expiry T , then the payoff is infinite.”. In this section I will shortly outline
how to make use of Simple Variance Swaps and use a similar replication technique as
discussed in the previous section to value a Simple Variance Swap in the presence of
jumps. Following Martin (2013), a Simple Variance Swap is an agreement to exchange
(
S1 − S0
F0,0
)2
+
(
S2 − S1
F0,1
)2
+ ...+
(
ST − ST−1
F0,T−1
)2
(3.7)
Chapter 3. Theory and construction of SVIX 12
for a pre-determined strike V at time T , where F0,t is the forward price of the underlying
index or stock at time t, known at time 0.
The continuous time pricing and hedging of a simple variance swap in the ∆ → 0 limit
simplifies to4
SSV IX2 =
2erT
T
{∫ F0,T
0
1
F 20,T
put0,T (K) dK +
∫ ∞
F0,T
1
F 20,T
call0,T (K) dK
}
(3.8)
For those interested in the complete derivation and theory of Simple Variance Swap
pricing and hedging, please see Martin (2013).
The difference between the continuous SVIX and continuous SSVIX definition is that
the weights in SSVIX contains F 20,T instead of K
2. This can easily be seen by comparing
Equations (3.6) and (3.8)
3.2 Practical construction of SVIX and SSVIX
We saw the continuous models of SVIX and SSVIX in the previous section. In this
section I will outline how one can use the discretized versions to achieve the desired
results. I will describe both the SVIX and SSVIX methods, which are very similar up
to the point of the variance calculation which will be clear in the following sections.
3.2.1 Step 1. Selection of options to include
Options included in the calculation of the SVIX are out-of-the-money OMXS30 calls
and puts centered around an at-the-money strike price K0. For each day SVIX is to
be calculated, this reference strike needs to be determined first. Since options with
exactly 30 days to maturity are usually not available, a 30-day variance will be inter- or
extrapolated from options with shorter/longer maturities. The options are divided into
”near-term” and ”next-term” expiring in the two nearest months. CBOE uses options
with more than 23 days and less than 37 days to expiration, including both standard 3rd
Friday expiration options as well as weekly options. In 2014, CBOE started to include
weekly options, something I cannot do in this paper due to data limitations.
The first step is to determine a forward OMXS30 level, which is done by identifying the
smallest absolute difference between calls and puts at each specific strike price. Prices are
4Please note that what I refer to as SVIX in this paper, is the ”normal” implied volatility index based
on variance swap replication and should not be confused with what Martin (2013) refers to as SVIX. In
his paper, his SVIX is based on the Simple Variance Swap replication, which in this paper is referred to
as SSVIX.
Chapter 3. Theory and construction of SVIX 13
mid-quoted bid/ask prices. Using the defined strike where the put/call price difference
is smallest for near-term and next-term options, the forward level F is calculated as
F = Strike price + eRT × (callprice− putprice) (3.9)
where:
• Strike price is the identified option strike which minimizes the | call price− put price |
• R is the continuously compounded risk free interest rate
• T is time to maturity in days
• call price is the call option price identified at Strike price
• put price is the put option price identified at Strike price
The forward OMXS30 level, can be seen as an expected at-the-money expiry level of
the options. Given the reference point, F , K0 can be determined - the strike price
immediately below the forward index level F for both near- and next-term options. We
define the strike price of the ith option as Ki and the price corresponding to that strike
as Q(Ki). Using the criteria below, we chose put options for strike prices below K0, and
call options for strike prices above K0. At K0, both call and put prices are chosen and
their prices are averaged. That is,
Q(Ki) =



put price for Ki < K0
(put price + call price)/2 for Ki = K0
callprice for Ki > K0
3.2.2 Step 2a. Calculation of variance for SVIX
When the correct option strips are chosen according to the Q(Ki), the variance for near-
and next term options can be calculated where σ21 is calculated using near-term and σ
2
2
calculated using next-term options. The below equations are discretized versions of (3.6)
where σ21 and σ
2
2 are calculated using near-term and next-term options respectively.
σ21 =
2
T1
∑
i=1
∆Ki
K2i
eRT Q(Ki) −
1
T1
[
F1
K0
− 1
]2
(3.10)
σ22 =
2
T2
∑
i=1
∆Ki
K2i
eRT Q(Ki) −
1
T2
[
F2
K0
− 1
]2
(3.11)
Chapter 3. Theory and construction of SVIX 14
where T is time to expiration for the ith option, K2 is the squared strike of the ith option
and ∆K is the difference between the strike of Ki and Ki−1 defined as follows
∆Ki



Ki − Ki−1 for Ki = Kmax
(Ki+1 − Ki−1)/2 for Kmax > Ki > Kmin
Ki+1 − Ki for Ki = Kmin
Hence, a single option’s contribution to the variance is proportional to ∆K and to the
price of the option, and inversely proportional to the square of that option’s strike price.
The last term to the right in the equation adjusts for the difference between F and the
nearest available strike.
3.2.3 Step 2b. Calculation of variance for SSVIX
The discretized version of the Simple Variance Swap replication for SSVIX will be very
similar to the one used to calculate SVIX. The only difference is the denumerator in the
sum part, which is exchanged for Fi. Since Fi is a constant for each given T the variance
calculation simplifies to
σ21 =
2
T1
∑
i=1
∆Ki
F 2T,0
eRT Q(Ki) −
1
T1
[
F1
K0
− 1
]2
(3.12)
σ22 =
2
T2
∑
i=1
∆Ki
F 2T,0
eRT Q(Ki) −
1
T2
[
F2
K0
− 1
]2
(3.13)
These equations are discretized versions of (3.8) and apart from the variance construc-
tion, SSVIX follows the same steps as SVIX. The same rule is applied here, σ21 and σ
2
2
are calculated using near-term and next-term options respectively.
