The Fine Structure of Variance:
Consistent Pricing of VIX Derivatives
Nicole Branger∗
Clemens Völkert‡
This version: August 22, 2012
Abstract
This paper provides a tractable framework for consistently modeling and pricing the two most actively traded options on the Chicago Board Options Exchange (CBOE), namely SPX and VIX options. We derive the dynamics of
the CBOE volatility index and give semi-closed form solutions for derivatives
on it in a general affine jump-diffusion setup. We compare the implications of
several special cases of the general model with the major empirically observed
properties of VIX derivatives and the time-series behavior of the VIX. We
show that commonly used affine jump-diffusion models cannot reproduce the
basic patterns observed in the data. The fine structure of the variance process
is essential to reconcile the empirical regularities with the theoretical models.
We find that both variance jumps and a stochastic volatility of variance seem
to be important factors in this respect.
Keywords: Jump-diffusion model, volatility derivatives, VIX options
JEL: G13
∗
Finance Center Münster, Westfälische Wilhelms-Universität Münster, Universitätsstr. 14-16,
48143 Münster, Germany. E-mail: nicole.branger@wiwi.uni-muenster.de.
‡
Finance Center Münster, Westfälische Wilhelms-Universität Münster, Universitätsstr. 14-16,
48143 Münster, Germany. E-mail: clemens.voelkert@wiwi.uni-muenster.de.
We thank seminar participants at the university of Münster and participants of the Annual
Meeting of the Swiss Society for Financial Market Research, 2012 for helpful comments and suggestions.
1
Introduction
One important source of information about market participants’ perception of aggregate stock market uncertainty is the volatility implied in equity index option
prices. Starting in 1993, the Chicago Board Options Exchange (CBOE) published
its volatility index. The VIX expresses the market expectations of the 30-day volatility implied in equity index options. Initially, this volatility index was based on S&P
100 at-the-money (ATM) put and call option prices. Implied volatilities were calculated by inverting the prominent Black and Scholes (1973) option pricing model. In
2003, the CBOE switched to a model-free calculation method which uses the entire
strike price range of S&P 500 (SPX) options.
A major reason for the revision of the VIX was to create a suitable underlying
for tradable volatility products. Volatility derivatives were proposed almost 20 years
ago by Whaley (1993).1 Before exchange traded volatility derivatives were introduced, the most direct way for investors to trade volatility were over-the-counter
variance and volatility swaps. VIX futures were listed at the CBOE Futures Exchange (CFE) in 2004, VIX options started trading at the CBOE two years later.
Using these volatility derivatives, investors can easily implement long or short positions in volatility. The volume and open interest of both VIX futures and options
have increased rapidly since their introduction. By now, VIX options are the second
most actively traded contract at the CBOE. The most liquid options are written
on the S&P 500. There is a close connection between these two contracts, as the
underlying of VIX options essentially represents a portfolio of SPX options.
The tight link between the VIX and the underlying S&P 500 return dynamics
requires a consistent modeling and pricing of VIX derivatives. In contrast to the
model-free replication of the VIX itself, the valuation of VIX derivatives is modeldependent. We consider models from the affine class of Duffie et al. (2000). These
1
A survey of the literature is given in Carr and Lee (2009).
1
models have been found to capture important stylized facts of returns and equity
index option prices. We show that typically used jump-diffusion models have problems in explaining the major characteristics of VIX related derivatives while being
consistent with the underlying return dynamics. We take a closer look at the fine
structure of the variance process and evaluate several extensions of the commonly
employed square-root variance dynamics.
Concerning the modeling perspective of the underlying volatility process, there
are at least two approaches. Among others Grünbichler and Longstaff (1996), Detemple and Osakwe (2000), and Mencia and Sentana (2012) directly model the dynamics
of the volatility index, effectively decoupling it from the underlying return process.
The main advantage of this approach is tractability, since closed-form solutions are
often available. However, the return dynamics and the VIX are related. While SPX
options are written on the S&P 500 index, a portfolio of SPX options resembles
the VIX. Directly modeling the dynamics of the VIX does not guarantee that the
portfolio of SPX options replicates the volatility index.
A consistent modeling of VIX related derivatives with the underlying S&P 500
return dynamics introduces an additional layer of complexity. The volatility index
needs to be derived from the assumed return dynamics. Afterwards, this quantity
can be used as underlying for derivatives. From a theoretical perspective, this modeling approach is superior since inconsistencies in the pricing of related financial
instruments can be avoided.2 Among others, Lin and Chang (2009, 2010) and Sepp
(2008b,a) adopt this approach. They use commonly employed affine option pricing
models with return and variance jumps. The general affine jump-diffusion framework
we use nests these models. Due to its tractability and flexibility, the affine jumpdiffusion setup of Duffie et al. (2000) has been widely used in derivatives pricing.3
2
Another modeling approach is pursued by Bergomi (2008) and Cont and Kokholm (2012).
They model the joint dynamics of forward variance swap rates and the underlying return process.
3
There is a vast literature on affine option pricing models. Important contributions include
Heston (1993), Bakshi et al. (1997), Bates (2000), Pan (2002), and Eraker (2004).
2
It also offers a suitable framework to analyze volatility related derivatives. However,
volatility derivatives have a nonlinear payoff function and the transform analysis of
Duffie et al. (2000) does not apply directly. To obtain semi-closed form solutions for
volatility related derivatives, we rely on the methodology in Lewis (2000, 2001) and
Chen and Joslin (2012). We consider several special cases of the general model and
show which features are necessary to reconcile the patterns observed in the data.
Preceding the analysis of the theoretical models, we give a detailed treatment
of the empirically observed properties the models ought to match. Concerning the
time-series behavior of volatility, some well-known characteristics are upward jumps,
heteroscedasticity, and mean-reversion. Using VIX options, we are able to go one
step further and infer information beyond what is available from the stock market
and the time-series of the VIX, i.e. we can extract information from the tails of
the volatility distribution. We compute the risk-neutral distribution and analyze
the option-implied moments. VIX options have the following characteristics: 1. the
implied volatility smile is upward sloping, 2. implied volatilities are higher and the
slope is more pronounced for shorter times to maturity, and 3. there is considerable
time-variation in VIX option smiles. The upward sloping implied volatility smile
indicates that when stock prices fall, both the volatility and the volatility of volatility increase. We find that the time-series behavior of the VIX and the empirically
observed patterns in the VIX option market can be explained by a combination
of variance jumps and a stochastic volatility of variance. Variance jumps primarily
induce right skewness and are most important for shorter times to maturity, while
a stochastic volatility of variance generates excess kurtosis and also has a strong
impact on longer times to maturity. In addition, this factor produces the necessary
amount of heteroscedasticity found in the first differences of the VIX. The weak performance of the standard square-root variance specification in pricing VIX options
shows that it is questionable to apply standard equity index option pricing models
to volatility derivatives.
3
The remainder of this paper is organized as follows. In Section 2, we provide
some background information about the VIX. We discuss derivatives on the VIX and
describe basic characteristics of the implied volatility smile and the option-implied
distribution in Section 3. Afterwards, we introduce the model setup. In Section 5,
we present the empirical results. Section 6 concludes.
2
The VIX and the Connection to Variance Swaps
In September 2003, the CBOE revised its volatility index. The new calculation
methodology makes it feasible to replicate the VIX based on a portfolio of SPX
options. The CBOE calculates the VIX in the following way
v
u
2
u 2 X ∆Ki
1 Ft (T )
rτ
t
VIXt = 100
e Ot (Ki , T ) −
−1 ,
τ i Ki2
τ
K0
(1)
where Ft is the S&P 500 forward index level derived from SPX index option prices,
r is the risk-free interest rate, Ki is the strike price of the i-th SPX option, ∆Ki =
Ki+1 −Ki−1
2
is the interval between strike prices, Ot denotes the mid price of the out-
of-the-money (OTM) SPX option, and K0 is the first strike price below the S&P
500 forward index level. The time to maturity τ = T − t is set to one month.4
The expression under the square-root in Equation (1) is a discretized version
of the fair value of variance of Demeterfi et al. (1999), or equivalently, the model-free
implied variance of Britten-Jones and Neuberger (2000) and Jiang and Tian (2005).
