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Pricing the CBOE VIX Term Structure and VIX Futures with
Realized Volatility
Zhuo Huang∗ Chen Tong† Tianyi Wang‡
May 2017
Abstract
Using an extended LHARG model proposed by Majewskia et al. (2015), we derive the closed-form
pricing formulas for both the CBOE VIX term structure and VIX futures with different maturity.
Our empirical results suggest that the quarterly and yearly components of lagged realized volatility
should be added into the model to capture long-term volatility dynamics. With the realized volatility
based on high frequency data, the proposed model provide superior pricing performance compared
to the classic Heston-Nandi GARCH model, both in-sample and out-of-sample. The improvement is
more pronounced during high volatility periods.
Keywords: Implied volatility, Volatility term structure, VIX futures, Realized volatility
JEL classification: C19;C22;C80
∗National School of Development, Peking University, Beijing 100871, P.R. China, Email: [email protected]. ZhuoHuang acknowledges financial support from the National Natural Science Foundation of China (71671004).†National School of Development, Peking University, Beijing 100871, P.R. China, Email:[email protected].‡Corresponding author, Department of Financial Engineering, School of Banking and Finance, University of Interna-
tional Business and Economics, Beijing 100029, P.R. China, Email: [email protected]. Tianyi Wang acknowledgesfinancial support from the Youth Fund of National Natural Science Foundation of China (71301027), the Ministry ofEducation of China, Humanities and Social Sciences Youth Fund (13YJC790146), and the Fundamental Research Fundfor the Central Universities in UIBE(14YQ05).
1
1 Introduction
The well-known Chicago Board Options Exchange (CBOE) VIX index, computed from a panel of
options prices, is a model-free measure of expected average variance for next 30 days under the risk
neutral measure. The index has become the benchmark for stock market volatility and it is used as
the “investor fear gauge” for the equity markets. With the launch of VIX futures in 2004 and VIX
options in 2006, volatility derivatives have received increasingly attention in the market as the market
average daily volume of VIX options and VIX futures have expanded to over 137 and 25 times in the
last decade. The VIX-linded products essentials creates a volatility market that enables investors to
trade volatility directly just like to equity or fixed income securities1. In addition to the VIX index that
measures 1-month implied volatility, CBOE has also lunched a series of implied volatility indices across
different maturities in recent years, to reflect volatility term structure under the risk neutral measure.
The CBOE S&P 500 3-Month Volatility Index (Ticker:VXV) was lunched on November 2007. The
CBOE Mid-Term Volatility Index (Ticker: VXMT), a measure of the expected volatility of the S&P
500 Index over a 6-month time horizon, was launched on November 20132.
Zhang and Zhu (2006) gave the first attempt to price VIX futures based on the classic continuous-
time Heston model. Lin (2007) extended the model with simultaneous jumps in both returns and
volatility to price VIX futures with approximation formula. Adding jumps into mean-reverting process
are also investigated by Sepp (2008), Zhang et al. (2010) and Zhu and Lian (2012) etc. Under the
discrete-time volatility framework, Wang et al. (2016) derived the pricing formulas for both VIX and
VIX futures using the discrete-time Heston-Nandi GARCH model. Inspired by Hao and Zhang (2013)
and Kanniainen et al. (2014), the model is estimated in a joint fashion where both the underling
dynamics as well as the VIX futures prices are taken into account in the objective function. Results
show that the model can yield good in-sample and out-of-sample prices when VIX or VIX futures are
involved in estimation.
Since the seminal work of Andersen et al. (2003), the realized volatility computed from high frequency
intra-day returns have proved to be an accurate measurement of the latent volatility process. Models
using the realized measures to model and forecast volatility have attracted great attention in recent years.
Such models include but not limited to the heterogenous autoregressive (HAR) model of Corsi (2009),
the MEM model of Engle and Gallo (2006), the HEAVY model of Shephard and Sheppard (2010) and
the Realized GARCH model of Hansen et al. (2012). Among these models, the HAR model is the most
popular in applied research on volatility modelling due to its estimation simplicity and good forecasting
performance. The model introduces a cascade structure into the linear autoregression framework in
1Luo and Zhang (2014) provide a good discuss of market for volatility derivatives.2CBOE reported historical data of VXMT back to January 2008.
2
which the current daily realized variance is regressed on the lagged realized variance over the past
day, past week and past month. Empirical studies show the HAR model provide a parsimonious but
good approximation for the long memory process of volatility. For option pricing purpose, Corsi et al.
(2013) extended the HAR model with an Gamma innovation (namely the Heterogenous Autoregressive
Gamma, HARG) and specified an exponentially affine pricing. Based on that, Majewskia et al. (2015)
further developed the model by allowing more flexible leverage components (LHARG) and derived the
analytical pricing formula for European options.
While most studies focus on the performances of the HAR model and its extensions in forecasting
volatility or realized volatility under the physical measure, Corsi et al. (2013) and Majewskia et al. (2015)
have shown the HAR framework is also capable to match the volatility information implied by option
prices, i.e., under the risk neutral measure. In this paper, we derive the analytical formulas for VIX term
structure and VIX futures based on an modified version of LHARG model. The empirical results suggest
we should add the quarterly and yearly average of lagged realized volatility into the LHARG model
to capture volatility dynamics in longer horizons in order to price volatility index and its derivatives.
Compared with the pricing formula under the classic Heston-Nandi GARCH model in Wang et al. (2016),
our proposed model provides superior performance in pricing VIX term structures and VIX futures. The
improvement is more pronounced during high volatility periods, when the realized volatility contains
relatively more accurate information about underlying volatility. Our empirical findings are robust in
out-of-sample analysis.
The remainder of the paper is organized as follows. Section 2 discusses the model setup and derive
the pricing formula for VIX term structure and VIX futures. Section 3 discusses model estimation using
different datasets. Section 4 presents the empirical results and Section 5 concludes.