3.2.4 Step 3. Inter- or extrapolation of variance
To make sure that the SVIX index reflects a 30-day market expected variance, the
variance calculations needs to be inter- or extrapolated for options with shorter or longer
maturities
σ2 = [T1σ
2
1w + T2σ
2
2(1− w)]
365
30
(3.14)
Chapter 3. Theory and construction of SVIX 15
where T1 and T2 are time to maturity in years for the two contract months σ21 and σ
2
2
are variances annualized for the periods corresponding to T1 and T2 and w is a weight
assigned to the first option strip to get a 30-day variance
w =
T2 −30 /365
T2 − T1
(3.15)
For those days where both the near-term and next-term options are maturities larger
than 30 days, the weights will be larger than one or less than zero, since the inter- or
extrapolation is of linear shape.
3.2.5 Final calculation of SVIX and SSVIX
After the inter- or extrapolation, SVIX and SSVIX can finally be calculated as
SV IX = 100
√
σ2 (3.16)
SSV IX = 100
√
σ2 (3.17)
Chapter 4
Data
4.1 Input data for SVIX calculations
In this part I will present the data needed for the SVIX calculations and modifications
needed to be done to ensure the reliability and quality of the data.
To construct an IV index, one need to obtain option prices for a wide range of strike
prices. I obtained data on OMXS30 option prices from SIX1 which includes daily ob-
servations on a large range of strike prices for put and calls. Prices are divided into bid,
ask and close, where information exists, and obviously there are far more bid/ask than
close prices. The moneyness of the options serve as an indicator of how liquid they are.
E.g. deep in-the-money options, are far less liquid and therefore have fewer close prices
observed. The sample period ranges from January 2005 to June 2015. The options
used are of European style, i.e. they can only be exercised at maturity. Table A.3 in
Appendix shows the amount of options used to calculate SVIX and SSVIX.
A proxy for the risk-free rate is used and matched with each observation date. The
proxy used for risk-free rate is the STIBOR-1M obtained from Sveriges Riksbank and is
expressed as continuously compounded.2
4.1.1 Input price data: Bid-Ask or Close prices?
When constructing the index, one has to chose whether to use an average of bid-ask
or the actual closing prices. There are pros and cons using both methods, but for this
1SIX is a leading provider of financial information within Europe and specializes in processing and
distributing financial data for institutional investors http://www.six.se/en/about-us.
2In theory, a risk-free rate with exact match of the days to maturity should be used, but due to
practical limitations I have made use of the STIBOR-1M as an approximation.
16
Chapter 4. Data 17
paper I have chosen to work with an average of bid-ask, and will explain the rationale
for this below.
For the VIX construction, CBOE uses midquoted bid-ask prices. Previous studies by
Fleming et al. (1995) conclude that to avoid negative autocorrelation implied in close
prices caused by the bid-ask bounce, it is better to use midquoted bid-ask prices. Fur-
ther, Lamoureux and Lastrapes (1993) make use of the midpoint of bid-ask prices and
states that ”actual transaction prices may include price pressure effects, and (latest)
transaction prices from both markets at a fixed point in time (such as closing) will al-
ways be asynchronous”. On the other hand Skiadopoulos (2004) uses close prices as he
finds much stronger negative autocorrelation in the case when GVIX is constructed by
the average of bid-ask prices and suggests that close prices should be superior.
I find the synchronicity argument to be of most importance mainly due to the fact
that OMXS30 options are not that liquid. Consider a day with large index movements,
observing an option with moneyness far from unity. It may be the case that there has
been only one settlement in the morning and after that the index has moved significantly.
The close price will then poorly represent the index movements and settlement level,
while an average of bid-ask prices at the end of the day will better capture this change
and settlement level. Another, more practical reason, for using midquoted bid-ask prices
is the fact that my dataset contains far more bid-ask prices than close prices. If I were
to make use of close prices, there would be a significant amount of missing days in the
SVIX index.
4.1.2 Data restrictions and missing observations
There are days where neither SVIX nor SSVIX can be constructed due to missing ob-
servations in the dataset. This may be due to that no matching put and calls are found
to identify Fi, or that Fi has been identified, but there are only one observed put or call
at either side of Fi. There needs to be at least two observed option prices on either side
of Fi (or K0) in order to come up with a ∆K, hence if there is only one observed option
price, no difference can be calculated3. For VIX, CBOE uses options with at least eight
days of maturity, since options closer to maturity might be affected by expiry effects and
thus introduce noise to the index. However, since my dataset does not include that many
options, I set the restriction at two days of maturity to not lose to many observations.
3I chose to drop days with less than two differences in either side. For the call options this is 20 days,
and for the puts only seven days, i.e. a very small part of the entire sample. More on this can be found
in section A.6 Appendix.
Chapter 5
Results
5.1 The SVIX series
In the first part of this Chapter I will go through the SVIX series together with the
underlying index, OMXS30, for the time period and point out events which have had
a significant impact on the SVIX level. Figure 5.1 shows the OMXS30 on the left-
hand axis vs. SVIX on right-hand for the period 2005-01-03 to 2015-06-11. The time
period has been divided into six sub periods which are characterized by low, medium
or high volatility environments and also selected in combination with signifiant periods
characterizing bull/bear markets for the OMXS30. The overall SVIX level mean is 22.89
with a sample standard deviation of 9.46. I will not cover the SSVIX index in this first
part since its movements in relation to the OMXS30 are very similar to those of SVIX.
The topic of interest regarding the SSVIX, is how it relates to SVIX since they are
computed in two different ways. This will be covered later in the result section.
5.1.1 Sub period 1 Jan-2005 - May 2006 - Low volatility regime
The first period is characterized by a stable SVIX level with a low sub period mean
of 15.07. Observing the OMXS30 we can see a steady increase in this period of 36%.
By the end of this period the OMXS30 drops from the peak in April 2006 at 1076 to
levels slightly below 900. Looking at the SVIX, we find a dramatic increase in the SVIX
relating to this period with a first peak at 42.75. The sub period standard deviation of
SVIX (the volatility of volatility) is the lowest for the entire sample at 1.80.
