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: zhuohuang@nsd.edu.cn. 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:tongchen@pku.edu.cn.‡Corresponding author, Department of Financial Engineering, School of Banking and Finance, University of Interna-

tional Business and Economics, Beijing 100029, P.R. China, Email: tianyiwang@uibe.edu.cn. 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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Corsi, F., N. Fusari, and D. L. Vecchia (2013). Realizing smiles: Options pricing with realized volatility.

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Hao, J. and J. E. Zhang (2013). Garch option pricing models, the cboe vix, and variance risk premium.

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valuation. Journal of Banking and Finance 43, 200–211.

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19

Luo, X. and J. Zhang (2014). Instantaneous squared vix and vix derivatives. Working Paper, Zhejiang

University and University of Otago..

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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

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221

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21195

21621

21420

21128

21118

21813

21666

21513

21444

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21034

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21364

21326

21494

21790

21722

21540

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20031

21074

21208

21054

20752

21473

21605

21367

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426

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19985

20733

20722

20852

20785

21088

21106

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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