Essentially, it resembles the 30-day variance swap rate. Variance swaps were the
first volatility derivatives introduced at the market. The payoff of a long position
in a variance swap is equal to the difference between the sum of squared daily log
price changes over the life of the contract and the variance swap rate, multiplied by
4
For a detailed description about the construction of the CBOE volatility index see Jiang and
Tian (2007), CBOE (2009), and Whaley (2009).
4
the notional amount of the swap. By definition, the variance swap rate is set such
that the contract has zero market value at initiation. Thus, the variance swap rate
is the risk-neutral expectation of the realized variance, i.e. the price of the realized
variance. We use a continuous time framework to model the stock price and the
variance dynamics and consequently replace the sum of squared log price changes
(with daily sampling) by the quadratic variation (with continuous sampling) and
set the floating leg of the swap equal to the latter. The variance swap rate is
1
V St (T ) = EQ
τ t
Z
t+τ
2
(d ln Ss )
t
2
= erτ
τ
Z
∞
0
Ot (K, T )
dK + ε,
K2
(2)
where S is the stock price. The first summand in Equation (2) is the expression
for the model-free implied variance, i.e. for a 30-day contract the squared VIX. ε
denotes the approximation error due to price jumps.5
The time-series behavior of the VIX from 1990 to 2011 is displayed in Figure 1,
with the associated descriptive statistics given in Table 1.6 The mean level of the VIX
over the sample period is about 20, with a historical high of 80.86 on November 20th,
2008. The volatility of the index is high and the unconditional distribution is rightskewed and leptokurtic. We observe frequent spikes in the level of the VIX during
periods of market stress. When markets recover, volatility gradually reverts back
to its mean level. The mean-reversion property can also be seen from the negative
autocorrelation of the first differences in Table 1. Looking at the VIX first differences,
we observe significant departures from normality. Especially the volatility clustering
in the first differences of the VIX is notable. The heteroscedasticity is confirmed
by the quantile-quantile plot and the scatter plot in Figure 2. The strong positive
relation between the VIX and the absolute value of its first differences indicates that
the VIX is far more volatile when it reaches high levels.
5
The approximation error is given in Jiang and Tian (2005) and Carr and Wu (2009). They
find that it is negligible for stock market indices in commonly employed option pricing models.
6
See also Dotsis et al. (2007) and Whaley (2009).
5
Furthermore, it is well-known that changes in the VIX have a negative correlation with the returns on the S&P 500 index. In turbulent times, when stock
prices decline, the demand for equity index (put) options rises sharply. This increases the premium necessary to compensate protection sellers. Thus, we observe
higher implied volatilities at the market. Investors who want to protect or diversify
their portfolios can exploit this inverse relationship to hedge downward movements
in stock prices and increases in volatility using VIX derivatives.
3
Derivatives on the VIX
VIX futures and options are contracts on the forward 30-day implied volatility of
SPX options. The underlying of VIX derivatives is thus not the spot VIX but the
expected, or forward, value of the VIX at expiration. VIX futures started trading
at the CBOE Futures Exchange (CFE) on March 26, 2004. European style options
on the VIX followed on February 24, 2006.
3.1
Data
VIX futures and options data are obtained from the CBOE’s website and Market
Data Express, respectively. The sample period ranges from February 24, 2006 to
December 31, 2011. The sampling frequency is daily. We apply several filters to the
options data. We eliminate option quotes that do not satisfy standard no-arbitrage
conditions. We also check for negative bid-ask spreads, zero bids, and filter out all
options with zero open interest and those options where the implied volatility could
not be computed. Options with time to maturity shorter than 7 days are discarded
to reduce pricing anomalies that might occur close to expiration. Options with ma-
6
turity of more than one year are also excluded.7 We use the mid of the bid and ask
prices in the following. Moneyness is defined as the strike price divided by the VIX
futures price. For VIX futures, we employ daily settlement prices. Constant maturity Treasury bill yields are treated as a proxy for the risk-free discount rate. Table
2 displays information about the VIX options data. It shows implied volatilities,
the number of observations, the average volume, the average open interest, and the
average percentage bid-ask spread for different moneyness and maturity categories.8
3.2
Empirical properties of VIX derivatives
From the contract specifications of VIX options, we know that the underlying is
the forward value of the VIX at expiration. The VIX futures price converges to the
spot VIX when the contract approaches expiration. Thus, VIX options can also be
treated as options on VIX futures. To get a better picture of the actual underlying
of VIX options, we briefly cover some properties of VIX futures.9 Historically, the
volatility of VIX futures is much lower than of the VIX itself. This can be seen by
comparing the time-series of the VIX in Figure 1 with the VIX futures in the upper
left corner of Figure 3. Consistent with the mean-reversion property of volatility,
we typically observe that the longer the time to maturity the lower the volatility of
volatility. The term structure of VIX futures has experienced various shapes over the
sample period: upward sloping, hump-shaped, and downward sloping.10 On average,
the term structure is upward sloping.
7
Currently, there are no VIX options with maturity beyond one year. Prior to the last changes
in the contract specifications, up to three near-term months and up to three additional months on
the February quarterly cycle were listed.
8
The statistics for the longest maturity category are somewhat misleading. VIX options with
long times to maturity were primarily traded in 2006 and 2007. This biases the associated statistics
towards a time period with relatively stable economic conditions and lower levels of volatility.
9
The empirical properties of VIX futures have been studied by Zhang and Zhu (2006) and
Zhang et al. (2010), among others.
10
Before the financial crisis in 2006 and 2007, when implied volatility was historically low, the
term structure was upward sloping. In October 2008 the VIX exceeded several times its long term
average and the VIX term structure was downward sloping.
7
The empirical properties of VIX options and especially the implied volatility
smile have not been studied in great detail. In order to have a clear understanding
what the theoretical models ought to match, we take a closer look at these options.11
The trading volume and open interest of VIX options increased rapidly after their
introduction. There is more trading activity in VIX derivatives when the uncertainty
in the market is already high. In general, VIX call options are more heavily traded
compared to put options. One reason for this is that market participants use OTM
VIX call options to protect their portfolios against sharp decreases in stock prices
and increases in volatility. Table 2 shows that based on the average volume and
the average open interest, in-the-money (ITM) options are less liquid than OTM
and ATM options. The percentage bid-ask spreads for ATM and ITM options are
typically below 10%. The percentage spreads can be quite wide for OTM options.
In the following, we look at the implied volatility of VIX options. The implied
volatility is the volatility parameter that, plugged into the Black (1976) option
pricing formula, makes market and model prices agree. By computing the implied
volatility using the Black (1976) model, the benchmark is that the VIX follows a
lognormal distribution. Empirically, the implied volatility smile for VIX options is
upward sloping. OTM call options on the VIX provide protection against high levels
of uncertainty in the market and are relatively expensive, compared to ATM and
ITM call options. This is the flipside of what we observe for equity index options.
For these options, it is well-known that implied volatilities are usually downward
sloping and sometimes there is a moderate U-shape. OTM equity index put options
provide insurance against sharp declines in stock prices. Similarly, due to the inverse
relationship between stock prices and volatility, OTM call options on the VIX are a
“disaster” insurance on the overall equity market. The implied volatilities in Table 2
provide more detail about the cross-section and the term structure of VIX options.
The average implied volatilities are calculated by equally weighting all call or put
11
See also Wang and Daigler (2011) and Mencia and Sentana (2012).
8
options in a given moneyness-maturity bucket. For each maturity category, implied
volatilities are upward-sloping. For short times to expiration, there is a U-shape.
For longer times to maturity, we observe lower implied volatilities and the smile
flattens out. These moneyness and maturity related biases indicate that assuming
lognormality might be inadequate.
