Received: 2 July 2019
Revised: 18 October 2019
Accepted: 18 June 2020
DOI: 10.1002/ijfe.1922
RESEARCH ARTICLE
Implied and realized volatility: A study of distributions and
the distribution of difference
M. Dashti Moghaddam
| Jiong Liu |
Department of Physics, University of
Cincinnati, Cincinnati, Ohio
R. A. Serota
Abstract
We study distributions of realized variance (squared realized volatility) and
Correspondence
R. A. Serota, Department of Physics,
University of Cincinnati, Cincinnati, OH
45221-0011.
Email: serota@ucmail.uc.edu
squared implied volatility, as represented by VIX and VXO indices. We find
that generalized beta distribution provide the best fits. These fits are much
more accurate for realized variance than for squared VIX and VXO—possibly
another indicator that the latter have deficiencies in predicting the former. We
also show that there are noticeable differences between the distributions of the
1970–2017 realized variance and its 1990–2017 portion, for which VIX and
VXO became available. This may be indicative of a feedback effect that implied
volatility has on realized volatility. We also discuss the distribution of the difference between squared implied volatility and realized variance and show
that, at the basic level, it is consistent with Pearson's correlations obtained
from linear regression.
KEYWORDS
Beta prime distribution, implied/realized volatility, inverse gamma distribution, stable
distribution, VIX/VXO
1 | INTRODUCTION
CBOE introduced its volatility index VIX (presently
VXO) in 1993 and reintroduced it in 2003 (presently
VIX) (Carr & Wu, 2006). Both indices are published on
CBOE site from 1990 to present day (CBOE, 2020). The
main purpose of volatility indices is to estimate future
realized volatility (RV). The original VIX was based on
the inverted Black–Scholes formula (Whaley, 1993),
which assumes that volatility does not have a stochastic
component (that is, it is constant or its time dependence is continuous). By then, however, it was already
realized that volatility is stochastic in nature, and several models of stochastic volatility emerged
(Heston, 1993; Nelson, 1990) prompting the need for an
index that would be agnostic to specific assumptions
about stochastic volatility (Bollerslev, Mathew, &
Zhou, 2011; Zhou & Chesnes, 2003). This need was
answered by the new VIX (CBOE, 2003). It was based
Int J Fin Econ. 2021;26:2581–2594.
on research in (Demeterfi, Derman, Kamal, & Zou, 1999),
where a closed-form formula for the expected value of RV
(Barndorff-Nielsen & Shephard, 2002) was derived using
call and put prices.
The original VIX used S&P 100 near-term, at-themoney options to calculate a weighted average of volatilities. The new VIX uses a far more representative S&P
500 index, both near-term and next-term options, and a
range of strike prices, which is broader than the original
(CBOE, 2003). Both indices are published daily and
intend to measure expectations for volatility over the next
30-day timeframe and are annualized to 365 days
(CBOE, 2003). Of note, in this regard, is that RV—which
is based on daily changes of stock prices, calculated from
the closing prices on consecutive trading days—is traditionally annualized to 252 (Kurella, 2013; RealVol, 2017;
Shu & Zhang, 2003), the number of trading days in a
year; on a monthly basis, the latter corresponds to
roughly 21–22 days (Degiannakis, 2018).
wileyonlinelibrary.com/journal/ijfe
© 2020 John Wiley & Sons, Ltd.
2581
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MOGHADDAM ET AL.
The question of how well volatility indices predict
future RV remains of great interest to researchers
(Chrstensen & Prabhala, 1998; Kownatzki, 2016;
Russon & Vakil, 2017; Vodenska & Chambers, 2013).
While most previous research concentrated on regression
analysis, we compare distributions of implied volatility
indices and RV. In our previous article (Dashti
Moghaddam, Liu, & Serota, 2019), we visually compared
the probability density function of realized variance
(RV2)—squared RV—and squared implied volatility, as
represented by VIX and VXO.1 We also studied the distribution of the ratio of RV2 to VIX2 and to VXO2, which
provided additional insights relative to qualitative comparison and simple regression analysis. Here, we specifically address the form of these distributions. We also
investigate the distributions of VIX2 − RV2 and
VXO2 − RV2 due to recent interest in looking at the time
series of VIX − RV—see Figure 1—which is equivalent to
the one shown in the Wall Street Journal (Sindreu &
Bird, 2018).