18
Chapter 5. Results 19
Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-160
1000
2000 OMXS30 vs. SVIX
0
50
100
Subperiod1
Subperiod2 Subperiod4 Subperiod6
Subperiod5
Subperiod3
Figure 5.1: OMXS30 and SVIX development 2005-2015
5.1.2 Sub period 2 May 2006 - Aug 2008 - Mid volatility regime
Apart from the first month, where the OMXS30 drops some 100 points and SVIX reaches
its first peak, this period was quite stable. Looking at the OMXS30 we see a steady
increase up to the highs before the financial crisis of 2008-09 and relatively stable volatil-
ity. Although we find some spikes in the SVIX together with increased variance of the
SVIX by the end of 2007, the mean SVIX level during this period remained at 24.34,
i.e. slightly above average. SVIX sub period standard deviation remained 5.56.
5.1.3 Sub period 3 Aug 2008 - Jul 2009 - High volatility regime
Bear markets following the financial crisis of 2008 pushed the SVIX index to the highest
levels in my sample, which not surprisingly coincide with the bottom of the crisis. The
first peak, found at September 15th 2008 coincides with Lehman Brothers filing for
Chapter 11 bankruptcy protection. After the Lehman crash, implied volatilities reached
extreme levels which made the SVIX reach a new peak of 81.70. During this period the
SVIX mean was 43.14, i.e. 88% higher than the overall mean. SVIX sub period standard
deviation was 12.88, i.e. this period was characterized by both volatile asset prices as
well as high volatility of volatility. We will see later that the differences between SVIX
and SSVIX during this time period is by far the largest ones.
Chapter 5. Results 20
5.1.4 Sub period 4 Jul 2009 - Jul 2011 - Mid volatility regime
During this sub period, the OMXS30 increased by 43%. The SVIX sub period mean was
23.01, i.e. at the overall mean level. There are some smaller spikes in the SVIX level in
mid-2010 but both the mean and variance of SVIX remained relatively low overall. This
was a period of medium volatility in line with the average SVIX level, but less volatile
SVIX level compared to the entire sample.
5.1.5 Sub period 5 Jul 2011 - Jun 2012 - High volatility regime
In the wake of the financial crisis of 2008, many European Union countries faced severe
problems to refinance their government debt or to bail-out national banks. Market tur-
moil over great sovereign debts and possible insolvency of the financial system, together
with poor economic outlooks, made markets take a down-turn by the end of summer
2011. We can also see that the SVIX peaks at 60.12 and then continue to stay at quite
high levels, around 40, for some time before a drop in the beginning of 2012. During
this period, the negative relationship between the OMXS30 and SVIX becomes evidently
clear with the last peaks of the SVIX around April 2012. The SVIX volatility was 7.46
which is slightly below the sample volatility.
5.1.6 Sub period 6 Jun 2012 - Jun 2015 - Low volatility regime
The last period of the sample contains a relatively stable period for the SVIX level and
and upward-trending bull-market for the OMX30, reaching all-time-high levels in April
2015. During this period, the SVIX mean was 17.02 while the OMXS30 increased by
59%. By observing this period, we get a hint of the asymmetrical relationship between
implied volatility indices and their underlying indices. While OMXS30 grew steadily,
the SVIX remained fairly constant with a lower mean compared to the entire period.
The asymmetrical relationship will be examined more rigorously later in this chapter.
The volatility of SVIX during this period was low, at 3.24.
Chapter 5. Results 21
5.2 Descriptive statistics
Table 5.1 shows the overall summary statistics for all indices and each respective differ-
enced series. VSTOXX is an implied volatility index calculated using the same method
as VIX, but with EURO STOXX 501 as underlying. Comparing SVIX with SSVIX we
see that SSVIX has a slightly lower mean but are still very close to each other. SVIX
has also had a higher mean than VIX but lower than VSTOXX during the sample pe-
riod. Finally we compare SVIX with SIXVX and find that SVIX has higher mean and
standard deviation for the leveled series. For the differenced series, SVIX has more than
double standard deviation compared to SIXVX. The reason for higher standard devia-
tion in my series (SVIX and SSVIX) might be due to some noise introduced by weak
restrictions on e.g. maturity but also due to fewer data points2.
Table 5.1: Summary statistics
level series differenced series
index N mean std dev N mean std dev
SVIX 2,505 22.89 9.46 2,504 0.001 3.925
SSVIX 2,505 22.48 9.43 2,504 0.001 3.715
VIX 2,608 19.81 9.79 2,607 0.000 1.848
VSTOXX 2,655 23.52 9.59 2,654 0.004 1.829
SIXVX 2,500 20.80 9.11 2,499 0.002 1.877
5.3 Tests of statistical properties
This part will present some basic statistical tests commonly performed when working
with time series. First I will evaluate whether the level and differenced series follow a
normal distribution, are stationary and if they exhibit autocorrelation, volatility clus-
tering and mean-reversion.
Figure A.1 in Appendix A shows histograms for the SVIX and SSVIX level and differ-
enced series. The distributions of the level series are right-skewed, while the distribu-
tions for the differenced series are leptokurtic. To ensure the statistical significance of
the non-normality I perform a Jarque-Bera test under the null-hypothesis that the series
are normally distributed. The alternative hypothesis is that the data is non-normally
1EURO STOXX 50 is a stock index designed by STOXX, an index provider owned by Deutsche Borse
Group and SIX Group. I obtain SIXVX data from SIX Group. Data on VIX and VSTOXX is obtained
from Bloomberg.
2A more thorough discussion about the relationship between SVIX and SIXVX is found in Appendix
A.4.
Chapter 5. Results 22
distributed. The null is rejected at the 1% significance level for all series. These results
are in line with previous studies on the well documented non-normality of volatility.