Using the result in Breeden and Litzenberger (1978), we can infer the implied
risk-neutral distribution of the forward VIX.12 The upward sloping implied volatility
function reflects positive skewness and leptokurtosis in the implied risk-neutral distribution, i.e. implied volatilities are increasing in the strike price, which results in a
fat right tail of the empirical risk-neutral VIX distribution (relative to the lognormal
distribution). Figure 3 shows the time-series of the implied risk-neutral moments for
VIX options with a fixed time to maturity of 60 days.13 As required, the values of
the implied mean and the corresponding VIX futures are about the same. The riskneutral volatility moves in lockstep with the mean volatility. Both shoot-up during
the turbulent times during the recent financial crisis. In the aftermath, they gradually mean-revert to their pre-crisis levels. The risk-neutral volatility of the VIX
is more erratic than the mean level. First order autocorrelations are high, exceeding 0.95 for both series. Looking at the higher moments, we observe a pronounced
right skewness and excess kurtosis. When the VIX is already at high levels, skewness decreases due to mean-reversion of volatility. Skewness and kurtosis are highly
correlated. After the crisis, both skewness and kurtosis increase, showing that market participants believed that large upward movements in volatility are likely (or
investors became more risk-averse). The time-variation in the implied moments indicates that the shape of the implied volatility smiles considerably changed over the
sample period.
12
Details are given in Appendix A.
We focus on the 60-day series to maximize liquidity. Results for option-implied moments with
different horizons are available upon request.
13
9
Concerning the modeling of the downward sloping implied volatility smile for
equity index options, researchers include jumps in the return process and stochastic
volatility to capture non-normality. For plain vanilla options, both stochastic volatility and jumps can generate a volatility smile. These two effects can be disentangled
because of the maturity pattern.14 For VIX options a similar result holds true.
4
Affine Models for Volatility Derivatives
Stochastic volatility, jumps in prices, and jumps in volatility have been found to be
important components in explaining stock market returns and equity index option
prices.15 We assume that the price and the variance dynamics follow a general affine
jump-diffusion process that allows for all of these components. In Sections 4.4 and
4.5 we focus on specific models. As we are mainly concerned about derivative pricing,
we directly specify the models under the risk-neutral measure Q.
4.1
Return process and volatility state variables
We follow Duffie et al. (2000) and assume that there are n state variables that follow
affine jump-diffusion processes. We let the instantaneous variance Vt = ϑXt depend
on the vector of state variables Xt , which follows
dXt = µ(Xt )dt + Σ(Xt )dWtX + ZX dNtX .
(3)
ϑ is a selection vector, WtX is a n-dimensional Brownian motion, and NtX is a
Poisson process. The drift µ(Xt ), the variance-covariance matrix Σ(Xt )Σ(Xt )0 , the
14
Bakshi et al. (1997) and Das and Sundaram (1999) find that jumps have a greater impact on
short times to maturity, while stochastic volatility dominates for longer maturities.
15
See e.g. Bakshi et al. (1997), Bates (2000), Pan (2002), Eraker et al. (2003), Eraker (2004),
and Broadie et al. (2007).
10
jump intensity λ(Xt ), and the risk-free discount rate r(Xt ) are affine in Xt . More
specifically, for the drift we set µ(Xt ) = M + KXt , M ∈ Rn , K ∈ Rn×n and for the
P
variance-covariance matrix Σ(Xt )Σ(Xt )0 = h + ni=1 Hi Xt,i , h ∈ Rn×n , Hi ∈ Rn×n .
Jumps have intensity λ(Xt ) = λ0 + λ1 Xt , λ0 ∈ Rn , λ1 ∈ Rn×n , and jump sizes
0
are characterized by the transform (moment generating function) %(u) = E[eZX u ],
ZX ∈ Rn×n . The risk-free rate has the form r(Xt ) = r0 + r1 Xt , r0 ∈ R, r1 ∈ Rn .
We assume that the log stock price also follows an affine jump-diffusion process
d ln St
p
1 2
=
r(Xt ) − σS Vt − λS µ̄J dt + σS Vt dWtS + ZS dNtS .
2
(4)
The intensity of the Poisson process NtS is affine in the state vector λS = λS0 + λS1 Xt .
Jump sizes are denoted by ZS , µ̄J is the mean percentage price change due to jumps.
4.2
Model implied VIX
In order to price derivatives on the VIX, we proceed in two steps. First, we derive the
variance swap rate based on the assumed log price process in Equation (4). We rely
on the connection between the variance swap rate and the squared VIX to obtain an
exact expression for the volatility index. Afterwards, we price futures and options
on the model implied VIX.
According to Equation (2), the variance swap rate is the annualized expected
quadratic variation of log price changes under the risk-neutral measure. Given the
log price process in Equation (4), this yields
Z t+τ
1 Q
2
2
V St (T ) = Et
σS Vu du + ZS dNu
τ
t
Z t+τ
2 S Q
2 S
1 2
Q
σS ϑ + Et ZS λ1 Et
Xu du + EQ
=
t Z S λ0 .
τ
t
11
(5)
We solve for the expected integrated state vector as in Egloff et al. (2010)
EQ
t
Z
t+τ
Xu du = AIV + BIV Xt ,
(6)
t
with
−1
K + EQ [ZX ] λ1
M + EQ [ZX ] λ0 ,
−1 (K+EQ [Z ]λ1 )τ
X
K + EQ [ZX ] λ1
e
−I ,
AIV
= (BIV − Iτ )
BIV
=
where I denotes an n-dimensional identity matrix. From Equations (5) and (6), it
follows that the variance swap rate is an affine function of the state variables
V St (T ) = AV S + BV S Xt ,
(7)
where
2 S 1 2
2 S
Q
Z
AV S = EQ
λ
+
σ
ϑ
+
E
t
t ZS λ1 AIV ,
S
0
τ S
2 S
1 2
BV S =
σS ϑ + EQ
t ZS λ1 BIV .
τ
Setting the time to maturity of the variance swap equal to one month, the variance
swap rate resembles the squared VIX. According to Equation (2), there is a small
difference between the two in the presence of price jumps. We adjust for the jump
Q
2
induced error by replacing EQ
t [ZS ] with 2 Et [exp(ZS ) − 1 − ZS ] and thus, we repli-
cate the VIX exactly.16 The variance swap rate only depends on the specification
of the drift components. When looking at the term structure of variance swaps, the
specification of the innovations is irrelevant. However, the variance swap rate is an
16
For conditionally normally
distributed price jumps proposed by Duffie et al. (2000), i.e.
ZS |ZV ∼ N µJ + ρJ ZV , σJ2 , where ZV ∼ exp(µV ) is the jump size in the instantaneous variance,
this yields EQ
t [exp(ZS ) − 1 − ZS ] = µ̄J − (µJ + ρJ µV ), with µ̄J = exp(µJ +
12
2
σJ
2
)/(1 − ρJ ) − 1.
affine function of the state variables. The innovations drive the behavior over time
and are important for pricing derivatives on the VIX.
4.3
VIX derivatives
VIX futures and options are written on the forward VIX. Setting T̄ − T = 30/365,
V ST (T̄ ) denotes the 30-day variance swap rate at time T . Furthermore, the 30-day
variance swap rate is identical to the squared VIX, i.e. VIX2T = V ST (T̄ ). The price
of a futures contract expiring at T is equal to the risk-neutral expectation of the
forward VIX, i.e. EQ
t [VIXT ]. The payoff of a call option on the VIX with maturity
in T and strike price K is (VIXT − K)+ = max [(VIXT − K) , 0]. Using risk-neutral
valuation, we can determine the price of a call option
h RT
i
+
− t rs ds
e
.