2 THE MODEL
2.1 LHARG Model and Risk Neutralization
In this paper, we denote the original LHARG model as LHARG-M since it contains volatility components
up to monthly average. It is extended with quarterly average (63 trading days) to LHARG-Q and with
both quarterly and yearly average (252 trading days) to LHARG-QY. If properties are applied to all
three models, we use LHARG to save space. The LHARG-QY model is given by:
Rt+1 = r + λRVt+1 −1
2RVt+1 +
√RVt+1εt+1, εt+1 ∼ i.i.dN(0, 1) (2.1)
RVt+1|Ft ∼ Γ(δ,Θ(RVt,Lt), θ)
3
Θ(RVt,Lt) =1
θ(d+βdRV
(d)t + βwRV
(w)t + βmRV
(m)t + βqRV
(q)t + βyRV
(y)t +
αd`(d)t + αw`
(w)t + αm`
(m)t + αq`
(q)t + αy`
(y)t )
We define components as follows:
RV(d)t = RVt `
(d)t = ε2t − 1− 2γεt
√RVt
RV(w)t =
1
4
4∑i=1
RVt−i `(w)t =
1
4
4∑i=1
(ε2t−i − 1− 2γεt−i√RVt−i)
RV(m)t =
1
17
21∑i=5
RVt−i `(m)t =
1
17
21∑i=5
(ε2t−i − 1− 2γεt−i√RVt−i)
RV(q)t =
1
41
62∑i=22
RVt−i `(q)t =
1
41
62∑i=22
(ε2t−i − 1− 2γεt−i√RVt−i)
RV(y)t =
1
189
251∑i=63
RVt−i `(y)t =
1
189
251∑i=63
(ε2t−i − 1− 2γεt−i√RVt−i)
The Rt, RVt, r denote the log-return of underlying index, the realzied volatility and the risk free rate
respectively. λ captures the equity risk premium. The coditional distribution of RVt+1 features a
noncentral gamma distribution (denoted as Γ(·)) with shape and scale parameters equals δ and θ. The
location parameter is given by Θ(RVt,Lt). It is easy to see that LHARG-Q is nested in LHARG-QY
and the original LHARG-M can be recovered by setting βq, βy, αq and αy equal to zero.
The specification of the leverage function has been inspired by Christoffersen et al. (2008) and
enriched by a heterogeneous structure3. Unlike the the Heston-Nandi type leverage function, Θ(RVt,Lt)
is no longer guaranteed to be positive. Nevertheless, Majewskia et al. (2015) provided numerical evidence
of the effectiveness of the analytical results in describing a regularized version of the model.
According to the properties of noncentral gamma distribution, the conditional expectation of RVt+1
in physical (P ) measure is
EPt [RVt+1] = θδ + θΘ(RVt,Lt)
= θδ + d+ βdRV(d)t + βwRV
(w)t + βmRV
(m)t + βqRV
(q)t + βyRV
(y)t +
αd`(d)t + αw`
(w)t + αm`
(m)t + αq`
(q)t + αy`
(y)t
3Majewskia et al. (2015) denoted this leverage function as “Zero-Mean” and found it has the best performance in optionpricing because of its less constrained leverage allows the process to explain a larger fraction of the skewness and kurtosisobserved in real data.
4
Therefore, the stationary condition for LHARG-QY model is
πP = βd + βw + βm + βq + βy < 1
The unconditional expectation of RV under the P measure is given by:
EP [RVt] =θδ + d
1− βd − βw − βm − βq − βy
When θδ + d is positive and the stationary condition is satisfied, the unconditional variance will be
positive.
We can rewrite the formula of Θ(RVt,Lt) as
Θ(RVt,Lt) =1
θ(d+
p∑i=1
βiRVt+1−i +
q∑j=1
αj`t+1−j)
where
`t+1−j = ε2t+1−j − 1− 2γεt+1−j√RVt+1−j
RVt ≡ (RVt, RVt−1, ..., RVt+1−p) Lt ≡ (`t, `t−1, ..., `t+1−q)
βi =
βd i = 1
βw/4 2 ≤ i ≤ 5
βm/17 6 ≤ i ≤ 22
βq/41 23 ≤ i ≤ 63
βy/189 64 ≤ i ≤ 252
αj =
αd j = 1
αw/4 2 ≤ j ≤ 5
αm/17 6 ≤ j ≤ 22
αq/41 23 ≤ j ≤ 63
αy/189 64 ≤ j ≤ 252
Applying the stochastic discount factor:
Mt,t+1 =exp(−v1RVt+1 − v2Rt+1)
EPt [exp(−v1RVt+1 − v2Rt+1)]
we obtain the following risk neutral (Q) dynamics:
Rt+1 = r − 1
2RVt+1 +
√RVt+1ε
∗t+1, ε∗t+1 ∼ i.i.dN(0, 1)
RVt+1|Ft ∼ Γ(δ∗,Θ∗(RVt,L∗t ), θ∗)
Θ∗(RVt,L∗t ) =
1
θ∗(d∗ +
p∑i=1
β∗iRVt+1−i +
q∑j=1
α∗j`∗t+1−j)
5
The risk neutral parameters are linked to physical parameters through:
β∗i = [βi + αi(2γλ+ λ2)]/(1 + θy∗)2, α∗j = αj/(1 + θy∗)2
δ∗ = δ, d∗ = d/(1 + θy∗)2, θ∗ = θ/(1 + θy∗), γ∗ = γ + λ
ε∗t = εt + λ√RVt
`∗t+1−j = ε∗2t+1−j − 1− 2γ∗ε∗t+1−j√RVt+1−j
L∗t ≡ (`∗t , `
∗t−1, ..., `
∗t+1−q)
We define v1 as a time-invariant variance risk parameter and y∗ = (λ− 1/2)2/2 + v1 − 1/8 to maintain
the LHARG structure under the Q dynamics. The corresponding stationary and positivity condition
under Q are:
πQ = β∗d + β∗w + β∗m + β∗q + β∗y < 1 θ∗δ∗ + d∗ > 0
2.2 The VIX Pricing Formula
In line with Hao and Zhang (2013), the VIX can be calculated as the annualized arithmetic average of
the expected daily variance over the following month under the risk-neutral measure:(VIXt
100
)2
=1
n
n∑k=1
EQt [RVt+k]×AF (2.2)
Where AF is the annualizing factor that converts daily variance into annualized variance. The implied
volatility term structure at time t with maturity of n, is defined as the average expected volatility over
the next n trading days:
Vt(n) =1
n
n∑k=1
EQt [RVt+k] (2.3)
Assume that there are 22 trading days in a month and 252 trading days in a year, the model implied
VIX, VXV and VXMT are:
VIXt = 100√
252Vt(22) (2.4)
VXVt = 100√
252Vt(63) (2.5)
VXMTt = 100√
252Vt(126) (2.6)
Proposition 1: If the return of S&P500 follows LHARG model, then the model implied term structure
6
at time t can be presented by a weighting average of the long-run varaince µQ and lagged RV s with
some zero mean leverage adjustments:
Vt(n) = (1−p∑i=1
ξi)µQ +
p∑i=1
ξiRVt+1−i +
q∑j=1
ωj`∗t+1−j (2.7)
Where
ξi =1
n
n∑k=1
ξ(k)i ωj =
1
n
n∑k=1
ω(k)j µQ ≡ EQ(RVt) =
θ∗δ∗ + d∗
1−∑p
i=1 β∗i
In LHARG-QY model the p and q are both equal to 252, and the ξ(k)i and ω
(k)i could be obtained by
an iterative relationship:
ξ(k+1)i =
ξ(k)i+1 + ξ
(k)1 β∗i 1 ≤ i < p
ξ(k)1 β∗i i = p
ω(k+1)j =
ω(k)j+1 + ξ
(k)1 α∗j 1 ≤ j < q
ξ(k)1 α∗j j = q
with initial conditions :ξ(1)i = β∗i , ω
(1)j = α∗j .
Proof: See the Appendix.