Table 5.2: Normality tests
Jarque-Bera Skewness Kurtosis
SVIX 5,151*** 1.98 8.80
SSVIX 3,703*** 1.82 7.71
∆SV IX 9,183*** 0.68 32.64
∆SSV IX 4,341*** 0.07 23.40
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
Stationarity has also been a widely documented property of volatility and is a necessity
for mean reversion. To test whether SVIX, SSVIX and their differenced series contain a
unit root I run Dickey-Fuller tests. The null-hypothesis in the Dickey fuller is that the
series contain a unit root, whereas the alternative hypothesis is that the series does not
contain a unit root. Table 5.3 shows the outcome of each respective test; the null of a
unit root is rejected in each case.3
Table 5.3: Unit root tests
test stat(obs) N
SVIX ADF (lag = 1) -10.11*** 2,505
SSVIX ADF (lag = 1) -9.71*** 2,505
∆SV IX ADF (lag = 1) -72.13*** 2,504
∆SSV IX ADF (lag = 1) -70.20*** 2,504
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
Figure 5.2 shows the autocorrelation structure of the level SVIX series and the first
differenced SVIX series. As seen from the sample autocorrelation of the level SVIX series,
the autocorrelations decay extremely slow which is in line with the well documented
property of volatility being long a memory process. The first differenced series exhibit
negative autocorrelation which may be supporting evidence of mean-reversion. It might
be important to point out that mean reversion is a stylized fact in asset price volatility
and not in asset returns. Commonly, the result of negative autocorrelation in asset
returns is taken as evidence of mean-reversion, although it must not be the case.4 I
will not dig deeper into this topic since previous research has established that volatility
3I re-ran the adftest using a larger number of lags. The null hypothesis is rejected for lags up to 28,
which all are significant at a 5% level.
4For the interested reader I refer to Choe et al. (2007) and Hillebrand (2003) for further discussion on
mean reversion of asset returns. Choe et al. (2007) state that the necessity of negative autocorrelation
in one-period returns is an artifact of assuming an AR(1) process for the transitory components of the
underlying asset. They show that the first-order autocorrelation sign can be positive under a different
lag specification of the transitory components of asset prices.
Chapter 5. Results 23
processes are mean reverting. My results of negative first-order autocorrelation in the
differenced volatility series points towards the same result. These results are in line with
results found by e.g. Andersen and Bondarenko (2007) and Ahoniemi (2006).
Lag0 20 40 60 80 100
Samp
le Auto
corre
lation
-0.2
0
0.2
0.4
0.6
0.8
1 SVIX
Lag0 20 40 60 80 100
Samp
le Par
tial Au
tocor
rela
tions
-0.2
0
0.2
0.4
0.6
0.8
1 SVIX
Lag0 20 40 60 80 100
Samp
le Auto
corre
lation
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 SVIX differenced
Lag0 20 40 60 80 100
Samp
le Par
tial Au
tocor
rela
tions
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1 SVIX differenced
Figure 5.2: Autocorrelation structure of SVIX and first differenced SVIX
Table 5.4 shows the first three autocorrelation coefficients for each respective series to-
gether with the Ljung-Box Q statistic. The Ljung-Box test for general autocorrelation
in the residuals under the null-hypothesis that the residual series do not exhibit autocor-
relation. The results clearly state that the null that all autocorrelation coefficients up to
the specified lag are simultaneously zero is rejected, i.e. the series exhibit autocorrelation
up to the specified number of lags. In the differenced series there is significant negative
autocorrelation for the first lag at 1% level and for the second lag at 5% level, which is in
line with the previously mentioned mean reversion property of volatility. The Ljung-Box
test statistics also show that we can reject the null of all autocorrelations coefficients up
to 7, 25 or 50 lags simultaneously being zero. The evidence of high autocorrelation in
the SVIX/SSVIX series indicates that the SVIX series exhibit volatility clustering.
Table 5.4: Autocorrelation test
coefficients lag 1-3 Ljung-Box Q stat
ρ(1) ρ(2) ρ(3) lags (7) lags (25) lags (50)
SVIX 0.922*** 0.899*** 0.886*** 13,339*** 40,160*** 69,303***
SSVIX 0.928*** 0.903*** 0.887*** 13,478*** 40,122*** 69,553***
∆SV IX -0.351*** -0.067** 0.035 355*** 385*** 477***
∆SSV IX -0.327*** -0.058** -0.009 297*** 327*** 395***
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
Chapter 5. Results 24
With the results from tests in this section, we can establish that the SVIX series is
stationary, exhibits non-normality, volatility clustering and mean reversion.
5.4 Forecast quality of the SVIX index
Numerous studies have been conducted on whether implied volatility is a good predic-
tor of future realized volatility. The results have been pointing in different directions
depending on time and data for the study. I compute the annualized 30 day5 realized
volatility according to
RVt,t+30 = 100×
√
√
√
√365
30
30∑
i=1
[
log
(
St+i
St+i−1
)]2
(5.1)
To evaluate the information content and quality of SVIX in predicting future realized
volatility, I follow the procedure of Christensen and Prabhala (1998) and Christensen
and Hansen (2002). I run the following regressions to asses the information content in
implied volatility of future volatility on a non-overlapping sample with 84 observations.
RVt+1,t+30 = α0 + β1SV IXt + et (5.2)
RVt+1,t+30 = α0 + β2SSV IXt + et (5.3)
RVt+1,t+30 = α0 + β3RVt−29,t + et (5.4)
RVt+1,t+30 = α0 + β4SIXV Xt + et (5.5)
RVt+1,t+30 = α0 + β1SV IXt + β3RVt−29,t + et (5.6)
RVt+1,t+30 = α0 + β2SSV IXt + β3RVt−29,t + et (5.7)
RVt+1,t+30 = α0 + β4SIXV Xt + β3RVt−29,t + et (5.8)
where RVt+1,t+30 is the realized future volatility at time t, SVIXt is the implied 30 day
expected volatility at the beginning of period t, SSVIXt is the implied 30 day expected
volatility derived from simple variance swap replication at time t and RVt−29,t is the
historical realized volatility at time t. Christensen and Prabhala (1998) states that there
are three hypotheses that can be tested from these regressions. The first is to asses if
implied volatility contain any information about future realized volatility, β1 should be
5At day t, the historical volatility is the sum of days t-29 up to day t, and future volatility at day t
is the sum of days t+1 upp to t+30. Equation (5.1) is linked to Equation (3.1) since it is the Realized
variance under a certain period which enter into the Variance swap contract.