2
Realized variance (index) is defined as follows:
RV 2 = 1002 ×
r i = ln
(VIX - RV)
1
0.5
0
-0.5
-1
2016
2017
2018
F I G U R E 1 VIX − RV, from January 1, 2014, to December
29, 2017 [Colour figure can be viewed at wileyonlinelibrary.com]
10 4
4
n=1
n=2
n=3
n=4
PDF
PDF
n=1
n=7
n = 14
n = 21
3
2
1
2
1
0
1
2
RV 2
3
4
10 -4
n
P
PDFs of n1 r 2i
i=1
for n = 1,2,3,4 (left) and
n = 1,7,14,21 (right) [Colour
figure can be viewed at
wileyonlinelibrary.com]
FIGURE 2
0
0
ð2Þ
10 4
4
3
Si
Si − 1
are daily returns and Si is the reference (closing) price on
day i. This is an annualized value, where 252 represents
the number of trading days. Specifically for monthly
returns, n ≈ 21, 365/252 ≈ 30/21 ≈ 1.4. Since VIX and
VXO are evaluated daily to forecast RV for the following
month and are annualized to 365, to properly compare
the distributions of RV2 to VIX2 and to VXO2, one should
rescale the distribution of RV2 with the ratio of the mean
of VIX2 and VXO2 to that of RV2 (Dashti Moghaddam
et al., 2019), which is usually close to 1.4.2
Since RV2 is based on the sum of realized daily variances,
the obvious questions for understanding its distribution are
what is the distribution of daily variances and what are the
correlations between them? A study of intraday returns, interpreted in terms of intraday jumps, (Behfar, 2016) points to fattailed / 1/xμ + 1 distributions with 1 < μ < 2. Here, our own
fitting of daily realized variance RV2 seems to correspond to
μ
similarly tailed distributions of returns, that is / 1=x 2 + 1 with
μ close to the values in (Behfar, 2016). However, none of
the distributions used here—all based on continuous
models of stochastic volatility—are a good fit to daily RV2.
This is not surprising since all of continuous models are best
suited for bell-shaped distributions. However, as is obvious
form Figure 2 it takes an addition of several days of daily
RV2 do develop the bell shape. Nonetheless, Generalized
Beta Prime distribution (see below) provides “the best of
the worst” fit to daily returns and is based on a non-meanreverting stochastic volatility model (Hertzler, 2003).
1.5
2015
ð1Þ
where
101
-1.5
2014
n
252 X
r2
n i=1 i
0
1
2
RV 2
3
4
10 -4
MOGHADDAM ET AL.
2583
differential processes. We use Maximum Likelihood Estimation (MLE) and obtain the list of fitting parameters using the
Kolmogorov–Smirnov (KS) values to compare goodness of
fits. We also examine the evolution of the power-law tail
exponents—including a direct comparison of the tails—and
KS values as a function of n in connection with Figures 2 and
3. In Section 3, we examine the distributions of differences of
VIX2 with scaled RV2 and VXO2 with scaled RV2 vis-a-vis simple correlation between the indices established by linear
regression. Finally, in the Appendix A we look at the distributions of RV, VIX and VXO and the corresponding difference
distributions, which is done because market observers and
researchers are more familiar with these indices.
We use 1970–2017 S&P 500 stock price data to calculate
RV and variance. Unless explicitly mentioned that we use
the 1990–2017 subset of the data, the full set is used below.
Had the realized variances been uncorrelated, the
monthly realized variance would have been expected, by the
generalized central limit theorem, to approach a stable distribution. However, Figure 3 indicates otherwise, where the initial fast power-law drop-off of correlations is followed by a
slow exponential decay with the time constant of about
120 days. Consequently, we are reduced to empirical fitting of
the distribution function of RV2 with heavy-tail distributions,
including stable probability distribution. We will concentrate
specifically on monthly returns—the timeframe tied to the
30-day forward volatility expectation from VIX and VXO.
Figure 2, however, shows that RV2 quickly approaches its limiting form at n ≈ 5–7—approximately the same number of
days over which power law yields to exponential in Figure 3.
This paper is organized as follows. In Section 2 we identify the list of distributions used for fitting of the probability
distribution functions (PDF) of RV2, VIX2 and VXO2 and discuss their role as steady-state distributions of stochastic
2 | P D F OF RV 2 , V I X 2 AND V X O 2
0.25
0.2
r2 Autocorrelation
As mentioned in Section 1, we do not have analytical predictions for the distribution functions, barring the expectation that they will express fat tails. For empirical fitting, we
use the distributions collected in Table 1: generalized beta
prime (GB2), beta prime (BP), generalized inverse gamma
(GIGa), inverse gamma (IGa), generalized gamma (GGa)
and Gamma (Ga). Here, p, q, α and γ are shape parameters
and β is a scale parameter. We also use stable distribution
(S) (Nolan, 2018), S(x; α, β, γ, δ), but it does not, in general,
reduce to a closed-form expression. For S, α and β are shape
parameters, γ is a scale parameter and δ is a location parameter. Two right columns in Table 1 show the power-law
exponents of the front end and of the tails respectively. GGa
and Ga are included as distributions with short tails. Notice
also that they are related to GIGa and IGa as distributions
of the inverse variable.