The proposition 1 shows that the implied volatility term structure is determined by the weighting
average of the past realized volatilities, leverage components and long-run volatility level. Plugging
Equation (2.7) in Equation (2.4) - (2.6), we can obtain the VIX, VXV and VXMT pricing formula
respectively.
2.3 The VIX Futures Pricing Formula
Following expression given by Proposition 1 of Zhu and Lian (2012), the VIX futures price in time t
with maturity time T can be presented as the conditional expectation of VIX in Q measure.
F (t, T ) = EQt [VIXT] = EQt [100√
252VT (22)] =100√
252
2π
∫ ∞0
1− EQt [exp(−sVT (22))]
s3/2ds
The last term EQt [exp(−sVT (22))] is the moment-generating function of the VT (22) in Q measure.
Proposition 2 : Under risk neutral measure, the moment-generating function of the VT (22) has
the following form
f(z, k,RVt,L∗t ) = EQt [exp(zVt+k(22))] = exp
Ω(k) +
p∑i=1
φ(k)i RVt+1−i +
q∑j=1
Γ(k)j `∗t+1−j
7
where Ω(k), φ(k)i and Γ
(k)j can be obtained by following iteration
Ω(k+1) = Ω(k) +A(φ(k)1 ,Γ
(k)1 )
φ(k+1)i =
φ(k)i+1 + Bi(φ(k)1 ,Γ
(k)1 ) 1 ≤ i < p
Bi(φ(k)1 ,Γ(k)1 ) i = p
Γ(k+1)j =
Γ(k)j+1 + Cj(φ(k)1 ,Γ
(k)1 ) 1 ≤ j < q
Cj(φ(k)1 ,Γ(k)1 ) j = q
with initial values:
Ω(1) = zµQ(1−p∑i=1
ξi) +A(zξ1, zω1)
φ(1)i =
zξi+1 + Bi(zξ1, zω1) 1 ≤ i < p
Bi(zξ1, zω1) i = pΓ(1)j =
zωj+1 + Cj(zξ1, zω1) 1 ≤ j < q
Cj(zξ1, zω1) j = q
The definition of µQ, ξi and ωj can be found in proposition 1 and A(·),Bi(·), Cj(·) are given as:
A(η, s) = −1
2log(1− 2s)− s− δW(χ, θ∗) + d∗V(χ, θ∗)
Bi(η, s) = V(χ, θ∗)β∗i Cj(η, s) = V(χ, θ∗)α∗j
V(χ, θ∗) =χ
1− θ∗χ, W(χ, θ∗) = log(1− θ∗χ), χ(η, s) = η +
2s2γ∗2
1− 2s
Proof: See the Appendix.
Replace EQt [exp(−sVT (22))] with f(−s, T − t,RVt,L∗t ), the analytical pricing formula is obtained.
2.4 COMPETING MODEL
Our model is compared with the Heston-Nandi GARCH model (HNG hereafter) presented by Heston and
Nandi (2000). It is a GARCH type model with explicit VIX and VIX futures pricing formula. However,
it does not utilize the realized variance and it is not adapted to fit long-run volatility dynamics. The
physical dynamic of the HNG model is
Rt+1 = r + λht+1 −1
2ht+1 +
√ht+1εt+1
ht+1 = ω + βht + α(εt − γ√ht)
2
The risk neutral counterpart is
Rt+1 = r − 1
2ht+1 +
√ht+1ε
∗t+1
ht+1 = ω + βht + α(ε∗t − γ∗√ht)
2
8
where ε∗t = εt+λ√ht, γ
∗ = γ+λ. The persistence parameters under P and Q measure are πP = β+αγ2
and πQ = β + αγ∗2 respectively. Pricing formulas for HNG can be found in Wang et al. (2016).
3 DATA AND ESTIMATION
3.1 Data description
We collect data for S&P 500 index returns and the panel of CBOE volatility index including VIX,
VXV and VXMT. The RV in this paper is calculated with realized kernel (see Barndorff-Nielsen et al.
(2008)). The panel of VIX futures prices with various maturities is also collected4. Since the VXMT
data starts at January 2008, our full-sample spans a nine-year period for January 2008 to March 2017.
The realized kernel series is trimmed following Majewskia et al. (2015) by removing the most extreme
observations (out of a four standard deviations threshold defined by a rolling window of 200 days)5. The
realized kernel is also re-scaled to match the sample variance of daily close-to-close returns. According
to Zhu and Lian (2012), several filters are applied to the VIX futures prices data. Firstly, VIX futures
that are less than 5 days to maturity are removed. Secondly, VIX futures data with the associated open
interest less than 200 contracts are excluded to avoid any liquidity-related bias. Finally, in order to
match the time horizon in VIX term structure data, we drop all VIX futures whose time to maturity
are longer than six months. The whole sample include 2270 daily observations for the underling data
and 13349 observations for VIX futures prices. Table 1 reports the summary statistics of our data set.
For VIX term structure, we find that: 1) The average volatility levels are higher in longer maturities.
2) The standard deviation of the series decreases almost monotonically with respect to maturity, which
can be found in Figure 1. 3) All series are skewed and leptokurtic. But, the skewness and kurtosis are
decreasing as maturity increases.
[Insert Table 1 here]
[Insert Figure 1 here]
3.2 Estimation Method
We estimate LHARG model with joint likelihood of the physical dynamics (index return and its real-
ized kernel), volatility indices (VIX, VXV and VXMT) and the derivative prices (VIX futures). We
assume normal distribution for pricing errors associated with volatility indices and VIX futures6. The
4The realized measures are collected from the Realized Library at the Oxford-Man institute and other data can befound on CBOE’s website.
5This procedure is proposed by Majewskia et al. (2015), and affected about 1.48% of the observations. As a comparison,this procedure affect 1.5% of the observations in their paper.
6Let T be the number of observations in returns, realized variance and volatility indices and N be the number of VIXfutures prices.
9
corresponding likelihood functions are:
• The ability to replicate underling dynamics
`R = −T2
log(2π)− 1
2
T∑t=1
log(RVt) + [Rt − r − λRVt +1
2RVt]
2/RVt
`RV = −
T∑t=1
(RVtθ
+ Θ(RVt−1,Lt−1))
+
T∑t=1
log( ∞∑k=0
RV δ+k−1t
θδ+kΓ(δ + k)
Θ(RVt−1,Lt−1)k
k!
)
• The ability to replicate volatility index
`VIX = −T2
log(2πs2VIX)− 1
2s2VIX
T∑t=1
(VIXModt −VIXMkt
t )2
`VXV = −T2
log(2πs2VXV)− 1
2s2VXV
T∑t=1
(VXVModt −VXVMkt
t )2
`VXMT = −T2
log(2πs2VXMT)− 1
2s2VXMT
T∑t=1
(VXMTModt −VXMTMkt
t )2
• The ability to replicate VIX futures prices
`Fut =− N
2log(2πs2Fut)−
1
2s2Fut
N∑i=1
(FutModi − FutMkt
i )2× T
N
Where is Γ(·) the Gamma function and the s2 is estimated with sample variance of pricing errors.