Chapter 5. Results 25
significantly different from zero. Second, if implied volatility should provide unbiased
forecasts of future realized volatility, the intercept α0 should be zero and the coefficient
for implied volatility β1 should equal unity. Third, if implied volatility provides efficient
forecasts, the residuals et should be white noise and not correlated with any variable
in the information set. Figure 5.3 shows a plot with volatility on y-axis of my different
volatility series with the 84 observations.
Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-160
10
20
30
40
50
60
70
80
90 HRV, FRV, SVIX, SSVIX & SIXVX plot
Figure 5.3: Historical RV in red, Future RV in blue, SVIX in dotted black, SSVIX in
dotted purple and SIXVX in dotted cyan
Table 5.5 shows the separate regression results from the equations above. As mentioned
before, some earlier studies concluded that implied volatility is a poor forecast of future
volatility. However, some of these studies have been criticized since they make use
of overlapping data. I run the forecasting regressions and divide my sample into 84
observations6. From the DW-statistic we see that there is no autocorrelation among
the residuals as the DW statistics are not significant at any level. From Equation (5.2)
we can see that the coefficient for SVIXt is 0.789 and significant at a 1% level, with
an adjusted R2 at 0.542. This result implies that SVIX contains information about
future volatility, however it appears to be a biased estimate. By looking at the F-value,
we can reject the joint hypothesis of α0 = 0 and β1 = 1. From results on Equation
(5.3) we can see that the coefficient for SSVIXt is 0.862 and significant at 1% level with
an adjusted R2 of 0.538, thus SSVIX also contains information about future volatility.
SSVIX compared to SVIX seems to provide a slightly less biased forecast since the
coefficient is closer to unity, however the adjusted R2 drops. Results from Equation
(5.4) shows that historical volatility also contains information on future volatility with
a coefficient of 0.697 and significant at a 1% level. However, compared to SVIX and
6I end up with 84 observations where I have data for all 84 days for all variables.
Chapter 5. Results 26
SSVIX, historical volatility provide a more biased forecast with a lower adjusted R2.
Results from Equation (5.5) show that SIXVX seem to provide the least biased estimate
with a coefficient of 0.948 and significance at a 1% level and highest R2 of 0.635. Based
on the results from Equations (5.2) - (5.5) it looks like that the implied volatility indices
are superior to historical volatility in predictive power of future volatility.
When SVIX is included as an additional regressor to historical volatility, the coefficient
for historical volatility drops to 0.279 and is only significant at a 10% level with an
adjusted R2 of 0.571. Similar results are found from Equation (5.7), when SSVIX is
included as an additional regressor. The coefficient for historical volatility drops to
0.283 and shows only significance at a 10% level, with an R2 of 0.566. When SIXVX
is included as a regressor additional to historical volatility, the coefficient for historical
volatility is not significant at a 10% level. However, the SIXVX coefficient is still close
to unity and significant at a 1% level with a R2 of 0.644. Also these results point
towards that the implied volatility indices are superior to historical volatility since the
significance of RVt.29,t drops to 10% and the coefficient drops by more than 50%.
Table 5.5: Regression results - Volatility forecasting using non-overlapping data
Eq. intercept SVIXt SSVIXt RVt−29,t SIXVXt adj R2 DW F-value N
(5.2) 5.701*** 0.789*** 0.542 1.73 94.5 84
(2.246) (0.101)
(5.3) 4.397** 0.862*** 0.538 1.77 57.6 84
(2.429) (0.114)
(5.4) 7.454*** 0.697*** 0.497 1.98 79.1 84
(2.443) (0.113)
(5.5) 4.330*** 0.948*** 0.635 1.77 218.9 84
(1.414) (0.064)
(5.6) 5.037** 0.531*** 0.279* 0.571 1.90 50.9 84
(2.564) (0.125) (0.161)
(5.7) 4.234* 0.574*** 0.283* 0.566 1.92 54.2 84
(2.597) (0.145) (0.159)
(5.8) 4.307*** 0.012 0.935*** 0.644 1.92 351.7 84
(1.602) (0.169) (0.1481)
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
In summary, my results show evidence of implied volatility, represented by SVIX, SSVIX
and SIXVX, being superior to historical volatility in predicting future volatility. All of
the explanatory variables provide efficient forecasts. However, all of them provide biased
forecasts with adjusted R2 between 50-60%. These results are in line with Dahlman and
Wallmark (2007) who also finds SVIX to be superior to historical volatility with biased
forecast and R2 around 50%. Having established that IV contain information on future
volatility, I will move on to investigate the relationship in the next section.
Chapter 5. Results 27
5.5 Relationship between OMXS30 returns and SVIX
The negative relationship between IV indices and their underlying indices has been
established by many previous studies. Figure 5.1 gives a hint of that the same result
holds for SVIX and OMXS30. The correlation between ∆SVIX and OMXS30 (cc) daily
returns expressed as OMXS30rett = 100× (lnOMXS30t − lnOMXS30t−1), for the
entire sample is −0.32 (p-vale 0.000) and ranges around the same value for all sub
periods as seen in Table A.1. These results show slightly less correlation than results
presented by Dahlman and Wallmark (2007) which find negative correlation between
SVIX and OMXS30 of -0.43 while Carr and Wu (2006) finds negative correlation of
-0.78 between VIX and S&P500 returns. On the other hand, Skiadopoulos (2004) finds
as low correlation as -0.17 between the Greek IV index and FTSE/ASE-20 returns.
Dahlman and Wallmark (2007) states that lower correlation might be an effect of lower
liquidity in the option markets. It is reasonable to assume that less liquid markets will
yield fewer option observations to match the index, and thus lead to lower correlation.
Following Skiadopoulos (2004) I also perform a Granger causality test to establish if
OMXS30 returns are caused by changes in SVIX or the other way around. It may be
reasonable to assume that changes in the OMXS30 level causes option prices to change,
and hence SVIX, although causality is highly philosophical. The null under the Granger
causality test states that y does not granger cause x, and if the F-value is larger than the
critical value we can reject the null. As seen in Table 5.6 the results are unidirectional,
i.e. that changes in OMXS30 return cause changes in the SVIX level and not vice-versa.