data
fitted curve
0.15
0.1
0.05
0
-0.05
0
100
200
300
400
500
Lags
F I G U R E 3 Autocorrelation function of daily realized variance
(dots) and the best fit with c × xb − 1 × exp(−a * x), a = 0.0088,
b = 0.73, c = 0.18 [Colour figure can be viewed at
wileyonlinelibrary.com]
T A B L E 1 Analytic form of
probability density functions for fitting
RV2, VIX2 and VXO2
Type
PDF
Front exponent
S(x;α, β, γ, δ)
GB2(x;p, q, α, β)
BP(x;p, q, β)
Tail exponent
− (α + 1)
−p −q
x α
x − 1 + pα
β
β
αp − 1
− (αq + 1)
−p− q x −1 + p
β
p−1
− (q + 1)
αð1 + ð
ð1 + βxÞ
ÞÞ
ðÞ
β Bðp,qÞ
ðÞ
β Bðp,qÞ
GIGa(x;α, β, γ)
γe
β γ
− x
− (αγ + 1)
ð Þ ðβÞ1 + αγ
x
βΓðαÞ
IGa(x;α, β)
GGa(x;α, β, γ)
β
− (α + 1)
1+α
e − x ðβx Þ
βΓðαÞ
γe
γ
− x
β
ð Þ ðxÞ −1 + αγ
β
αγ − 1
βΓðαÞ
Ga(x;α, β)
e
− x x −1 + α
β
β
ðÞ
α−1
βΓðαÞ
Note: * Bðp,qÞ = ΓΓððppÞΓ+ðqqÞÞ: beta function; Γ(α): gamma function.
2584
MOGHADDAM ET AL.
It should be pointed out that all of these distributions
are steady-state distributions of stochastic processes used
to describe stochastic volatility. In particular, Ga, IGa
and BP are the steady-state distributions of the meanreverting Heston (Dragulescu & Yakovenko, 2002;
Heston, 1993), multiplicative (Bouchaud & Mézard, 2000;
Nelson, 1990), and combined Heston-multiplicative
(Dashti Moghaddam & Serota, n.d.) models respectively.
GIGa (Ma, Holden, & Serota, 2013; Ma & Serota, 2014),
GGa and GB2 (Hertzler, 2003) are the steady states of
non-mean-reverting stochastic processes. Namely, consider a stochastic differential equation.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dx = −η x −θx 1− α dt + κ 22 x 2 + κ 2α x 2 − α dW t
p=
1
2ηθ
−1 + α + 2
α
κα
ð5Þ
1
2η
1+ 2
α
κ2
ð6Þ
and
q=
The steady-state distribution of (3) is GIGa for κα = 0
and GGa for κ 2 = 0. For α = 1 we have mean-reverting
models which yield a BP steady-state distribution in general and IGa and Ga for κ 1 = 0 and κ 2 = 0, respectively.
ð3Þ
2.1 | Monthly data
where dWt is a Wiener process. Its steady-state distribution is (Hertzler, 2003) GB2(x; p, q, α, β) in Table 1 with
β=
2=α
κα
κ2
ð4Þ
10-3
3.5
data
GGa
S
BP
GIGa
GB2
2.5
2.5
PDF
1.5
2
1.5
1
1
0.5
0.5
0
500
1000
1500
2000
data
GGa
S
BP
GIGa
GB2
3
2
0
10-3
3.5
3
PDF
Fits of monthly data with the distributions from Table 1
are shown in Figure 4. Parameters of the distribution fits
in Figure 4 and their KS statistics are shown in
Tables 2-5. Smaller KS numbers correspond to better
0
2500
0
500
1000
RV2
10
3.5
-3
data
GGa
S
BP
GIGa
GB2
1.5
1.5
1
0.5
0.5
0
1500
2
VIX
2000
data
GGa
S
BP
GIGa
GB2
2
1
1000
2500
-3
2.5
2
500
2000
3
PDF
PDF
2.5
0
10
3.5
3
0
1500
RV2
2500
0
500
1000
1500
2000
2500
VXO2
Clockwise: PDF of monthly RV2 from January 2, 1970, to December 29, 2017, and PDFs of monthly RV2, VIX2 and VXO2
from January 31, 1990, to December 29, 2017 [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 4
MOGHADDAM ET AL.
2585
fits. For RV2, GB2, BP and GIGa fits are at or close to a
95% confidence level (Knuth, 1998). Obviously, GGa
and Ga fit much worse than any of the fat-tailed distributions. Notice also that the power-law exponents of the
front end of GB2 and BP are very large, indicating that
MLE results for RV2
from January 2, 1970, to December
29, 2017
TABLE 2
MLE results for RV2
from January 31, 1990, to December
29, 2017
TABLE 3
MLE results for VIX2
from January 31, 1990, to December
29, 2017
TABLE 4
MLE results for VXO2
from January 31, 1990, to December
29, 2017
TABLE 5
the front end is highly suppressed. This explains why
GIGa provide nearly as good a fit as GB2. Interestingly,
the fits of VIX2 and VXO2 are not nearly as precise as
RV2, which confirms that VIX and VXO are not a very
good gauge for predicting RV.