Noticed that the log-likelihood in `RV cannot be applied directly since it contains an infinite number
of terms. To implement the maximum likelihood estimator, we truncate the infinite sum up to its 90th
order following Majewskia et al. (2015). The T/N in `Fut is used to adjust the observation number
imbalance between VIX futures prices and other series.
Since the variance premium parameter v1 is not identifiable without risk neutral information, maxi-
mize the joint log-likelihood function `R,RV = `R+ `RV using returns and realized volatility information
is not an option for our model. This is not the case for traditional GARCH models where λ determines
both equity and variance premium. The risk neutral information can be extracted from either volatility
index or VIX futures quotes. Therefore, we have the following estimation methods:
10
• Maximize joint likelihood with volatility index7:
`R,RV,VIX = `R + `RV + `VIX `R,RV,VXV = `R + `RV + `VXV `R,RV,VXMT = `R + `RV + `VXMT
• Maximize joint likelihood with VIX futures:
`R,RV,Fut = `R + `RV + `Fut
For the following empirical investigation, we only keep daily leverage component and constrain other
leverage components to zero to make the model more concise. The main results doesn’t change when
additional leverage components are added.
4 EMPIRICAL RESULTS
4.1 Parameter Estimation
In table 2, we report the parameter estimation results for different LHARG models with different estima-
tion methods. The suffix “-Q” is for the augmented LHARG with quarterly components and “-QY” is
for the augmented LHARG with quarterly and yearly components. “VIX1M” (CBOE VIX), “VIX3M”
(CBOE VXV), “VIX6M” (CBOE VXMT) and “Fut” (VIX futures) indicating the Q information used
in estimation along with index returns and realized kernels.
[Insert Table 2 here]
For the LHARG-M model in column (1) - (4), compared with parameters in P measure (see Table 8
in appendix C), the monthly component βm increases significantly and weekly coefficient decreases. This
is due to the fact that volatility index is smoother than the realized volatility8 since it is an average of
expected volatility over a certain period. Therefore, long-term moving average of realized measures can
provide more valuable information than short-term moving average when volatility index is modeled.
Similar pattern can be found when VIX futures prices is used in estimation. This point is reinforced
by the fact that monthly coefficient βm exceed the weekly coefficient βw when VIX3M or VIX6M is
used in estimation. This is not the case when only P information is considered, Table 8 in appendix C
shows little difference in term of likelihood value when long-term components are added. In line with
previous literatures, the persistence parameter πQ is bigger than πP and close to 1, which indicates
7It is possible to construct a likelihood with all three volatility indices simultaneously, however, preliminary resultsshow that the improvement can only be found when compared with (R,RV,VIX) case. One possible explanation is thatthe variance premium parameter v1 might be different for different terms. Therefore, we estimate parameters with each ofthe three indices one at a time.
8See Table 1 for the standard deviation of Ann.RK and VIXs.
11
the high persistence in risk neutral dynamic. The risk premium parameter v1 is negative in all cases.
As shown in Corsi et al. (2013), a negative v1 is consistent with existing literatures on variance risk
premium such as Bakshi and Kapadia (2003), Carr and Wu (2009) etc. The absolute value of variance
premium parameter decreases while maturities increases while the equity risk premium λ remains in a
stable level. This fact suggest that equity premium and variance risk premium are quite different and
providing additional motivation of separating v1 from λ.
For the LHARG-Q model in column (5) - (8), the quarterly coefficient βq is positive and signifi-
cant across different estimation methods. The likelihood value of `R,RV changes slightly while large
improvements are found in `R,RV,VIX and `R,RV,Fut. This indicates the importance of the newly added
components. Similar results can also be found for LHARG-QY model in column (9) - (12).
[Insert Table 3 here]
Table 3 reports the results for HNG. Since there is no room for an independent variance risk pre-
mium within a single shock in the HNG models, the equity risk premium λ fluctuates violently across
different methods. Using risk neutral information, the model provides larger equity premium parameter
λ compared with its physical counterpart (see “Ret” method). This suggests that the estimated equity
risk premium has to be inflated to fit the Q information.
4.2 Volatility Indices and VIX Futures Pricing
Table 4 reports the volatility indices and VIX futures pricing performance of different models and
methods. Panels A through C are associated with VIX, VXV and VXMT respectively while panel D
summarizes the performance for VIX futures pricing.
[Insert Table 4 here]
For VIX and VXV, all the LHARG models outperform the HNG in terms of RMSE. When pricing
VXMT, HNG model outperforms LHARG-M model for all estimation methods. It is not the case for our
augmented LHARG models. Especially, the LHARG-QY always provides smaller RMSE than other two
LHARG model and dominates the HNG model. For VIX futures pricing, we still find that LHARG-QY
is the best model while HNG dominates LHARG-M and delivers similar results as LHARG-Q. Those
findings indicate: 1)It is important to include realized measures when volatility index or VIX futures
are priced. 2)The model structure is also important, a model without explicit long-term information
(i.e. yearly component) can be outperformed by a model without realized measures.
[Insert Table 5 here]
In Table 5, total RMSE for VIX futures pricing is decomposed by the VIX level and the time to
maturity when the price is quoted. The first dimension is linked to the model’s ability to generate
appropriate variance risk premium. The second dimension is linked to the model’s ability to track
volatility dynamics at different time horizon. The models here are estimated by index return, realized
12
kernel (for LHARG models) and VIX futures prices jointly. The total RMSE shows that LHARG-
QY has the smallest pricing error. The performance of the LHARG-M model and HNG model are
mixed: the LHARG-M model performs better on short-term VIX futures while HNG does better in
long-term ones. A possible explanation is that the high-frequency data, while providing useful realtime
information on the latent volatility, is overacting to short term shocks in the view of pricing long-term
VIX futures. Taking average of realized measures over a much longer time span will significantly reduce
the sensitivity of the model in reacting to temporary shocks.
[Insert Figure 2 here]
Figure 2 presents the relationship between VIX futures pricing error (RMSE) and maturity in
different volatility interval for HNG and LHARG-QY model. The models here are also estimated by
index return, realized kernel (for LHARG models) and VIX futures prices jointly. The results show that
LHARG-QY model outperforms HNG at almost all aspects, especially for high volatility cases.
4.3 Out-of-sample Pricing
The superior performance of LHARG-QY combined with the fact that LHARG-M and LHARG-Q is
nested in it calls for out-of-sample pricing check to ensure no over-fitting is involved. The out-of-sample
performance evaluation based on a rolling window of 252 trading days, with the parameters updated on
the monthly basis. We evaluate the out-of-sample pricing errors from 200902-201703 (the observations
in 2008 are used as pre-sample to get the first parameter for the out-of-sample analysis) with 2018
volatility indices and 11909 VIX futures prices.