Table 5.6: Granger causality tests
Null hypothesis F-value Crit-val lags
∆SVIX does not Granger cause OMXS30ret 2.81 3.85 10
OMXS30ret does not Granger cause ∆SVIX 34.89*** 2.38 10
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
Whether there exists an asymmetric effect of volatility indices on asset returns has been
widely discussed with different conclusions. As discussed in the previous research section,
there might be an asymmetric component in implied volatility indices which Figure 5.1
might show evidence of. During times when the SVIX level remain fairly constant,
OMXS30 seems to yield positive returns, while when changes in SVIX are positive, the
OMXS30 seems to drop and yield negative returns. If losses in the OMXS30 lead to an
increase in the SVIX which are larger than equally sized positive returns, there might be
evidence for an asymmetric relationship. To assess whether there exists an asymmetric
Chapter 5. Results 28
relationship I follow the methodology of Whaley (2000). I run the following regression
to test this relationship
∆SV IXt = α0 + β1OMXrett + β2OMXret
−
t + t (5.9)
Where OMXret−t is equal to OMXrett if OMXrett < 0 and zero otherwise.
If there should exists an asymmetric relationship, the coefficient for OMXret−t should be
positive and significantly different from zero. The coefficient for OMXrett express the
standard relationship between daily OMXS30 returns and changes in SVIX level.
As seen in Table 5.7, the coefficient for OMXrett is −0.768 and significant at 1% level.
I.e. if OMXS30 yield positive (cc) daily return of 1% unit, SVIX decreases on average
0.768 ppts. The coefficient for OMXret−t is negative and not statistically significant at
any level. Table A.2 in Appendix shows complete results for every sub period. I run
the same test in each sub period to see if the results are different in low/high volatility
regimes. The OMXrett coefficient is negative significant in most cases at 5% level, while
OMXret−t is only positive and significant for sub period 6.
Table 5.7: Regression results - asymmetric relationship
intercept OMXrett OMXret
−
t adj R
2 N
-0.022 -0.768*** -0.095 0.103 2,505
(0.083) (0.129) (0.175)
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
The results from the entire period clearly point out the negative relationship between
index returns and changes in SVIX. These results are in line with many other studies
on the negative relationship between index returns and their volatility indices, e.g. Giot
(2002) and Whaley (2000). However, the coefficient for OMXret−t is not significantly
different from zero for the entire sample period, suggesting that there should be no
significant asymmetric effect between the two. However, sub period six, which is a
low volatility regime, show results of an asymmetric component since the OMXret−t is
positive and significant at a 1% level. This means that for days when OMXS30 yield
negative returns, the combined effect on SVIX from both independent variables is 2.034,
while when OMXS30 yield positive returns the effect on SVIX level is only 1.139.
Results derived by Dahlman and Wallmark (2007) implies that there exists an asym-
metric component during low volatility regimes. An explanation for this might be that
demand for put options as insurance rises in low volatility regimes, but not when volatil-
ity already is high. Giot (2005) states that the rationale behind this is that traders are
unwilling to bid more aggressive prices during high volatility regimes even if markets
Chapter 5. Results 29
are falling. This reasoning could be applied to results derived in sub period 6, however,
looking at the entire period there is not enough evidence to conclude that there exists
an asymmetric component in the SVIX index considering the negative and insignifi-
cant OMXret−t coefficient. Similar results was concluded by Giot (2005) for the VXN
index and by Dahlman and Wallmark (2007) who do not find enough evidence for an
asymmetric relationship when looking at the entire sample period.
5.6 Spillover effects in volatility indices
In this section I examine the possible volatility transmission between the US and the
European markets to the Swedish one. There is no obvious causality between which
market causes volatility in other markets, however, it is reasonable to assume that
changes in VIX have more impact on changes in SVIX than the other way around. By
graphical inspection we can conclude that all the three series share the same major
trends, which can be seen from Figure A.3 in Appendix. The full sample correlation
between SVIX and VIX is 0.87 and for SVIX and VSTOXX it is 0.85. To more formally
assess whether VIX or VSTOXX have any effect on SVIX, I run the following regressions
∆SV IXt = α0 + β1∆V IXt + β2∆V IXt−1 + t (5.10)
∆SV IXt = α0 + γ1∆V STOXXt + γ2∆V STOXXt−1 + t (5.11)
∆SV IXt = α0 + β1∆V IXt + β2∆V IXt−1 + γ1∆V STOXXt + γ2∆V STOXXt−1 + t
(5.12)
Results are reported in Table 5.8. Equation 5.10 tests whether there is a contemporane-
ous and lead effect in VIX on SVIX. Equation 5.11 tests the same thing for VSTOXX on
SVIX and 5.12 includes both VIX and VSTOXX with first lags. ∆VIX has the highest
contemporaneous effect on ∆SVIX with a coefficient equal to 0.317 and significant at a
1% level. The first lag of ∆VIX is also included to test whether there is a lead effect of
VIX at time t− 1.7 Since there is a time-zone difference between the US and European
markets, it might be good to clarify the contemporaneous and lead concepts. Since clos-
ing prices are used for all indices, the contemporaneous effect is the same calendar day
even if Swedish and US markets are only coinciding for a few hours. Changes in VIX
together with the first lag seems to explain some variation in the SVIX series, however
the adjusted R2 is very low, at 0.027, indicating that the independent variables do not
explain much of the variation in SVIX. The results from the second regression are a bit
7I re-ran the same regression including up to three lags, however only the first lag shows any signifi-
cance and the coefficients are small.
Chapter 5. Results 30
surprising as one could expect that the implied volatility in the Swedish market should
be affected by changes in European implied volatility indices, however, my results do
not point at this. There seems to be neither any contemporaneous effect nor a lead
effect from the VSTOXX on SVIX since both coefficients are insignificant at a 10% level
together with an adjusted R2 near zero. However, these results are in line with some
previous research e.g. Ciffarelli and Paladino (2012) on volatility spillovers and Aboura
(2003) on implied volatility index correlations. My results are also similar to the one
derived by Skiadopoulos (2004) that there is a contemporaneous spillover of changes in
GVIX and VIX, but no lead effect.