Type
Parameters
Front exp
Tail exp
KS test
Stable
S(0.9686, 1.0000, 84.0679, 175.8546)
−1.9686
0.0202
GB2
GB2(15.9183, 1.8735, 1.0150, 23.7045)
15.1570
−2.9016
0.0115
BP
BP(17.2160, 1.9116, 21.9595)
16.2160
−2.9116
0.0116
GIGa
IGa
GIGa(2.5562, 625.4491, 0.8023)
−3.0508
0.0138
IGa(1.7394, 319.7392)
−2.7394
0.0203
GGa
GGa(5.0882, 11.1902, 0.4812)
1.4484
0.0786
Ga
Ga(1.1391, 364.0363)
0.1391
0.1330
Type
Parameters
Front exp
Stable
S(0.9033, 1.0000, 91.7350, 168.7239)
GB2
GB2(14.1895, 3.1115, 0.6613, 15.6843)
BP
BP(16.2164, 1.8349, 16.19619)
GIGa
IGa
GGa
GGa(4.2662, 17.0561, 0.4900)
1.0904
0.0652
Ga
Ga(1.0295, 436.8326)
0.0295
0.1163
Type
Parameters
Front exp
Stable
S(0.9548, 1.0000, 92.1996, 234.4670)
GB2
GB2(63.3797, 1.3249, 1.4751, 18.1068)
BP
BP(44.1482, 2.6245, 16.1142)
GIGa
Tail exp
KS test
−1.9033
0.0289
8.3835
−3.0576
0.0134
15.2164
−2.8349
0.0137
GIGa(3.8505, 2,195.2527, 0.5631)
−3.1682
0.0140
IGa(1.4149, 245.5728)
−2.4149
0.0296
Tail exp
KS test
−1.9548
0.0486
92.5294
−2.9544
0.0363
43.1482
−3.6245
0.0407
GIGa(1.4520, 325.9344, 1.3814)
−3.0058
0.0375
IGa
IGa(2.5156, 667.9832)
−3.5156
0.0402
GGa
GGa(6.8529, 3.1634, 1.0607)
6.2689
0.0693
Ga
Ga(1.8988, 230.0093)
0.8988
0.0882
Type
Parameters
Front exp
Stable
S(0.9554, 1.0000, 104.3782, 232.7193)
GB2
GB2(58.4930, 2.6432, 0.8839, 8.1216)
BP
BP(44.1507, 2.1309, 12.5195)
Tail exp
KS test
−1.9554
0.0564
50.7020
−3.3363
0.0392
43.1507
−3.1309
0.0401
GIGa
GIGa(3.0721, 1,092.4445, 0.7954)
−3.4435
0.0423
IGa
IGa(2.0448, 519.0907)
−3.0048
0.0483
GGa
GGa(5.599, 16.0437, 0.5327)
1.9826
0.0713
Ga
Ga(1.6328, 283.4215)
0.6328
0.0922
2586
MOGHADDAM ET AL.
An obvious qualitative difference between RV2
1970–2017 and RV2 1990–2017 in Figure 4 is that the
latter is “choppier,” which is reflected by its consistently higher KS numbers (poorer fit). In this vein,
VIX2 and VXO2 are far choppier still, which explains
their already mentioned far poorer fits by continuous
distributions. We also observe that for GB2 fitting the
front end (low volatilities) of the 1970–2017 RV2 is
suppressed relative to 1990–2017. The tail exponents,
on the other hand, are much closer to each other,
which points to that low volatility may have increased
at the expense of mid-volatility—possibly a feedback
effect on RV from the introduction of VIX (this needs
to be further studied by direct fitting of front ends,
similarly to the tails below). The front ends of VIX and
VXO are greatly suppressed relative to RV2, while tails
are more similar, indicating that volatility indices
0.5
GGa
S
BP
GIGa
GB2
IGa
Ga
KS statistic
0.4
0.3
0.2
0.1
0
0
5
10
15
20
25
underestimate low volatility and overestimate mid-volatility—compare with Figure 9 and results in (Dashti
Moghaddam et al., 2019). The latter may be important
for volatility traders.