[Insert Table 6 here]
Table 6 provides an out-of-sample version of Table 4. For short-term volatility index such as VIX
and VXV, all LHARG model still outperforms HNG regardless of the estimation method. For long-term
volatility index, the HNG only performance slightly better than LHARG-QY (the reduction of RMSE
is less than 1%) when parameter are estimated with VIX1M and is dominated by LHARG-QY for the
rest cases. For VIX futures prices, LHARG-QY model delivers the smallest pricing errors in most of
the cases. Results for LHARG-M and LHARG-Q are mixed with HNG. An decomposed version of
out-of-sample VIX futures pricing errors can be found in Table 7. Results suggesting that LHARG
models out preforms HNG model and in most cases LHARG-QY has the smallest RMSE. In short, we
do not find strong empirical evidence suggesting over-fitting for LHARG-QY model.
[Insert Table 7 here]
13
5 CONCLUSION
In this paper, we augmented the LHARG model proposed in Majewskia et al. (2015) by adding quarterly
and yearly components. The analytical pricing formulas for both the CBOE VIX term structure and
VIX futures with different maturity are derived based on the extended model. Several estimation
methods with volatility over variety terms as well as VIX future quotes are applied to an sample
spanning nine years and our empirical results suggest that the quarterly and yearly components of
lagged realized volatility should be added into the model to capture long-term volatility dynamics.
With the realized volatility based on high frequency data, the proposed model provide superior pricing
performance compared to the classic Heston-Nandi GARCH model, both in-sample and out-of-sample.
The improvement is more pronounced during high volatility periods. Although long-term components
seems irrelevant under the physical measure, they do play an important role in modeling volatility index
and VIX futures in addition to realized measures. Such findings indicating model VIX derivatives with
realized measures and long-run volatility components.
14
Appendix A: Proof of Proposition 1
Following the definition of Gamma distribution, we have:
EQt [RVt+1] = θ∗δ∗ + θ∗Θ∗(RVt,L∗t )
= θ∗δ∗ + d∗ +
p∑i=1
β∗iRVt+1−i +
q∑j=1
α∗j`∗t+1−j
Let µQ ≡ EQ(RVt) = (θ∗δ∗ + d∗)/(1−∑p
i=1 β∗i ), then
EQt [RVt+1]− µQ =
p∑i=1
β∗i (RVt+1−i − µQ) +
q∑j=1
α∗j`∗t+1−j
Suppose the formaula of EQt [RVt+1]− µQ is
EQt [RVt+k]− µQ =
p∑i=1
ξ(k)i (RVt+1−i − µQ) +
q∑j=1
ω(k)j `∗t+1−j
When k = 1, ξ(1)i = β∗i , ω
(1)j = α∗j .
When k 6= 1, we have
EQt [RVt+k+1]− µQ = EQt [EQt+1[RVt+k+1 − µQ]]
= EQt [
p∑i=1
ξ(k)i (RVt+2−i − µQ) +
q∑j=1
ω(k)j `∗t+2−j ]
= EQt [ξ(k)1 (RVt+1 − µQ) + ω
(k)1 `∗t+1] +
p∑i=2
ξ(k)i (RVt+2−i − µQ) +
q∑j=2
ω(k)j `∗t+2−j
=
p∑i=1
ξ(k)1 β∗i (RVt+1−i − µQ) +
q∑j=1
ξ(k)1 α∗j`
∗t+1−j +
p−1∑i=1
ξ(k)i+1(RVt+1−i − µQ) +
q−1∑j=1
ω(k)j+1`
∗t+1−j
=
p∑i=1
ξ(k+1)i (RVt+1−i − µQ) +
q∑j=1
ω(k+1)j `∗t+1−j
ξ(k+1)i =
ξ(k)i+1 + ξ
(k)1 β∗i 1 ≤ i < p
ξ(k)1 β∗i i = p
ωj =
ω(k)j+1 + ξ
(k)1 α∗j 1 ≤ j < q
ξ(k)1 α∗j j = q
15
So the Vt(n) can be expressed as
Vt(n) =1
n
n∑k=1
EQt [RVt+k]
=
p∑i=1
ξiRVt+1−i +
q∑j=1
ωj`∗t+1−j + µQ(1−
p∑i=1
ξi)
where ξi = 1n
∑nk=1 ξ
(k)i , ωj = 1
n
∑nk=1 ω
(k)j .
Appendix B: Proof of Proposition 2
Vt(n) = µQ(1−p∑i=1
ξi) +
p∑i=1
ξiRVt+1−i +
q∑j=1
ωj`∗t+1−j
= a∗ +
p∑i=1
b∗iRVt+1−i +
q∑j=1
c∗i `∗t+1−j
Where a∗ = µQ(1 −∑p
i=1 ξi), b∗i = ξi, c
∗j = ωj . Using the results in Corsi et al. (2013), the moment-
generating function of RVt+1 is
EQt [exp(ηRVt+1)] = exp(η
1− ηθ∗θ∗Θ∗(RVt,L
∗t )− δlog(1− ηθ∗))
Then we have
EQt [exp(zRVt+1 + s`∗t+1)] = EQt [exp(zRVt+1 − s− 2sγ∗ε∗t+1
√RVt+1 + sε∗2t+1)]
= EQt [exp(zRVt+1 − s)EQ[exp(−2sγ∗√RVt+1ε
∗t+1 + sε∗2t+1)|RVt+1]
= EQt [exp(zRVt+1 − s−1
2log(1− 2s) +
2s2γ∗2RVt+1
1− 2s)]
= exp(−1
2log(1− 2s)− s− δlog(1− χθ∗) +
χ
1− χθ∗θ∗Θ∗(RVt,L
∗t ))
= exp(A(z, s) +
p∑i=1
Bi(z, s)RVt+1−i +
q∑j=1
Cj(z, s)`∗t+1−j)
Where
A(z, s) = −1
2log(1− 2s)− s− δW(χ, θ∗) + d∗V(χ, θ∗)
Bi(z, s) = V(χ, θ∗)β∗i
Cj(z, s) = V(χ, θ∗)α∗j
16
V(χ, θ∗) =χ
1− θ∗χ, W(χ, θ∗) = log(1− θ∗χ), χ(z, s) = z +
2s2γ∗2
1− 2s
Combining the above two equation, EQt [exp(zVt+1(n))] is
EQt [exp(zVt+1(n))] = EQt [exp(z(a∗ +
p∑i=1
b∗iRVt+2−i +
q∑j=1
c∗j`∗t+2−j))]
= EQt [exp(zb∗1RVt+1 + zc∗1`∗t+1)]× exp(za∗ +
p−1∑i=1