Table 5.8: Regression results - Volatility transmission
Eq. intercept ∆V IXt ∆V IXt−1
∆
VSTOXXt
∆
VSTOXXt−1
adj
R2
F-val N
(5.10) 0.039 0.317*** -0.083** 0.027 34.4 2,419
(0.075) (0.042) (0.042)
(5.11) 0.012 0.055 -0.059 0.000 2.02 2,482
(0.073) (0.041) (0.041)
(5.12) 0.049 0.388*** 0.005 -0.117* -0.112 0.031 15.1 2,397
(0.060) (0.106) (0.134) (0.080) (0.121)
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
5.7 The relationship between SVIX and SSVIX
As pointed out in the Theory section, there are some limitations in the replication
technique of variance swaps. Following Martin (2013) I construct the SSVIX index,
which relies on simple variance swap replication, an instrument which can be priced and
hedged even if we allow for jumps in the price of the underlying asset. According to
Martin (2013), an index constructed in this way is a better measure of market volatility,
compared to SVIX, which rather measures risk-neutral entropy of simple returns. In my
sample, 91% of my SVIX observations are higher than SSVIX, which is a result in line
with Martin (2013). After cleaning out my dataset I am left with 36,348 call options
and 55,101 put options for the variance calculation, while Martin has 1,165,585. By
having far less observations it might be reasonable to assume that some noise have been
introduced into the SVIX and SSVIX series which might account for some days where
SSVIX actually is higher than SVIX8. Figure 5.4 shows the 10-day moving average of
SVIX and SSVIX to the left and SVIX minus SSVIX to the right. From the right figure,
8This also relates to the discretization error since options cannot be traded at continuos strikes, so
the integrals needs to be approximated by a sum. When there are non-tradable or non-existent strikes,
the approximation error increases. More on this can be found in Martin (2013).
Chapter 5. Results 31
it is obvious that SVIX spikes were of larger magnitude than SSVIX during market
turmoil periods. The MA differenced series reaches a max value of around 8, while in
absolute terms for the ordinary series, differences are up to 20 ppts.
Jan-04 Jan-06 Jan-08 Jan-10 Jan-12 Jan-14 Jan-1610
20
30
40
50
60
70 SVIX and SSVIX
Jan-04 Jan-06 Jan-08 Jan-10 Jan-12 Jan-14 Jan-16-1
0
1
2
3
4
5
6
7
8
9 SVIX less SSVIX
Figure 5.4: 10-day MA of SVIX and SSVIX and their differences
The fact that SVIX is higher in almost every case is evidence that we do not live in
a log-normal world. According to Martin (2013), SVIX is more sensitive to the left
tail of the return distribution, while SSVIX is more sensitive to the right tail. By
comparing Equation (3.6), which points out that SVIX loads more on out-of-the-money
puts, with (3.8), which shows that the SSVIX loads equally on options of all strikes,
we get a hint of this. In the case when the left tail of the return distribution of the
underlying asset is fatter than the right tail, SVIX will be higher than SSVIX. Therefore,
as showed theoretically by Martin (2013), the difference between SVIX and SSVIX will
be a measure of non-normality. He also points out that this evidence is much stronger
than, what might be more familiar, the observation that histograms of log returns are
not Normal.
Martin continues with connecting the SSVIX to the equity premium. He states that
SSVIX can provide a lower bound on the equity premium and also is a measure of
forward-looking expected excess return on the market under the true, as opposed to
risk-neutral, probability distribution. What do these findings mean in terms of portfolio
choices and trading strategies? If one finds the lower bound on the equity premium,
according to the derivation in Martin (2013), to be very high compared to previous
periods, one would receive extraordinary high premiums if selling options. Following the
results derived by Martin, that SSVIX actually represents the real-world risk measure,
we could develop a profitable trading strategy during times when SVIX-SSVIX difference
spikes. We would then receive larger premiums than we should, under a real-world risk
measure.
Chapter 6
Conclusion
6.1 Concluding remarks
In this paper I construct and implied volatility index, SVIX, from OMXS30 European-
style options measuring the market expected 30 day forward volatility. My first objective
was to assess whether this index could be constructed using the CBOE method for the
Swedish market between January 3 2005 to June 11 2015. After having constructed the
index, I wanted to outline its statistical properties, how it relates to the underlying index,
how well it forecasts future volatility and whether it is affected by other international
volatility indices. In addition, I construct a second index called SSVIX with a slightly
different theoretical background, relying on simple variance swap replication.
SVIX is an index quite similar to other implied volatility indices. It is characterized by
common properties of volatility series, e.g. long memory components, non-normality,
mean-reversion and volatility clustering. I do not find any significant asymmetric effect
in SVIX for the entire sample. However, there seems to be some tendency of such an
effect in low-volatility regimes. The relationship between SVIX and OMXS30 is negative
and of unidirectional causality, where changes in OMXS30 cause changes in SVIX. SVIX
contains information about future volatility but provide a biased and inefficient forecast.
From my sample, there seems to be evidence of implied volatility being superior to
historical volatility, in forecasting future volatility.
There is a significant contemporaneous spillover effect on SVIX from VIX. In contrast
to previous studies on the same subject (Dahlman and Wallmark (2007)) I find stronger
effect from VIX than from the Eurozone implied volatility index, VSTOXX. When VIXt
and VIXt−1 are included as regressors, there seems to be both a contemporaneous and a
lead effect at a 5% significance level, while no such effects are found for VSTOXX running
32
Chapter 6. Conclusion 33
the separate regression. When both VIX and VSTOXX are included as regressors, only
the contemporaneous VIX effect is significant at a 1% level.
Finally, the difference between SVIX and SSVIX is an index of non-lognormality, which
exhibits spikes during market turmoil. The indices are computed based on two different
theoretical approaches where SVIX measures entropy and SSVIX measures variance1.