2.2 | Development of RV2 distribution as
function of number of days
Figure 2 shows that the distribution function of RV2
develops rapidly with the number of days n at about
n ≈ 5–7. Here, we take a more careful look at how the
parameters of the distribution fits depend on n. Figure 5
gives the n-dependence of KS statistics, which compares
the goodness of fits. Figure 6 shows the n-dependence of
power-law exponents. Figure 7 compares tails of fitted
distributions to the actual tail and its fit. The important
observations are as follows:
Figure 5 Gap between GIGa/IGa and GB2/BP KS
decreases with n, as front exponents of the latter grows
Figure 6 Front exponents are negative for GB2/BP
and GGa for daily RV2, reflecting the absence of bell
shape. Unlike GIGa/IGa, for S and GB2/BP tail exponents saturate rapidly from smaller (fatter) daily RV2
Figure 7 Tail fits by GIGa/IGa become much more
accurate and approach those of GB2/BP with the increase
of n, in agreement with Figure 5
Data represented in Figures 5–7 reflect the 1970–2017
period, but the 1990–2017 subset looks quite similar.
Notice that, unlike other distributions here, GGa does
not have a heavy tail. Notice also that not all of the distributions “make it” into every tail-fitting windows in
Figure 7.
n
0.1
0.08
KS statistic
3 | PDF OF DIF FERENCES
GGa
S
BP
GIGa
GB2
IGa
Ga
0.06
There is a simple relationship for the correlation ρ
between two time series ai and bi and their standard deviations and the standard deviation of the distribution of
the difference:
σ 2a − b = σ 2a + σ 2b −2ρσ a σ b
0.04
0.02
0
0
5
10
15
20
25
n
F I G U R E 5 KS statistics as function of n. Top and bottom
graphs are the same but for a different vertical scale [Colour figure
can be viewed at wileyonlinelibrary.com]
ð7Þ
Of course, knowledge of the distribution functions of
RV2, VIX2, VXO2 and that of the difference of VIX2 and
VXO2 with scaled RV2 opens up a possibility to gain far
richer information than a simple extraction of the correlation coefficient between the indices. The latter may
actually give new insights to option traders. Consequently, in this Section we study the distribution of the
differences of VIX2 and VXO2 with RV2, where the latter
is rescaled per the ratios in Table 6 (In other words,
1
2
0.99
1.95
0.98
1.9
( + 1)S
F I G U R E 6 Power-law
exponents, as per Table 1 as a
function of n [Colour figure can
be viewed at
wileyonlinelibrary.com]
2587
S
MOGHADDAM ET AL.
0.97
0.96
1.85
1.8
0.95
1.75
0.94
1.7
0.93
0
5
10
15
20
1.65
25
0
5
10
n
15
20
25
15
20
25
15
20
25
15
20
25
n
3
15
10
( q + 1) GB2
( p - 1 )GB2
2.8
5
2.6
2.4
2.2
0
2
0
5
10
15
20
25
0
5
10
n
n
3
15
2.8
2.6
(q + 1) BP
(p - 1) BP
10
5
2.4
2.2
2
0
1.8
-5
0
5
10
15
20
1.6
25
0
5
10
n
n
1.5
2.6
2.4
+ 1) GIGa
0.5
(
(
-1) GGa
1
0
2.2
2
1.8
1.6
1.4
-0.5
0
5
10
15
n
“VIX2 − RV2” in actuality means VIX2 − (mean(VIX2)/
mean(RV2))RV2).
In addition to the stable distribution, S(x;α, β, γ, δ),
discussed in Section 2, we use the three distributions
listed in Table 7: Normal (N) and two fat-tailed
20
25
0
5
10
n
distributions—Generalized Student's t (GST) and the distribution generalized from the Tricomi Confluent Hypergeometric (GCHU) (Dashti Moghaddam & Serota, n.d.).
The latter, to the best of our knowledge, has not been
previously used in the literature for fitting purposes. For
2588
MOGHADDAM ET AL.