zb∗i+1RVt+1−i +
q−1∑j=1
zc∗j+1`∗t+1−j)
= exp(A(zb∗1, zc∗1) +
p∑i=1
Bi(zb∗1, zc∗1)RVt+1−i +
q∑j=1
Cj(zb∗1, zc∗1)`∗t+1−j)
× exp(za∗ +
p−1∑i=1
zb∗i+1RVt+1−i +
q−1∑j=1
zc∗i+j`∗t+1−j)
= exp(Ω(1) +
p∑i=1
φ(1)i RVt+1−i +
q∑j=1
Γ(1)j `∗t+1−j)
Ω(1) = za∗ +A(zb∗1, zc∗1)
φ(1)i =
zb∗i+1 + Bi(zb∗1, zc∗1) 1 ≤ i < p
Bi(zb∗1, zc∗1) i = pΓ(1)j =
zc∗j+1 + Cj(zb∗1, zc∗1) 1 ≤ j < q
Cj(zb∗1, zc∗1) j = q
Assume EQt [exp(zVt+k(n))] adopts following foumula
EQt [exp(zVt+k(n))] = exp(Ω(k) +
p∑i=1
φ(k)i RVt+1−i +
q∑j=1
Γ(k)j `∗t+1−j)
When k = 1, it holds. When k > 1, we have
EQt [exp(zVt+k+1(n))] = EQt [EQt+1[exp(zVt+k+1(n))]]
= EQt [exp(Ω(k) +
p∑i=1
φ(k)i RVt+2−i +
q∑j=1
Γ(k)j `∗t+2−j)]
= EQt [exp(φ(k)1 RVt+1 + Γ
(k)1 `∗t+1)]× exp(Ω(k) +
p−1∑i=1
φ(k)i+1RVt+1−i +
q−1∑j=1
Γ(k)j+1`
∗t+1−j)
= exp(A(φ(k)1 ,Γ
(k)1 ) +
p∑i=1
Bi(φ(k)1 ,Γ(k)1 )RVt+1−i +
q∑j=1
Cj(φ(k)1 ,Γ(k)1 )`∗t+1−j)×
17
EQt [exp(zVt+k+1(n))] exp(Ω(k) +
p−1∑i=1
φ(k)i+1RVt+1−i +
q−1∑j=1
Γ(k)j+1`
∗t+1−j)
=exp(Ω(k+1) +
p∑i=1
φ(k+1)i RVt+1−i +
q∑j=1
Γ(k+1)j `∗t+1−j)
Ω(k+1) = Ω(k) +A(φ(k)1 ,Γ
(k)1 )
φ(k+1)i =
φ(k)i+1 + Bi(φ(k)1 ,Γ
(k)1 ) 1 ≤ i < p
Bi(φ(k)1 ,Γ(k)1 ) i = p
Γ(k+1)j =
Γ(k)j+1 + Cj(φ(k)1 ,Γ
(k)1 ) 1 ≤ j < q
Cj(φ(k)1 ,Γ(k)1 ) j = q
Appendix C: The physical parameter for LHARG models
In table 8, we report the parameter estimation results under P measure for LHARG models by maxi-
mizing `R,RV . For standard LHARG-M model, the coefficients show a decreasing impact of the lags on
the present value which is a sign of volatility cascade. Adding quarterly components increase likelihood
value slightly while the parameter βq is close to zero. For LHARG-QY model, the coefficient of yearly
component βy is positive and significant but does not improve the fitting of the underlying data in term
of the likelihood value.
[Insert Table 8 here]
18
References
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20
Table 1: Descriptive Statistics (2008-2017)
N Mean Std Skew Kurt Min Max
Panel A: S&P 500 Returns and VIX Term StructureReturn(×102) 2270 0.05 1.27 0.02 10.51 −9.47 10.96
Ann.RV 2270 16.27 11.84 3.00 13.60 2.96 108.65VIX1M 2270 20.69 9.85 2.34 7.01 10.32 80.86VIX3M 2270 22.27 8.54 1.99 4.72 12.24 69.24VIX6M 2270 23.58 7.50 1.68 3.16 14.04 61.47
Panel B: VIX FuturesTotall 13349 22.78 7.17 1.56 3.22 11.73 66.23
Partitioned by VIX LevelVIX<15 4244 16.95 1.95 0.41 1.30 11.73 26.60
15<VIX<20 4120 20.64 2.81 0.51 −0.29 14.55 30.8520<VIX<25 2301 25.08 3.05 0.21 −0.71 18.45 32.5525<VIX<30 1112 27.95 2.88 −0.18 −0.86 20.20 34.5530<VIX<35 523 31.45 2.58 −1.41 3.06 20.08 36.85
35<VIX 1049 39.95 6.92 0.54 3.91 17.80 66.23
Partitioned by VIX LevelDTM<30 2211 21.20 8.71 2.14 5.26 8.73 66.23
30<DTM<60 2195 22.06 7.65 1.76 3.46 12.70 57.6260<DTM<90 2352 22.79 7.20 1.63 3.13 13.45 59.7790<DTM<120 2323 23.21 6.60 1.34 1.82 14.25 52.26120<DTM<150 2163 23.60 6.20 1.13 0.99 14.85 47.07
150<DTM 2105 23.88 5.91 0.98 0.52 15.20 45.26
Kurt is excess kurtosis. Ann.RV is the annualized daily volatility calculated by√
252×Adj.RK × 100 andAdj.RK = RK × var(ret)/mean(RK). VIX1M, VIX3M, VIX6M are CBOE VIX, VXV and VXMT.
21
Tab
le2:
Joi
nt
Est
imat
ion
ofL
HA
RG
Mod
els
Mod
elL
HA
RG
-ML
HA
RG
-QL
HA
RG
-QY
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
Met
hod
VIX
1MV
IX3M
VIX
6MF
ut
VIX
1M
VIX
3M
VIX
6M
Fu
tV
IX1M
VIX
3M
VIX
6M
Fu
tλ
3.56
723.
5684
3.56
523.5
612
3.5
683
3.5
678
3.5
669
3.5
574
3.6
112
3.5
676
3.5
687
3.5
570
(2.6
881)
(1.8
593)
(1.6
070)
(0.9
184)
(5.1
150)
(1.5
429)
(2.4
606)
(0.5
519)
(0.2
536)
(2.4
540)
(6.1
262)
(1.8
552)
θ(10
−5)
3.3
511
3.47
523.
5034
3.5
084
3.2
758
3.3
464
3.4
001
3.4
550
3.2
935
3.3
617
3.4
237
3.4
874
(0.2
372)
(0.2
855)
(0.3
098)
(0.2
953)
(0.2
307)
(0.2
372)
(0.2
663)
(0.2
654)
(0.2
401)
(0.2
551)
(0.2
894)
(0.1
985)
δ1.
4971
1.61
201.
609
71.6
819
1.4
206
1.5
306
1.5
600
1.6
402
1.4
291
1.5
399
1.5
959
1.6
754
(0.0
525)
(0.0
508)
(0.0
545)
(0.0
501)
(0.0
537)
(0.0
500)
(0.0
496)
(0.0
562)
(0.0
573)
(0.0
517)
(0.0
511)
(0.0
545)
d(1
0−5)
-3.5
562
-4.1
392
-4.2
960
-4.2
669
-3.3
637
-3.7
428
-3.8
525
-3.9
100
-3.5
022
-3.9
739
-4.1
690
-4.2
499
(0.2
662)
(0.3
354)
(0.3
657)
(0.3
982)
(0.2
615)
(0.2
644)
(0.2
981)
(0.3
178)
(0.2
214)
(0.2
989)
(0.3
576)
(0.2
292)
βd
0.45
670.