Ct (K, T ) = EQ
(VIX
−
K)
T
t
Deriving an explicit formula for the price of a call option is not straightforward
since the VIX is not affine in the state variables. For volatility derivatives, we have a
nonlinear transform (square-root) for a process with a tractable conditional characteristic function (variance swap rate). We transform the payoff function to deal with
the nonlinearities. To do so, we rely on the Fourier inversion techniques in Lewis
(2000, 2001) and Chen and Joslin (2012).17 Sepp (2008a,b) uses a similar approach
to solve for volatility derivatives. Since VIX derivatives have some unique characteristics, we summarize some important properties of the pricing formulas. The VIX
futures price converges to the spot VIX as the time to maturity approaches zero
and to a constant for long times to maturity. Call option values initially increase in
time to maturity. However, due to mean-reversion, prices decline for longer times to
maturity and converge to zero for long times to maturity.
17
Details on the computation are provided in Appendix B.1. We test the accuracy of the pricing
formula using a Monte Carlo study in Appendix B.2.
13
4.4
Shortcomings of existing models
Regarding the return process and equity index option prices, the commonly used
affine jump-diffusion models are able to replicate major empirical properties. In
order to identify deficiencies of existing models concerning the variance dynamics,
we first explain the time-series behavior of the VIX in common affine jump-diffusion
models and then focus on the pricing implications for VIX derivatives.
Figures 1 and 2 indicate that there is volatility clustering in the first differences
of the VIX. This poses a challenge to the commonly used square-root instantaneous
variance specification.18 More specifically, changes in the instantaneous volatility in
the Heston (1993) model are Gaussian and homoscedastic. For the VIX the coefficients AV S and BV S change this result. Even in the square-root case changes in
the VIX are heteroscedastic and there is a positive relation between the VIX and
its first differences, as observed in the data. Although the effect has the right sign,
for reasonable values of the parameters it is too small. Another problem with the
square-root variance specification is the low volatility of volatility.
Figure 1 demonstrates that there are large spikes in the time-series of the
VIX. Especially the turbulent period after the Lehman crash in September 2008 is
difficult to match in a diffusion setting. Jumps in the instantaneous variance process
are a direct way to model this feature of the data. Eraker et al. (2003) extend the
square-root variance dynamics in this direction. Jumps are usually assumed to have
a constant arrival rate. However, Figure 1 indicates that jumps seem to occur more
often in high volatility periods. Wu (2011) confirms this conjecture. He finds that the
arrival rate of variance jumps is not constant and that a jump intensity proportional
to the variance level is more appropriate. In addition, a constant arrival rate does
not induce heteroscedasticity in the first-differences of the VIX and thus can only
18
Several studies, e.g. Christoffersen et al. (2010), find that the square-root variance dynamics
are incompatible with empirical observations.
14
partially eliminate the problems of the standard square-root model.
VIX options are very sensitive to the specification of the variance process. To
illustrate implications of a misspecified process, we use a very popular model from the
equity index option pricing literature, the double-jump model (SVCJ). This model
was introduced by Duffie et al. (2000) and has subsequently been applied to describe
stock returns and to price equity index options by Eraker et al. (2003) and Eraker
(2004), among others. It includes stochastic volatility and jumps in both returns and
variance. The SVCJ model nests the SVJ model by excluding variance jumps and
the SV model by only allowing for diffusive components. The instantaneous variance
process in the SVCJ model follows by setting Xt = vt and ϑ = 1
√
dvt = κv (v̄ − vt )dt + σv vt dWtv + Zv dNtS ,
where WtS and Wtv are correlated Wiener processes. Jumps in returns and variance
are assumed to occur contemporaneously with constant intensity. Jumps in the log
asset price are conditionally normally distributed, while variance jumps are exponentially distributed. In the following, we use the parameters from Eraker (2004).19
Figure 4 displays the implied volatility surface for the SV model. The squareroot variance specification is not able to replicate the upward sloping implied volatility smile for VIX options. Implied volatilities are decreasing in the strike price for
all times to maturity. The SVJ model adds constant intensity return jumps to the
SV model. In the SVJ model, jump components explain a part of the unconditional
return variance, resulting in a more damped volatility path. Looking at the formula
for the variance swap rate in Equation (7), we recognize that a specification with
a constant return jump intensity induces a higher lower bound on the value of the
VIX. As the part depending on the instantaneous variance makes up a smaller por19
The parameters were estimated using Markov chain Monte Carlo methods based on return
and derivative data for a sample period from 1987 to 1990.
15
tion of the total VIX, larger shocks are necessary to generate enough variation in
the level of the VIX over time.20 The square-root variance specification is not able
to achieve this. It is inconsistent with the VIX options data. If constant intensity
return jumps are included, the performance deteriorates even further.
For the SV model and the SVJ specification, the VIX spends relatively little
time at high values. Variance jumps might help in this respect, as they induce
additional right skewness and kurtosis in the instantaneous variance compared to the
non-central chi-squared distribution of the standard square-root model. In Figure 5,
we observe an upward sloping implied volatility smile for the SVCJ model. However,
there are still deficiencies of this specification. First, the overall level of the implied
volatilities is rather low. Second, for longer times to maturity implied volatilities
are either flat or decreasing in the strike price. Third, the constant jump intensity
implies very little time-variation in the implied volatility smile. Finally, in the SVCJ
model return and variance jumps occur simultaneously. However, among others, Wu
(2011) finds that jumps in volatility tend to arrive more frequently than return
jumps. Thus, a tight coupling of return and variance jumps might be problematic.
Summing up, although sufficient to model returns and to price equity index
options, commonly used affine jump-diffusion models seem to be inadequate for
modeling the time-series behavior of the VIX and for pricing VIX derivatives.
4.5
Variance jumps and stochastic volatility of variance
We extend the square-root variance specification along two lines. First, we introduce
jumps in the instantaneous variance process. Guided by the empirical evidence, we
allow for an arrival rate proportional to the level of variance. Second, we allow for
stochastic volatility of variance that is positively correlated with the variance process
20
A stochastic jump intensity for return jumps, which is proportional to the variance level,
would solve this issue. There has been mixed evidence concerning a stochastic jump intensity in
returns, see e.g. Bates (2000), Pan (2002), and Eraker (2004).
16
itself. The model follows from Equation (3) by setting Xt = [vt , qt ]0 and ϑ = [1, 0]
√
dvt = κv (v̄ − vt )dt + σv qt dWtv + Zv dNtv ,
q
√ dqt = κq (q̄ − qt )dt + σq qt ρvq dWtv + 1 − ρ2vq dWtq ,
where Wtv and Wtq are independent Wiener processes. ρvq denotes the correlation
between diffusive shocks to the instantaneous variance and its variance. The jump
intensity is assumed to be affine in the state variables λv,t = λv,0 + λv,1 vt . We restrict
our analysis to jump sizes with a positive support. More specifically, we use gamma
distributed jumps Zv ∼ Γ ν, µνv , where ν is the shape parameter and µv denotes
the mean jump size. The jump transform for gamma distributed jumps is
µv −ν
%v (u) = 1 − u
.
ν
By setting the shape parameter ν to 1, we recover the exponential distribution used
for example in Eraker et al. (2003) and Eraker (2004).
In the context of variance and volatility derivatives, exponentially distributed
variance jumps have been studied by Sepp (2008a,b) and Lin and Chang (2009,
2010). While for the exponential distribution skewness and kurtosis are fixed, the
gamma distribution can generate larger levels of right skewness and kurtosis if ν < 1.
Eraker and Shaliastovich (2008) use gamma distributed jumps in a general equilibrium asset pricing model. To the best of our knowledge, the stochastic volatility of
variance specification has not been employed to price VIX options. Fong and Vasicek
(1991) use a similar process to model interest rates, whereas Bollerslev et al. (2009)
apply it to model consumption volatility in a long-run risks asset pricing model.
A restricted version of the model with only variance jumps can be obtained by
imposing qt = vt . As only the drift components matter for the level of the VIX, the
expressions for the variance swap rate of the restricted and the unrestricted models
17
coincide. More specifically, we have V St (T ) = AV S + BV S [vt , qt ]0 , with
AV S
BV S
1 − e−(κv −λv,1 µv )τ
τ−
=
(κv − λv,1 µv )
−(κv −λv,1 µv )τ
21−e
= σS
, 0 .