-1
-1
Tail
Fit
S
BP
GIGa
GGa
GB2
-1.5
log 10(1-CDF)
log 10(1-CDF)
-1.5
-2
-2.5
-3
-0.5
Tail
Fit
S
BP
GIGa
GGa
GB2
-2
-2.5
0
0.5
1
-3
-0.5
1.5
0
2
-1
log 10(1-CDF)
log 10(1-CDF)
-2.5
-2
-2.5
0
0.5
1
-3
-0.5
1.5
0
log 10(RV2)
1.5
Tail
Fit
S
BP
IGa
GIGa
GGa
GB2
-1.5
log 10(1-CDF)
log 10(1-CDF)
1
-1
Tail
Fit
S
BP
IGa
GIGa
GGa
GB2
-1.5
-2
-2.5
-2
-2.5
0
0.5
1
-3
-0.5
1.5
0
2
0.5
1
1.5
log 10(RV2)
log 10(RV )
-1
-1
Tail
Fit
S
BP
IGa
GIGa
GGa
GB2
Tail
Fit
S
BP
IGa
GIGa
GGa
GB2
-1.5
log 10(1-CDF)
-1.5
log 10(1-CDF)
0.5
log 10(RV2)
-1
-2
-2.5
-3
-0.5
1.5
Tail
Fit
S
BP
GIGa
GGa
GB2
-1.5
-2
-3
-0.5
1
-1
Tail
Fit
S
BP
GIGa
GGa
GB2
-1.5
-3
-0.5
0.5
log 10(RV2)
log 10(RV )
-2
-2.5
0
0.5
1
1.5
2
log 10(RV )
-3
-0.5
0
0.5
1
1.5
log 10(RV2)
FIGURE 7
Tails of fitted distribution vis-a-vis the actual tail and its linear fit as a function of n. From left to right and top to bottom,
the plots are for n = 1, 2, 3, 4, 6, 12, 18, 21 days [Colour figure can be viewed at wileyonlinelibrary.com]
these functions, μ is a location parameter, σ is a scale
parameter and ν, p and q are shape parameters. The
results of fitting are shown in Figure 8 and the
parameters of the distributions and KS statistics derived
from MLE fitting are collected in Tables 8 and 9. Notice
that the location parameters for all are rather close to
MOGHADDAM ET AL.
2589
each other and that S, GST and GCHU KS numbers are
very close. In contrast, in Appendix A we find that S fit is
far more accurate than GST and GCHU for VIX-RV and
VXO-RV.
TABLE 8
Rescale Values for RV2 for VIX2 − RV2 (left) and
2
2
VXO − RV (right)
TABLE 6
meanðVIX
2
Þ
meanðRV 2 Þ
meanðVXO2 Þ
meanðRV 2 Þ
= 1:4075
Type
−
ðx −μÞ2
2σ 2
e pffiffiffi
ffi
2π σ
!ν +2 1
GST(x;μ, σ, ν)
ν
ðx − μÞ2
ν+
σ2
p
νσB ν2, 12
ffiffi
ð Þ
ðx − μÞ2
1
Γðq + 2ÞU q + 12, 32 − p, 2σ 2
pffiffiffiffi
GCHU(x;p, q, σ, μ)
2π σBðp,qÞ
KS
test
Normal
N(63.2773, 131.8926)
0.0751
GenStudent's t
GST(73.8714, 92.4056, 1.3310)
0.0280
Tricomi
GCHU(1.7775, 0.7367, 71.2039,
72.3703)
0.0262
Stable
S(1.1842, −0.1503, 86.5044, 77.5295)
0.0265
Parameters
KS
test
Normal
N(60.6005, 139.6113)
0.0607
GenStudent's t
GST(66.1158, 103.6974, 1.3909)
0.0245
Tricomi
GCHU(8.3382, 0.7080, 31.2797,
65.7904)
0.0236
Stable
S(1.2111, −0.0899, 95.8820, 69.2693)
0.0248
103
4
2
2
0
0
-2
-4
-4
-6
-8
-8
-10
90 92 95 97 00 02 05 07 10 12 15 17 20
data
N
S
GST
GCHU
4
3.5
3
2.5
2
3.5
3
2.5
2
1.5
1.5
1
1
0.5
0.5
-500
0
2
VIX - RV
500
2
1000
data
N
S
GST
GCHU
4
PDF
PDF
90 92 95 97 00 02 05 07 10 12 15 17 20
10-3
10-3
0
-1000
103
-2
-6
-10
MLE results for VXO2 − RV2
Type
VXO2 - RV2
VIX2 - RV2
4
Parameters
TABLE 9
PDF
N(x;μ, σ)
Type
= 1:4908
T A B L E 7 Analytic Form of Distributions For Fitting
VIX2 − RV2 and VXO2 − RV2
MLE results for VIX2 − RV2
0
-1000
-500
0
500
1000
VXO2 - RV2
PDF of VIX2 − RV2 (left) and VXO2 − RV2 from January 31, 1990, to December 29, 2017 [Colour figure can be viewed at
wileyonlinelibrary.com]
FIGURE 8
2590
MOGHADDAM ET AL.
10
3.5
0.1
Scaled RV
-3
Scaled RV2
VIX2
3
VIX
0.08
VXO
VXO2
2.5
PDF
PDF
0.06
0.04
2
1.5
1
0.02
0
0.5
0
0
20
40
60
80
Scaled RV & VIX & VXO
0
500
1000
1500
2000
2500
Scaled RV 2 & VIX 2 & VXO 2
Contour PDF plots of scaled RV, VIX and VXO (left) and scaled RV2, VIX2 and VXO2 from January 31, 1990, to December
29, 2017 (right) [Colour figure can be viewed at wileyonlinelibrary.com]
FIGURE 9
From Figures 8 and A11, it is obvious that VIX and
VXO do not anticipate surges in RV, the largest one being
during the financial crisis. They do respond to such
surges of RV with surges of their own but, again, they are
not a very good predictor of future RV. This is consistent
with the results in (Dashti Moghaddam et al., 2019),
where it was shown that, despite availability of more current information, implied volatility indices predict future
RV only very slightly better than past RV.