4738
0.470
20.4
790
0.4
365
0.4
217
0.4
322
0.4
561
0.4
347
0.4
100
0.4
202
0.4
428
(0.0
270)
(0.0
348)
(0.0
380)
(0.0
354)
(0.0
280)
(0.0
289)
(0.0
333)
(0.0
332)
(0.0
300)
(0.0
315)
(0.0
351)
(0.0
324)
βw
0.29
120.
1886
0.17
76
0.1
726
0.3
440
0.3
405
0.3
317
0.2
995
0.3
368
0.3
301
0.3
129
0.2
943
(0.0
318)
(0.0
450)
(0.0
476)
(0.0
544)
(0.0
310)
(0.0
376)
(0.0
399)
(0.0
320)
(0.0
280)
(0.0
385)
(0.0
423)
(0.0
370)
βm
0.16
040.
2454
0.26
73
0.2
457
0.0
877
0.0
609
0.0
243
0.0
256
0.0
906
0.0
898
0.0
778
0.0
616
(0.0
140)
(0.0
193)
(0.0
202)
(0.0
161)
(0.0
118)
(0.0
134)
(0.0
129)
(0.0
038)
(0.0
105)
(0.0
143)
(0.0
237)
(0.0
112)
βq
0.0
501
0.0
889
0.1
189
0.1
072
0.0
362
0.0
645
0.0
752
0.0
641
(0.0
041)
(0.0
084)
(0.0
111)
(0.0
052)
(0.0
037)
(0.0
066)
(0.0
080)
(0.0
025)
βy
0.0
248
0.0
282
0.0
306
0.0
355
(0.0
020)
(0.0
018)
(0.0
028)
(0.0
012)
αd(1
0−6)
2.51
291.
8996
1.55
361.8
688
2.5
470
1.6
505
1.3
982
1.6
018
4.0
996
3.2
609
3.0
711
3.4
598
(0.3
205)
(0.4
442)
(0.4
748)
(0.6
142)
(0.3
059)
(0.3
647)
(0.3
649)
(0.2
723)
(0.4
551)
(0.3
661)
(0.5
075)
(0.4
344)
γ32
3.1
311.
031
6.8
305.8
336.5
408.4
435.8
411.5
239.5
241.8
230.5
219.8
(32.
0)(3
5.3)
(44.
7)
(51.4
)(3
6.8
)(6
8.6
)(8
8.3
)(6
4.4
)(2
0.8
)(2
7.5
)(3
0.5
)(2
9.3
)υ1
-115
1.2
-903
.7-7
46.6
-919.1
-1150.8
-994.7
-926.9
-1084.8
-1078.8
-1009.3
-942.4
-1120.7
(486
.0)
(419
.5)
(563
.7)
(365.5
)(5
63.2
)(4
10.0
)(4
73.5
)(5
21.5
)(1
61.1
)(3
85.4
)(5
61.9
)(2
08.1
)πP
0.90
830.
9078
0.91
510.8
972
0.9
183
0.9
120
0.9
071
0.8
884
0.9
232
0.9
226
0.9
166
0.8
983
πQ
0.98
870.
9719
0.96
830.9
621
0.9
980
0.9
808
0.9
714
0.9
637
0.9
998
0.9
943
0.9
839
0.9
784
logL
` R,R
V26
909
2690
426
897
26904
26897
26905
26898
26904
26904
26915
26917
26922
` R,R
V,V
IX21374
2120
221
070
21195
21621
21420
21128
21118
21813
21666
21513
21444
` R,R
V,V
XV
2056
720971
2090
220950
21034
21445
21364
21326
21494
21790
21722
21540
` R,R
V,V
XM
T19
613
2063
420757
20635
20031
21074
21208
21054
20752
21473
21605
21367
` R,R
V,Fut
1968
720
426
2034
720458
19985
20733
20722
20852
20785
21088
21106
21313
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22
Table 3: Joint Estimation of Heston-Nandi GARCHMethod R R+VIX1M R+VIX3M R+VIX6M R+Fut
λ 4.0982 6.9808 8.2016 9.0196 6.1047(2.8937) (0.6112) (0.5582) (0.5896) (0.2907)
ω 9.36E-14 9.36E-14 9.36E-14 9.36E-14 9.36E-14(6.45E-14) (6.77E-15) (2.02E-14) (7.04E-15) (6.30E-15)
β 0.7353 0.6970 0.6672 0.6654 0.7686(0.0289) (0.0190) (0.0197) (0.0190) (0.0242)
α 5.56E-06 3.28E-06 2.06E-06 1.60E-06 1.36E-06(9.69E-07) (2.12E-07) (5.90E-08) (4.47E-08) (6.17E-08)
γ 199.5 293.2 391.7 446.9 403.5(24.7) (11.3) (12.2) (13.8) (12.9)
πP 0.9565 0.9786 0.9828 0.9843 0.9907πQ 0.9656 0.9921 0.9961 0.9973 0.9975logL`R 7339.7 7256.7 7228.4 7217.2 7186.1
`R,VIX -198.5 445.6 372.7 268.8 151.4`R,VXV -331.8 721.4 891.9 829.9 722.2`R,VXMT -431.2 615.6 1036.9 1138.3 1076.9`R,Fut -377.1 570.6 974.8 1091.3 1123.8
Note. Robust standard errors are in parenthesis. The first row indicates the information set used. For
simplicity, we name each column with the information set used for estimation. e.g. “R+Fut” means the
parameters are estimated with index returns and VIX futures prices. The bold number in Log-likelihood
indicates the largest within each row.
23
Table 4: RMSE for VIX, VXV, VXMT and VIX Futures Pricing (Full Sample)Method LHARG-M LHARG-Q LHARG-QY HNG
Panel A: VIX PricingR+RV+VIX1M 2.7710 2.4724 2.2785 4.8624R+RV+VIX3M 2.9823 2.7114 2.4430 4.9589R+RV+VIX6M 3.1520 3.0740 2.6161 5.1656
R+RV+Fut 2.9919 3.0961 2.7025 5.3659
Panel B: VIX3M PricingR+RV+VIX1M 3.9534 3.2023 2.6223 4.3062R+RV+VIX3M 3.3025 2.6813 2.3138 3.9451R+RV+VIX6M 3.3941 2.7703 2.3861 4.0345
R+RV+Fut 3.3326 2.8255 2.5909 4.1729
Panel C: VIX6M PricingR+RV+VIX1M 6.0178 4.9808 3.6371 4.5116R+RV+VIX3M 3.8298 3.1568 2.6602 3.7010R+RV+VIX6M 3.6178 2.9680 2.5125 3.5219
R+RV+Fut 3.8300 3.1848 2.7953 3.5692
Panel D: VIX Futures PricingR+RV+VIX1M 5.8249 5.0837 3.5838 4.6018R+RV+VIX3M 4.1980 3.6685 3.1518 3.8036R+RV+VIX6M 4.3358 3.6756 3.1303 3.5956
R+RV+Fut 4.1401 3.4804 2.8626 3.4963
Note. The bold number indicates the minimum RMSE within each row. For HNG model, the information
sets used don’t include RV. i.e. “R+RV+VIX1M” for HNG is actually “R+VIX1M”.