τ (κv − λv,1 µv )
σS2
(κv v + λv,0 µv )
τ (κv − λv,1 µv )
,
(8)
In order to appreciate the properties of the stochastic volatility of variance
model, we look at the instantaneous volatility in the restricted model
p
√ √
1
vt + Zv − vt dNtv ,
d vt = µ(vt )dt + σv dWtv +
2
where µ(vt ) denotes the drift of the instantaneous volatility process. Ignoring jumps,
the restricted model implies that changes in the instantaneous volatility are Gaussian and homoscedastic. Furthermore, constant intensity variance jumps do not affect the volatility of volatility. Unless there is an arrival rate proportional to the
instantaneous variance and jumps occur frequently, which is at odds with the notion that jumps are rare events, the model is unable to generate volatility clustering.
In contrast, shocks to the instantaneous volatility and its volatility are correlated
in the stochastic volatility of variance model. A positive shock to the instantaneous
volatility increases both the instantaneous volatility and its volatility.21
There are several other multi-factor variance structures that have been explored in the literature. Bates (2000) and Christoffersen et al. (2009) use a double
Heston (1993) model, while Duffie et al. (2000) introduce a stochastic central tendency factor. Both specifications imply a more flexible term structure of VIX futures.
However, they have a minor influence on implied volatility smiles of VIX options.22
21
Because of the coefficients AV S and BV S , the effect is not as clear-cut for the VIX. If AV S is
small, which is typically the case, the results do not change significantly.
22
We tested the stochastic central tendency model and found little impact on VIX options. In
particular, for reasonable parameter values the implied volatility smile is downward sloping.
18
5
Empirical Results
We analyze if variance jumps and a stochastic volatility of variance are able to
replicate the major stylized facts of VIX options. First, we describe some of the
properties of the models using “reasonable” parameter values. Afterwards, we calibrate the models to the average implied volatility smile. In the analysis, we exclude
return jumps. Alternatively, we could assume that return jumps have an arrival rate
proportional to the instantaneous variance. Similar to the parameter σS , this implies
an influence on the level of the VIX but not on the implied volatility smile.
5.1
The impact of jumps and stochastic volatility
We fix the mean-reversion levels at 1. The mean-reversion speed of the instantaneous
variance κv is set to 5. This value is consistent with the high autocorrelation of
variance and implies a half-life of about two month. For the volatility of variance σv ,
we choose a value of 1.5. The values of these parameters are equal across all models
considered. Note that a lower value of κv (shocks die out more slowly) leads to higher
implied volatilities. An increase in σv (a higher volatility of variance) makes the VIX
more volatile, leading to an overall increase in implied volatilities. We fix the state
variables at their unconditional means and assume that the risk-free rate is zero.
Table 3 shows implied volatilities for several parameter constellations. Two
values are considered for each of the jump parameters. Increasing the average jump
size induces additional right skewness and kurtosis in the distribution of the instantaneous variance. This leads to a higher level and also a more pronounced implied volatility smile. The effect is strongest for short times to maturity. Concerning
gamma versus exponentially distributed jumps, we know that the variance swap rate
in Equation (8) is identical for both distributions. A shape parameter smaller than
one implies a greater variance, skewness, and kurtosis compared to the exponential
19
distribution. The chances of ending up in the money are higher and consequently a
lower shape parameter increases the level of the implied volatilities and makes the
implied volatility smile more pronounced. From the formulas of the variance swap
rate in Equation (8), we can see that regarding the intensity of the jump process
there are two effects at play. First, a constant jump intensity implies a lower weight
on BV S . Second, constant intensity jumps contribute only a constant part to the
conditional variance. This leads to lower levels of the implied volatility smile. It also
implies less time-variation compared to a proportional jump intensity.
The variance swap rate of the stochastic volatility of variance model is the same
as for the restricted model. However, a stochastic volatility of variance has different
implications for the time-series behavior of the VIX and VIX option pricing. The
magnitude of the parameters can be roughly guided using the risk-neutral moments
in Figure 3. The risk-neutral volatility of the VIX is high and the autocorrelation
is close to one (a bit lower than for the risk-neutral mean). Furthermore, the riskneutral mean of the VIX and its volatility are highly correlated. These observations
imply that κq is higher than κv , a high value of σq , and ρvq close to one.
Table 4 shows implied volatilities for several parameter constellations. It is
apparent that while variance jumps mainly influence short times to maturity, the
effect is more evenly distributed for the stochastic volatility of variance model. Positive shocks to the instantaneous variance increase both the instantaneous variance
and its volatility. This effect is persistent and consequently also influences long-term
options on the VIX. The parameters ρvq and σq are important for the slope of the
implied volatility smile. In order to match the basic patterns, we require ρvq to be
close to one and a high value of σq . High values of σq and ρvq also lead to significant
heteroscedasticity in VIX changes, which is consistent with what we observe in Figures 1 and 2. A low value of κq increases the curvature. The volatility of variance
model can generate a sizeable dip for low strike prices. While upward jumps in the
20
instantaneous variance generate a long right tail of the risk-neutral VIX distribution,
the stochastic volatility of variance model is able to put more weight on both tails
of the distribution. This is beneficial for pricing short-term deep ITM call options.
5.2
Calibration
We calibrate the models to the average volatility surface. Compared with the classification in Table 2, we use a finer partition to focus on the most liquid options. We
consider 7 moneyness categories ranging from 0.5 to 2 (0.50-0.75, 0.75-0.90, 0.901.00, 1.00-1.10, 1.10-1.25, 1.25-1.50, and 1.50-2.00) and 4 maturity buckets (7-30,
30-60, 60-90, and 90-120 days). We average the implied volatilities, futures prices,
interest rates, and times to maturity in the respective moneyness-maturity categories. The implied volatilities are converted into call option prices using the Black
(1976) formula. Figure 6 shows the representative implied volatility curves.
To calibrate the models, we solve the following minimization problem
N
X
M
2
min
Ci (K, T ) − Ci (K, T ) .
i=1
N is the number of options, CiM (K, T ) denotes the i-th option price observed at the
market, and Ci (K, T ) is the corresponding model implied price. In each step of the
optimization, given the values of the structural parameters, we obtain the instantaneous variance by inverting the formula for the variance swap rate in Equation
(8).23 We constrain all one-factor models to satisfy the Feller condition.
The parameter estimates and the root mean squared pricing errors (RMSE) are
reported in Table 5. The RMSE are tabulated for all options and also separately for
each maturity category. Since the objective function assigns more weight to relatively
expensive options, it is expected that OTM call options have relatively large pricing
23
The initial value of the stochastic volatility of variance is treated as a parameter.
21
errors compared to options with lower levels of moneyness. The estimation procedure
forces all models to match the current value of the VIX. For the specifications with
variance jumps, the value of the parameter σS is much lower because variance jumps
contribute significantly to the level of the VIX.
Table 5 shows that the square-root (SR) variance dynamics are incompatible
with VIX option prices. The model generates the largest pricing errors and the
parameter estimates are rather unrealistic. It requires a very high value of σv to
match the average level of the implied volatilities and the Feller condition is binding.
It cannot replicate the upward sloping implied volatility curves, which implies that
the SR model severely misprices OTM and ITM options.
The parameters of the restricted model with only variance jumps are in line
with studies that use SPX options for parameter estimation. Among others, Eraker
(2004) and Broadie et al. (2007) estimate the risk-neutral parameters of a squareroot process with variance jumps using SPX options. The variance process in their
SVCJ model corresponds to the SREJ0 model in Table 5. The parameters of the
mean-reversion speed κv and the volatility of variance σv are comparable to these
studies. Variance shocks have a half-life of less than three months. Concerning the
jump parameters, the intensity and the mean jump size are also similar to existing
studies. On average there are about 1.5 jumps in the instantaneous variance every
year. The mean jump size is 3.646. This implies that if the instantaneous variance
is currently at its long-run mean, an average jump almost doubles the value of
the VIX.24 The SREJ1 variance specification lets jumps occur more frequently in
high volatility periods. Thus, the model can generate a more pronounced volatility
smile during periods of high uncertainty. We observe that the frequency of jumps
is about the same as in the constant intensity model, while the mean jump size is
lower. This is expected, since a proportional jump intensity puts a higher weight
24
Large daily percentage changes in the VIX are not uncommon, e.g. from August 3, 2011 to
August 8, 2011 the VIX more than doubled.