We also wish to underscore once again the importance of scaling RV to properly match the implied volatility (see Table 6). The unscaled figure identical to
Figure 1, which was published in WSJ (Sindreu &
Bird, 2018), is entirely misleading in representing implied
versus RV since, as explained in Section 1, the former is
calculated for the full year and the latter for the number
of trading days. In this regard, Figure 1 should be contrasted with Figure A11, which correctly represents the relationship between implied and realized volatilities.
4 | C ON C L U S I ON S
We set out to analyse the distribution function of realized
variance. We found that it saturates rapidly to its final
form after several days of adding daily realized variances.
This saturation is quite remarkable in that the daily distribution, with a maximum at low variance and a fat tail
with the exponent around 2, gives way to a bell-shaped
distribution with strongly suppressed low variance and a
fat tail with exponent around 3. The only explanation we
can offer is the rapid initial drop-off of correlations of
daily variances. We also found that for any number of
added days, GB2 distribution would give the best fit.
However, all the fitting distributions did poorly for daily
realized variance, which is not bell-shaped and is possibly
better described by jump models.
For monthly distributions, we found that squared
VIX and VXO distributions are fitted considerably less
accurately than the distributions of realized variance.
This may be one of the signs of misalignment between
implied and realized variances (and volatilities). We also
observe a noticeable difference between the 1970–2017
distribution and its 1990–2017 subset, which may indicate that the introduction of implied volatility index
influenced future RV. For once, 1970–2017 is more accurately fitted with continuous distributions. Additionally,
its front end (low volatilities) is suppressed more in GB2
fitting, pointing to possible increase in low volatility since
1990. In this regard, the front ends of all studied distributions are strongly suppressed, with VIX and VXO considerably more so. As explained in text, implied volatility
consistently statistically overestimates mid-level volatility
and underestimates low volatility, which may be of value
for volatility traders.
We analysed the dependence of the fitting parameters
of the distribution of realized variance on the number of
days over which the daily realized variances are added.
We find that the GB2 achieved the fat-tail exponent saturation over about the same number of days as the saturation of the whole distribution. We also find that it very
accurately describes the fat tails of the distributions of
realized variance.
Finally, we studied the distribution of difference
between squared implied volatility indices and scaled
realized variance and found that it is fitted equally well
by stable, generalized Student's-t and generalized Tricomi
distributions. Using the standard deviation of this distribution one can evaluate the correlation between implied
and realized variances, which is, of course, also possible
MOGHADDAM ET AL.
to achieve using regression analysis. However, knowledge
of the entire distribution opens up a possibility of establishing more intricate connections between the two,
which may be of value to traders of options indices. We
intend to explore this topic in a future publication.
ORCID
R. A. Serota
https://orcid.org/0000-0002-2619-4136
E N D N O T ES
1
Sums/integrals of squared quantities are used in theoretical calculation of implied and realized variance (see, for instance, (Dashti
Moghaddam et al., 2019)) and their distributions may be related
to those of their summands. This is the main reason for which we
study distributions of variances, although the latter can be also
used for options trading (Carr, Geman, Madan, & Yor, 2005;
Kurella, 2013). Volatilities are obtained from square root of variances are normalized to be easily comprehensible numbers.
2
Accordingly, in
more meaningful version of Fig. 1 RV would be
paffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rescaled with 365=252.
DATA AVAILABILITY STATEMENT
Originally, we downloaded our data from 1970 to 2017
for S&P 500 at https://finance.google.com/finance/
historical?q=INDEXSPX and for DJIA at http://www.
google.com/finance/historical?q=INDEXDJX Presently,
historic market data can be found for S&P 500 from
1950 at https://finance.yahoo.com/quote/%5EGSPC/
history?p=%5EGSPC and for DJIA from 1985 at https://
finance.yahoo.com/quote/%5EDJI/history?p=%5EDJI We
will be happy to provide data downloaded from those
sites on request.
ORCID
R. A. Serota
https://orcid.org/0000-0002-2619-4136
R EF E RE N C E S
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(Statistical Methodology), 64(2), 253–280.
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Applications, 461, 716–725.
Bollerslev, T., Mathew, G., & Zhou, H. (2011). Dynamic estimation
of volatility risk premia and investor risk aversion from optionimplied and realized volatility. Journal of Econometrics, 160,
235–245.