The difference between SVIX and SSVIX is positive in 91% of the cases which could be
taken as evidence that we do not live in a lognormal world. In theory, SSVIX can be
higher than SVIX and would occur when returns are lognormally distributed under a
risk-neutral measure.
1Following the derivation in Martin (2013)
Appendix A
Appendix
A.1 Normality tests
Figure A.1 shows histograms of SVIX and SSVIX and their differences series. As seen
in the figure, the level series are right-skewed while the differenced series exhibit sincere
kurtosis.
-50 0 50 1000
50
100
150
200
250
300 SVIX level series histogram
-20 0 20 40 60 800
50
100
150
200
250 SSVIX level series histogram
-40 -20 0 20 40 600
200
400
600
800
1000 SVIX differenced series histogram
-40 -20 0 20 400
100
200
300
400
500
600
700 SSVIX differenced series histogram
Figure A.1: Histograms of each respective serie
34
Appendix A. 35
A.2 Autocorrelation
By looking at ∆SVIX plotted in Figure A.2, we can expect the series to exhibit volatility
clustering. Large changes tend to cluster together and small changes tend to cluster
together, of either sign.
Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-16-40
-30
-20
-10
0
10
20
30
40
50
Figure A.2: ∆SVIX plot
A.3 Spillover effects in volatility indices - plot of VIX, VS-
TOXX and SVIX
Figure A.3 shows, as mentioned before, that VIX, VSTOXX and SVIX all share the
same major trends. Correlations among indices ranges around 85%,
Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-160
10
20
30
40
50
60
70
80
90 VIX, VSTOXX and SVIX plotted
Figure A.3: VIX in red, VSTOXX in blue and SVIX in dotted black
Appendix A. 36
A.4 Comparison of SVIX and SIXVX
The relationship between SVIX and SIXVX is very similar since both indices aim to
measure the same thing, the 30 day expected volatility derived from option prices. What
makes them two differ is the model and theoretical assumptions used for calculations.
SIXVX is calculated through extracting the implied Black-Scholes volatility from ATM
options. By extracting the implied volatility using Black-Scholes we rely on the same
assumptions being made to derive the model. We know that some of those assumptions
do not hold in practice. The one which is most significant is the constant volatility
assumption, that the underlying asset volatility remains constant over strikes and over
time. By assuming constant volatility, we find a ”volatility smile/skew” if we plot
the Black-Scholes implied volatility on the y-axis and moneyness on the x-axis. The
Black-Scholes model also assumes that returns are lognormally distributed, which Martin
(2013) shows is not the case and something we have seen many times before. These two
assumptions are the ones which are often shown to not hold in the real world. For a
full derivation of the assumptions of the Black-Scholes model, I refer to Hull (2000).
Figure A.4 shows an example of the implied volatility surface, with moneyness and
time-to-maturity on the axis.
Time to Matutity T
0.511.5
Implied Volatility Surface
22.53
Moneyness M = SK
2.521.51
0.4
0.9
0.3
0.5
0.8
0.7
0.6
Imp
lied
Vol
atil
ity
σ(T
,M)
Figure A.4: Implied volatility surface
As seen from Figure A.5 the movements between SVIX and SIXVX are very similar.
The sample correlation is 0.92. The bias in SIXVX comes from the fact that it is based
on Black-Scholes model. More on this bias can be found in Carr and Wu (2006) which
makes the same analysis on VIX and VXO.
Appendix A. 37
Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10 Jan-11 Jan-12 Jan-13 Jan-14 Jan-15 Jan-160
10
20
30
40
50
60
70
80
90
Figure A.5: SVIX and SIXVX
A.5 SVIX - OMXS30 relationship
Table A.1 shows correlation coefficients for different sub periods. Previous studies have
conducted that correlations between volatility and index returns are higher in high
volatility regimes. My results slightly indicates the same conclusion since sub periods
1 and 6 are low volatility regimes and have slightly lower correlation, while medium
and high sub periods (2, 3, 4 5) show larger correlation between ∆SVIX and OMXS30
returns.
Table A.1: sub period correlations between ∆SV IX and OMXS30 CC daily returns
sub period ρ pValue
#1 -0.30 0.000
#2 -0.31 0.000
#3 -0.33 0.000
#4 -0.38 0.000
#5 -0.34 0.000
#6 -0.27 0.000
Table A.2 shows the re-estimated asymmetry regression for each specific sub period. As
seen, the coefficient for OMXrett is negative in all cases, and significant at a 5% level
in all periods except for the first, where the it is only significant at a 10% level. Sub
period 6 is the only one showing a significant and positive asymmetry coefficient, and it
is a low volatility period.
Appendix A. 38
Table A.2: Regression results - asymmetry
subp. intercept OMXrett OMXret
−
t adj R
2 N
1. -0.056 -0.386* -0.395 0.096 340
(0.121) (0.197) (0.314)
2. -0.139 -0.572*** -0.246 0.096 552
(0.205) (0.173) (0.278)
3. -0.312 -0.871** -0.249 0.098 204
(0.869) (0.362) (0.599)
4. -0.222 -0.708*** -0.598* 0.145 502
(0.198) (0.189) (0.333)
5. -0.228 -0.462** -0.307 0.107 241
(0.331) (0.224) (0.351)
6. 0.357 -1.139*** 0.895*** 0.086 666
(0.133) (0.168) (0.277)
Numbers in parentheses denote t-statistics based on Newey-West HAC robust standard errors.
*, **, *** denotes significance at 10 percent, 5 percent and 1 percent respectively.
A.6 Option data used in the SVIX/SSVIX calculation
Table A.3 show the number of call an put options used to calculate my SVIX and SSVIX
series. As discussed in Chapter 4, I dropped those days where only one ∆K could be
calculated. For the call options, this was 20 days and for the puts seven days. None
of the days coincide, i.e. there is no day where this restriction makes it impossible
to calculate SVIX or SSVIX. Table A.3 shows the actual number of options used to
calculate my 2,505 SVIX/SSVIX values.
Table A.3: Option data for the SVIX/SSVIX
Calls Puts
OOTM 36,348 55,101
ATM/ITM 26,303 23,800
Total 62,651 78,901
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