22
on BV S . This implies that smaller shocks in the instantaneous variance process are
necessary to generate comparable upward spikes in the VIX. The SREJ1 model
produces lower pricing errors compared with the SREJ0 model. This shows that
a variance specification with a time-varying jump intensity is important for VIX
option pricing. The largest improvements are achieved for longer times to maturity.25
Concerning the jump size specification, the parameters are similar across models.
We do not find that the more flexible gamma jump size distribution can significantly
improve upon the special case of exponentially distributed jumps. Especially for the
shortest time to maturity, all one-factor models have problems in pricing deep ITM
call options, i.e. upward jumps in the instantaneous variance do not make the left
tail of the risk-neutral VIX distribution heavy enough.
We find that the stochastic volatility of variance (SVV) specification generates
lower pricing errors compared with the variance jump models. It performs well for all
times to maturity. The improvements arise due to a better fit to long-term options
and pricing short-term deep ITM call options more accurately. The pricing errors
of the SVV model for longer times to maturity are about half the pricing errors of
the one-factor models. Both models perform well for short-term OTM call options.
In the SVV model the variance process is relatively persistent, while the process for
the volatility of variance is short-lived and very volatile. The correlation between
diffusive shocks to the instantaneous variance and its variance is 0.87. While variance
jumps mainly introduce right skewness, the stochastic volatility of variance model
generates an instantaneous variance process that exhibits a greater kurtosis. Thus,
depending on the value of ρvq , OTM and ITM call options can be rather expensive.
Due to the high positive correlation between the instantaneous variance and its
variance, the implied volatility smile is mostly upward sloping with a moderate
U-shape for low levels of moneyness.
25
As we only calibrate the models to a single cross-section, the potential benefits of a proportional jump intensity for explaining changes in the shape of the implied volatility smile over time
are not investigated.
23
6
Conclusion
The market for exchange traded volatility derivatives has experienced a dramatic
upswing in recent years. However, accurate modeling and pricing tools are still in its
infancy. We develop a general framework for modeling and pricing VIX futures and
options. We investigate several special cases of the general model and find that the
fine structure of the variance process is essential for the time-series behavior of the
VIX and the shape of the implied volatility surface. In particular, the commonly used
square-root variance process is misspecified. We show that both variance jumps and
a stochastic volatility of variance are important to reconcile the empirical regularities
with the theoretical models. Jumps have a major impact on short times to maturity.
The effect dies out with increasing time to maturity. In contrast, the stochastic
volatility of variance model is able to match the basic pattern for longer maturities.
Concerning the assumed variance process, two possible extensions are a more
flexible jump specification and non-affine models. We restricted ourselves to Poisson
jumps. Wu (2011) uses a variety of Levy processes to model variance jumps. It would
be interesting to investigate the implications of these processes for VIX option prices.
Furthermore, there is a growing literature on non-affine models. Constant elasticity
of variance models have been studied by Christoffersen et al. (2010). This model
class has more desirable volatility time-series properties. Gatheral (2008) uses a
non-affine two-factor model to price VIX derivatives and is able to replicate the
basic patterns. The disadvantage of using non-affine option pricing models is that
they do not admit analytic solutions for VIX derivatives and option prices can only
be computed based on time-consuming Monte Carlo simulations.
Another promising topic for future research is to use the information content
of VIX options for estimation purposes. VIX options are very informative about the
variance process and using them jointly with SPX options and return data might
help to identify the true data generating process.
24
A
Risk-Neutral Distribution
Breeden and Litzenberger (1978) show that the second derivative of a European call option price
with respect to its strike price is the discounted risk-neutral probability of the future asset price
ending up at exactly the strike price of the option. The price of a VIX call option with strike price
K and maturity in T is
Z ∞ R
T
e− t rs ds (VIXT − K) qt (VIXT , T )dVIXT .
Ct (K, T ) =
K
Differentiating twice with respect to the strike price and rearranging yields the following expression
for the risk-neutral density
RT
∂ 2 Ct (K, T ) .
(9)
qt (VIXT , T ) = e t rs ds
∂K 2
K=VIXT
As a continuum of option prices is not available in practice, we follow the interpolation and extrapolation approach in Bliss and Panigirtzoglou (2002, 2004) and Figlewski (2010). We exclude
ITM options (puts with moneyness greater than 1.05 and calls with moneyness smaller than 0.95)
because they are less liquid than OTM and ATM options. To reduce the effect of a jump at the
transition point between call and put options, we blend the call and put implied volatilities in the
region around the at the money level as in Figlewski (2010). For each maturity, we interpolate
implied volatilities across delta using a cubic smoothing spline. The smoothing parameter is initialized at 0.99 and if necessary adjusted upwards to fit option prices within their bid-ask spreads.
To convert implied volatilities from the strike price space to the delta space, we use the Black
(1976) formula with the ATM implied volatility. The fitted spline leaves us with a narrowly spaced
set of implied volatilities across deltas. In the next step, the implied volatilities are converted into
call prices using the Black (1976) formula. A fine grid of 20,000 prices is used to approximate the
risk-neutral distribution in Equation (9) using finite difference methods. We numerically integrate
the appropriate function of the probability density to estimate the moments. Risk-neutral moments
for a fixed time to maturity are obtained by linear interpolation.
Concerning the tales of the distribution, we have to impose some distributional assumptions.
We follow Bliss and Panigirtzoglou (2002, 2004) and extrapolate beyond the maximum (minimum)
available strike price with the implied volatility of the highest (lowest) actually traded strike price.
This procedure effectively makes the tails lognormal. Note that the Black (1976) formula is solely
used to switch between option prices and implied volatilities. We do not presume that the formula
prices options correctly.
B
B.1
VIX Option Pricing
Pricing formula
Equation (7) shows that the variance swap rate is an affine function of the state variables
V ST (T̄ ) = AV S + BV S XT .
To price futures and options on the VIX, we apply the results in Lewis (2000, 2001) and Chen
and Joslin (2012). First, we derive the Fourier transform of the payoff functions. The forward and
25
inverse (generalized) Fourier transform of some function f [x] are
Z ∞
fˆ[z] =
eizx f [x]dx,
−∞
f [x]
=
1
2π
izi +∞
Z
e−izx fˆ[z]dz,
izi −∞
with transform variable z = zr + zi i, where zr and zi denote the real and imaginary parts of z.
The payoff of a call option on the square-root of the 30-day variance swap rate is
f1 V ST (T̄ ) =
q
+
V ST (T̄ ) − K
.
The forward transform is
√
fˆ1 [z] =
√
π 1 − erf(K −iz)
,
2(−iz)3/2
where erf denotes the error function of a complex valued argument, which can be evaluated using
the series approximation in Abramowitz
and Stegun (1972). A futures contract on the VIX has
p
the payoff function f2 V ST (T̄ ) = V ST (T̄ ). The forward transform is
√
fˆ2 [z] =
π
.
2(−iz)3/2
(10)
For both payoff functions the Fourier transform is well-behaved if zi > 0.