24
Table 5: RMSE for VIX Futures Pricing for Different Models (Full Sample)Model LHARG-M LHARG-Q LHARG-QY HNG
Total RMSE 4.1401 3.4804 2.8626 3.4963
Panel A.1: Partitioned by VIX LevelVIX<15 3.3299 2.9047 2.2097 2.6055
15<VIX<20 2.7629 2.4430 1.7648 2.466220<VIX<25 4.2264 3.9202 3.1264 2.865225<VIX<30 4.9867 4.3958 3.7861 3.466930<VIX<35 5.2612 4.3301 5.1030 6.5073
35<VIX 7.9961 5.7469 4.7619 7.2350Panel B.1: Partitioned by Maturity
DTM<30 3.9670 3.3440 2.9286 4.457530<DTM<60 4.1294 3.3304 2.6991 3.446660<DTM<90 4.0155 3.3131 2.6737 3.259890<DTM<120 4.0893 3.4153 2.7587 3.1064120<DTM<150 4.2803 3.6575 2.9665 3.2046
150<DTM 4.3695 3.8238 3.1518 3.3448
Note. This table only report the pricing performance for the “Fut” method (i.e. “R+Fut”for HNG and
“R+RV+Fut” for LHARG models.).
25
Table 6: RMSE for VIX, VXV, VXMT and VIX Futures Pricing (Out of Sample)Method LHARG-M LHARG-Q LHARG-QY HNG
Panel A: VIX PricingR+RV+VIX1M 2.1155 2.0387 1.9123 3.7222R+RV+VIX3M 2.2061 2.1564 2.0796 3.6813R+RV+VIX6M 2.5045 2.4524 2.3391 3.7831
R+RV+Fut 2.2399 2.4869 2.3355 4.0999
Panel B: VIX3M PricingR+RV+VIX1M 3.9659 3.5043 2.5741 3.9354R+RV+VIX3M 2.2956 2.0102 1.7458 3.2533R+RV+VIX6M 2.2685 2.0984 1.9134 3.1087
R+RV+Fut 2.5128 2.4070 2.2319 3.7068
Panel C: VIX6M PricingR+RV+VIX1M 7.8385 7.6182 4.6674 4.6516R+RV+VIX3M 3.6808 3.3154 3.0392 3.4884R+RV+VIX6M 2.5077 2.1154 1.7351 2.8519
R+RV+Fut 3.0533 2.6502 2.2525 3.6314
Panel D: VIX Futures PricingR+RV+VIX1M 8.1706 8.2251 4.3900 4.8872R+RV+VIX3M 4.0821 3.9573 3.8282 3.7099R+RV+VIX6M 3.3963 3.1214 2.8883 2.9189
R+RV+Fut 3.0395 2.6590 2.2233 3.2986
Note. The bold number in each panel indicates the minimum RMSE within each row. The out-of-sample
performance evaluation based on a rolling window of 252 trading days, with the parameters updated on the
monthly basis. We evaluate the out-of-sample pricing errors of 2018 trading days (i.e. 200902-201703, the
observations in 2008 are used as pre-sample to get the first parameter for the out-of-sample analysis.) with
2018 VIX,VXV,VXMT prices and 11909 VIX futures prices. For HNG model, the information sets used don’t
include RV. i.e. “R+RV+VIX1M” for HNG is actually “R+VIX1M”.
26
Table 7: RMSE for VIX Futures Pricing for Different Models (Out of Sample)Model LHARG-M LHARG-Q LHARG-QY HNG
Total RMSE 3.0395 2.6590 2.2233 3.2986
Panel A.1: Partitioned by VIX LevelVIX<15 2.0620 1.9154 1.2827 1.9238
15<VIX<20 2.7539 2.3763 1.7949 2.771520<VIX<25 3.1372 2.6007 2.4320 3.879025<VIX<30 3.3684 3.1611 3.3512 4.955730<VIX<35 4.1812 3.5364 4.1947 5.1390
35<VIX 6.8162 5.9028 4.2837 6.3484Panel B.1: Partitioned by Maturity
DTM<30 2.8977 2.8613 2.7989 3.629830<DTM<60 2.7286 2.5147 2.3107 3.383060<DTM<90 2.7289 2.3336 2.0042 3.258490<DTM<120 2.9938 2.4382 1.9081 3.1463120<DTM<150 3.2919 2.7177 1.9712 3.1393
150<DTM 3.5573 3.0648 2.2422 3.2087
Note. This table only report the out of sample pricing performance for the “Fut” method (i.e. “R+Fut”for
HNG and “R+RV+Fut” for LHARG models.).
27
Table 8: Parameter Estimation of LHARG Models in P MeasureModel LHARG-M LHARG-Q LHARG-QYλ 3.5692 3.5687 3.5680
(3.0503) (1.7490) (1.5256)θ(10−5) 3.4986 3.4917 3.5415
(0.2669) (0.2695) (0.2945)δ 1.6695 1.6677 1.6840
(0.0489) (0.0489) (0.0493)d(10−5) -3.9760 -3.9700 -4.3970
(0.3056) (0.3047) (0.3643)βd 0.4482 0.4475 0.4309
(0.0401) (0.0389) (0.0421)βw 0.3174 0.3178 0.3180
(0.0553) (0.0579) (0.0572)βm 0.1175 0.1110 0.1178
(0.0326) (0.0356) (0.0346)βq 0.0074 -0.0074
(0.0043) (0.0039)βy 0.0412
(0.0102)αd(10−6) 1.5735 1.5663 3.5223
(0.4526) (0.4803) (0.7352)γ 400.9 403.1 197.4
(77.7) (85.4) (40.2)πP 0.8832 0.8838 0.9006`R,RV 26923.1 26923.2 26932.1
Note. Robust standard errors are in parenthesis.
28
Figure 1: Time Series Data for Ann.RV, VIX, VXV and VXMT
Note: Reported are S&P500 annualized realized volatility, and time series data of VIX, VXV and VXMT.
29
Figure 2: VIX Futures Pricing under Different Volatility Interval (Full Sample)
Note: This graph only reports the pricing performance for the “Fut” method (i.e.“Ret+Fut” and
“Ret+RV+Fut” for HNG and LHARG models respectively). The number X in horizontal axis denotes the
maturity within (30(X-1),30X].
30