We use the results in Duffie et al. (2000) to compute the discounted characteristic function
of the state vector
h RT
i
− t rs ds uXT
ψ Q (u, Xt , t, T ) = EQ
e
t e
=
eα(τ )+β(τ )Xt ,
for u ∈ Cn . The coefficients satisfy the following set of ODEs
∂β(τ )
1
= r1 − K 0 β(τ ) − [β(τ )0 Hβ(τ )] − λ01 [%(β(τ )) − 1] ,
∂τ
2
(11)
∂α(τ )
1
= r0 − M 0 β(τ ) − β(τ )0 hβ(τ ) − λ00 [%(β(τ )) − 1] ,
∂τ
2
subject to β(0) = u and α(0) = 0. [β(τ )0 Hβ(τ )] denotes a n×1 vector with the i-th component given
by [β(τ )0 Hi β(τ )]. For the Heston (1993) model and certain other specifications of the instantaneous
variance, the ODEs can be solved analytically. Models with a state-dependent jump intensity or
gamma distributed jump sizes do not admit closed-form solutions. However, the ODEs can be
easily solved numerically using Runge-Kutta methods.
26
Putting the results together, the price of a call option on the VIX is given by
h RT
i
− t rs ds
Ct (K, T ) = EQ
f1 V ST (T̄ )
t e
R
Z izi +∞
Q
− tT rs ds 1
−izV ST (T̄ ) ˆ
= Et e
e
f1 [z]dz
2π izi −∞
Z izi +∞
1
=
e−izAV S ψ Q (−izBV S , Xt , t, T )fˆ1 [z]dz
2π izi −∞
#
√
Z ∞ "
1 − erf(K zi − izr )
1
(zi −izr )AV S Q
ψ ((zi − izr )BV S , Xt , t, T )
< e
= √
dzr .
π 0
2(zi − izr )3/2
The integration is performed along a straight line in the complex plane parallel to the real axis.
The price of a forward contract on the VIX can be computed by substituting the transformed
payoff in Equation (10) into the pricing formula and ignoring the discounting in Equation (11), i.e.
by setting r0 = 0 and r1 = 0. To determine put prices, we use the put-call parity relationship. The
characteristic function does not depend on the strike price. Thus, a cross-section of options can be
priced without evaluating the characteristic function in each step.
As the variance swap rate is an affine function of the state variables, we can easily obtain
its risk-neutral distribution by numerical integration. Using a change of variable, we obtain the
following density of the VIX
Z
i
2
2x ∞ h (zi −izr )AV S Q
< e
ψ ((zi − izr )BV S , Xt , t, T ) e(izr −zi )x dzr .
qt (x, T ) =
π 0
B.2
Monte Carlo tests
To demonstrate the accuracy of the solution technique, we simulate the SVCJ model (see Section
4.4) 1000 times, with parameters taken from Eraker (2004), and calculate VIX option prices for
various strike prices. Figure 7 compares the analytical prices using Equation (12) (solid lines) with
Monte Carlo option prices (asterisks) for 30 days (blue) and for 180 days (red). We consider three
values of the initial variance: long-run mean minus the average variance jump size (top panel),
long-run mean (middle panel), and long-run mean plus the average variance jump size (bottom
panel). Figure 7 demonstrates that option prices computed using the methodology outlined above
and options prices obtained using Monte Carlo simulation are practically indistinguishable.
27
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521–531.
31
Min
Max
Mean
Std. Dev.
Skewness
Kurtosis
AC1
VIX
9.31
80.86
20.57
8.28
1.94
9.70
0.98
VIX first differences
-17.36
16.54
0.00
1.57
0.60
21.50
-0.11
Table 1: Descriptive Statistics
The table shows descriptive statistics of the VIX and its first differences. The sample
period is from January 2, 1990 to December 31, 2011.
32
Moneyness
≤ 0.75
0.75 − 0.90
0.90 − 1.00
1.00 − 1.10
1.10 − 1.25
> 1.25
IV
Obs.
Avg Vol.
Avg OI
Spread
IV
Obs.
Avg Vol.
Avg OI
Spread
IV
Obs.
Avg Vol.
Avg OI
Spread
IV
Obs.
Avg Vol.
Avg OI
Spread
IV
Obs.
Avg Vol.
Avg OI
Spread
IV
Obs.
Avg Vol.
Avg OI
Spread
Call Options
Days to Expiration
< 60 60 − 180 > 180
93.36
55.51
38.87
7051
12770
1475
223
50
48
7193
1754
1323
4.18
5.44
7.82
71.08
57.07
45.62
5794
11246
1606
953
196
149
13283
4169
2345
6.30
7.03
9.45
78.77
61.94
49.06
3648
6343
964
2937
563
148
26175
7117
3561
7.05
7.96
10.53
86.15
64.44
51.47
3307
5834
916
5212
724
208
39415
9578
2806
8.33
9.18
12.31
94.27
67.49
53.30
4436
7699
1224
5401
719
163
46798
10351
3587
12.05
11.29
14.78
113.34
78.69
56.44
16642
39246
3066
2958
461
188
37595
6474
4372
31.54
32.22
28.81
Put Options
Days to Expiration
< 60 60 − 180 > 180
80.21
57.79
40.58
2605
10048
1091
1822
417
49
26921
8697
1364
43.49
44.46
54.71
73.44
57.68
45.26
5287
11191
1505
4099
697
207
35643
9616
2446
22.98
17.84
22.82
78.60
62.22
49.03
3647
6201
821
4601
585
225
38861
7885
4545
10.06
9.93
13.85
86.04
64.73
51.41
3287
5521
687
2418
297
52
32172
5049
883
7.03
7.86
11.68
94.01
68.10
54.20
4376
6756
852
624
120
6
16337
2431
201
5.75
6.00
8.50
122.68
77.18
55.69
19002
24414
2170
60
11
3
2309
432
73
2.64
3.16
4.87
Table 2: VIX Options Data
The table shows implied volatilities (in percent) using the Black (1976) formula,
the number of observations, the average volume, the average open interest, and the
average percentage bid-ask spread for different moneyness and maturity categories.
The average percentage bid-ask spread follows from dividing the bid-ask spread by
the ask price. Moneyness is defined as the strike price divided by the VIX futures
price. The sample period is from February 24, 2006 to December 31, 2011.
33
λv,0
1
1
1
1
3
3
3
3
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Parameters
λv,1 µv
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
2
0
2
0
2
0
2
0
1
0
1
0
1
0
1
1
1
1
1
1
1
1
1
3
1
3
1
3
1
3
1
1
2
1
2
1
2
1
2
1
1
1
1
1
1
1
1
ν
1
1
1
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
1
1
1
1
1
1
1
1
1
1
1
1
0.5
0.5
0.5
0.5
Maturity
Days
30
60
180
365
30
60
180
365
30
60
180
365
30
60
180
365
30
60
180
365
30
60
180
365
30
60
180
365
30
60
180
365
0.6
54.44
45.56
28.57
20.12
51.29
45.10
29.64
20.92
52.76
45.55
29.62
20.91
54.38
45.53
28.61
20.15
58.01
49.35
31.55
22.26
56.92
56.13
44.61
32.81
58.45
53.14
37.15
26.52
57.91
49.26
31.52
22.25
0.8
53.45
46.20
29.87
21.06
51.78
47.08
31.85
22.50
52.83
47.62
32.44
22.95
53.41
46.27
30.08
21.22
56.24
49.47
32.83
23.23
57.69
58.61
47.94
35.67
57.89
55.18
41.02
29.54
56.14
49.44
32.95
23.33
1.0
53.14
46.46
30.41
21.45
55.20
49.60
33.05
23.33
56.04
51.28
35.22
24.91
53.32
46.93
31.04
21.91
56.36
50.27
34.01
24.11
65.10
63.14
49.88
37.17
63.96
61.43
45.91
33.14
56.48
50.66
34.57
24.53
K/VIX
1.2
1.4
57.94 67.47
48.63 52.52
31.23 32.35
22.01 22.77
64.32 71.91
53.22 55.81
33.86 34.19
23.86 24.06
69.12 84.11
58.92 66.57
38.49 41.18
27.14 28.98
59.65 72.47
50.49 56.86
32.75 35.16
23.10 24.75
62.51 72.57
53.57 58.20
35.70 37.42
25.31 26.48
74.72 79.99
66.70 68.26
50.44 50.15
37.54 37.28
80.36 94.81
71.12 78.66
50.39 53.26
36.25 38.17
64.22 77.55
55.38 62.39
37.20 40.16
26.39 28.42
1.6
74.28
55.68
33.32
23.43
75.43
56.80
34.12
24.00
93.30
71.18
42.81
30.10
82.49
62.37
37.37
26.27
79.46
61.60
38.58
27.27
82.31
68.55
49.41
36.71
103.20
82.97
54.78
39.17
87.58
68.03
42.49
30.01
1.8
77.94
57.36
33.88
23.81
76.73
56.83
33.78
23.75
98.48
73.53
43.64
30.67
88.91
65.91
38.92
27.35
83.30
63.45
39.16
27.65
83.10
68.14
48.45
35.98
107.98
85.27
55.45
39.59
94.05
71.66
44.00
31.04
Table 3: Jump Components
The table shows implied volatilities (in percent) using the Black (1976) formula.
The risk-free interest rate is set to zero.
34
Parameters
κq σq ρvq
10 4 0.95
10 4 0.95
10 4 0.95
10 4 0.95
5
4 0.95
5
4 0.95
5
4 0.95
5
4 0.95
10 2 0.95
10 2 0.95
10 2 0.95
10 2 0.95
10 4
0.8
10 4
0.8
10 4
0.8
10 4
0.8
Maturity
Days
30
60
180
365
30
60
180
365
30
60
180
365
30
60
180
365
0.6
38.04
28.32
19.09
13.62
43.74
31.88
20.27
14.31
54.80
48.20
32.75
23.23
54.77
46.06
29.55
20.89
0.8
36.61
32.11
22.09
15.70
32.28
25.91
15.96
11.23
52.87
45.25
29.63
20.94
45.90
39.01
25.66
18.16
1.0
48.57
40.39
25.74
18.16
47.97
38.89
23.75
16.70
50.75
43.04
27.65
19.50
48.84
40.83
26.13
18.43
K/VIX
1.2
52.18
43.49
27.34
19.25
53.02
44.44
27.92
19.66
48.45
41.03
26.14
18.41
51.20
42.75
26.94
18.97
1.4
52.89
44.35
27.86
19.60
54.23
46.26
29.69
20.93
46.21
39.17
24.88
17.51
51.58
43.30
27.22
19.15
1.6
52.45
44.23
27.84
19.58
54.01
46.59
30.40
21.45
44.12
37.46
23.78
16.73
51.03
43.06
27.12
19.07
1.8
51.50
43.63
27.55
19.38
53.17
46.21
30.55
21.58
42.21
35.90
22.81
16.04
50.05
42.42
26.79
18.84
Table 4: Stochastic Volatility of Variance
The table shows implied volatilities (in percent) using the Black (1976) formula.
The risk-free interest rate is set to zero.
35
σS
κv
σv
λv,0
λv,1
µv
ν
κq
σq
ρvq
qt
RMSE
RMSE
RMSE
RMSE
RMSE
(all)
(1)
(2)
(3)
(4)
SR
0.312
1.710
1.849
—
—
—
—
—
—
—
—
SREJ0
0.180
2.870
1.906
1.535
—
3.646
—
—
—
—
—
0.314
0.350
0.304
0.281
0.315
0.132
0.213
0.062
0.065
0.126
Models
SREJ1 SRGJ0
0.201
0.169
4.923
2.912
1.648
1.993
—
2.442
1.702
—
1.811
2.802
—
0.522
—
—
—
—
—
—
—
—
0.124
0.209
0.060
0.045
0.109
0.130
0.211
0.060
0.064
0.123
SRGJ1
0.196
5.143
1.602
—
1.896
1.754
1.051
—
—
—
—
SVV
0.339
1.463
1.685
—
—
—
—
22.589
20.403
0.876
1.194
0.123
0.207
0.056
0.048
0.111
0.086
0.152
0.047
0.022
0.060
Table 5: Parameter Estimates and Pricing Errors
The parameters are estimated by minimizing the sum of squared dollar pricing
errors between market and model prices. The mean-reversion levels v̄ and q̄ are fixed
at one. SR denotes the square-root model. The SREJ0 and SREJ1 model include
constant and stochastic intensity exponentially distributed jumps, respectively. The
SRGJ specification features gamma distributed jumps. SVV denotes the stochastic
volatility of variance model. To assess the performance of the models, we report
the root mean squared pricing error (RMSE) for all options and separately for each
maturity category.
36
80
VIX
60
40
20
0
1990
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
1992
1994
1996
1998
2000
2002
2004
2006
2008
2010
2012
VIX First Differences
15
10
5
0
−5
−10
−15
1990
Figure 1: CBOE Volatility Index
The figure displays the VIX and its first differences. The sample period is from
January 2, 1990 to December 31, 2011. The shaded areas represent NBER recessions.
37
VIX First Differences Quantiles
VIX First Differences (Modulus)
20
10
0
−10
−20
−4
−3
−2
−1
0
1
Standard Normal Quantiles
2
3
4
20
15
10
5
0
0
10
20
30
40
50
VIX
60
70
80
90
100
Figure 2: VIX Time-Series Diagnostics
The upper panel of the figure shows a quantile-quantile plot of the sample quantiles
of VIX first differences versus the theoretical quantiles of a normal distribution. The
lower panel displays a scatter plot of absolute values of VIX first differences and the
VIX, including a least squares regression line. The sample period is from January
2, 1990 to December 31, 2011.
38
70
25
60
20
Volatility
Mean
50
40
30
15
10
20
5
10
2008
2010
0
2006
2012
2.5
9
2
8
Kurtosis
Skewness
0
2006
1.5
1
0.5
0
2006
2008
2010
2012
2008
2010
2012
7
6
5
4
2008
2010
3
2006
2012
Figure 3: Risk-Neutral Moments
The figure shows the risk-neutral moments for a fixed time to maturity of 60 days.
The dotted red line in the upper left corner is the VIX futures contract with a fixed
time to maturity of 60 days.
39
0.9
Implied Volatility
0.8
0.7
0.6
0.5
0.4
0.3
0.2
60
1
50
0.8
40
0.6
0.4
30
Strike Price
0.2
20
0
Time to Maturity
Figure 4: Implied Volatilities (SV Model)
The figure shows the implied volatility surface of the SV model. The parameters are
taken from Eraker (2004) and the initial variance is set to its long-run mean. The
risk-free rate is fixed at 5%.
40
0.6
Implied Volatility
0.5
0.4
0.3
0.2
0.1
70
60
1
50
0.8
0.6
40
0.4
30
Strike Price
0.2
20
0
Time to Maturity
Figure 5: Implied Volatilities (SVCJ Model)
The figure shows the implied volatility surface of the SVCJ model. The parameters
are taken from Eraker (2004) and the initial variance is set to its long-run mean.
The risk-free rate is fixed at 5%.
41
1.4
1.3
19 days
45 days
75 days
105 days
1.2
Implied Volatility
1.1
1
0.9
0.8
0.7
0.6
0.5
0.8
1
1.2
1.4
1.6
1.8
2
Moneyness
Figure 6: Average Implied Volatility Smiles
The figure shows average implied volatility smiles for 4 maturity categories. The
average times to maturity in the 4 maturity categories are 19, 45, 75, and 105 days.
Moneyness is defined as the strike price divided by the VIX futures price. The sample
period is from February 24, 2006 to December 31, 2011.
42
Option Price
15
10
5
0
20
25
30
35
40
45
50
25
30
35
40
45
50
25
30
35
Strike Price
40
45
50
Option Price
15
10
5
0
20
Option Price
20
15
10
5
0
20
Figure 7: Analytical versus Monte Carlo Option Prices
The figure shows VIX call option prices for the SVCJ model. The asterisks indicate
prices obtained by Monte Carlo simulation and the solid lines show analytically
computed values (in blue for 30 days and in red for 180 days). The parameters are
taken from Eraker (2004). The initial variance is set to its long-run mean minus the
average variance jump size (top panel), long-run mean (middle panel), and long-run
mean plus the average variance jump size (bottom panel). The risk-free rate is 5%.
43