Bouchaud, J.-P., & Mézard, M. (2000). Wealth condensation in a
simple model of economy. Physica A: Statistical Mechanics and
its Applications, 282(3), 536–545.
Carr, P., Geman, H., Madan, D. B., & Yor, M. (2005). Pricing
options on realized variance. Finance and Stochastics, 9(4),
453–475.
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Carr, P., & Wu, L. (2006). A tale of two indices. The Journal of
Derivatives, 13(3), 13–29.
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com/micro/vix/vixwhite.pdf.
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Chrstensen, B. J., & Prabhala, N. R. (1998). The relation between
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Dashti Moghaddam, M., Liu, Z., & Serota, R. (2019). Distributions
of historic market data—Implied and realized volatility.
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(4), 9–32.
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Quantitative Finance, 2, 445–455.
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the Australian Agricultural and Resource Economics Society,
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ed.) Boston: Addison Wesley.
Kownatzki, C. (2016). How good is the vix as a predictor of market
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Paper presented at CBOE Risk Management Conference, 2013.
Ma, T., Holden, J. G., & Serota, R. (2013). Distribution of wealth in
a network model of the economy. Physica A: Statistical Mechanics and its Applications, 392(10), 2434–2441.
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Physica A: Statistical Mechanics and its Applications, 398, 89–115.
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jpnolan/www/stable/chap1.pdf
RealVol.
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White
Paper,
https://www.realvol.com/
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How to cite this article: Moghaddam MD, Liu J,
Serota RA. Implied and realized volatility: A study
of distributions and the distribution of difference.
Int J Fin Econ. 2021;26:2581–2594. https://doi.org/
10.1002/ijfe.1922
AP P ENDI X : R V, V IX A N D VX O F I TT IN G A.
Here we fit RV, VIX, VXO and VIX-(scaled) RV and VXO(scaled) RV. First, for illustrative purposes, in Figure 9 we
show contour plots of PDF of scaled RV, VIX and VXO visa-vis that of scaled RV2, VIX2 and VXO2. As mentioned
above, it is clear VIX and VXO overestimate mid-volatility
and underestimate low volatility (Dashti Moghaddam
et al., 2019). Next, in Figure A10 we show fits of the RV,
VIX and VXO PDF, with the parameters of the distributions and KS statistics collected in Tables A10–A13. Finally,
in Figure A11 we show fits of VIX-RV and VXO-RV, with
the parameters of the distributions and KS statistics collected in Tables A14 and A15.
Two peculiarities should be noted. First, in contrast to
VIX2 − RV2 and VXO2 − RV2, where S, GST and GCHU
fitted equally well, here we find that S fit is more accurate
for VIX-RV and VXO-RV, probably because of a greater
skewness of the latter two. Second, BP fits are worse than
IGa for RV but are better for RV2. Obviously, distributions of RV and RV2 are not independent: under transformation x ! xr, r > 0 we have GB2(x; p, q, α, β) ! GB2(x;
p, q, αr, β1/r) and GIGa(x; α, γ, β) ! GIGa(x; α, γr, β1/r).
With the values of parameters for RV, IGa transforms
into GIGa, which fits RV2 distribution rather well, if not
as the best GIGa fit, while neither BP nor GB2 with RV2
parameters transforms into a GB2 that is close to BP.
0.1
0.1
data
GGa
S
BP
GIGa
GB2
0.08
0.08
0.06
PDF
PDF
0.06
data
GGa
S
BP
GIGa
GB2
0.04
0.04
0.02
0.02
0
0
20
40
60
0
80
0
20
RV
40
60
80
RV
0.1
0.1
data
GGa
S
BP
GIGa
GB2
0.08
data
GGa
S
BP
GIGa
GB2
0.08
PDF
0.06
PDF
0.06
0.04
0.04
0.02
0.02
0
0
20
40
VIX
60
80
0
0
20
40
60
80
VXO
F I G U R E A 1 0 Clockwise: PDF of monthly RV from January 2, 1970, to December 29, 2017, and PDFs of monthly RV, VIX and VXO
from January 31, 1990, to December 29, 2017 [Colour figure can be viewed at wileyonlinelibrary.com]
MOGHADDAM ET AL.
TABLE A10
2593
MLE results for RV
1970–2017
TABLE A11
MLE results for RV
1990–2017
TABLE A12
MLE results for VIX
1990–2017
TABLE A13
1990–2017
MLE results for VXO
Type
Parameters
Front exp
Stable
S(1.3278, 1.0000, 3.4936, 13.8773)
GB2
GB2(15.8782, 1.8724, 2.0309, 4.8757)
31.2470
26.1723
Tail exp
KS test
−2.3278
0.0171
−4.8026
0.0115
BP
BP(27.1723, 6.7415, 4.001)
−7.7415
0.0275
GIGa
GIGa(2.5562, 25.0090, 1.6047)
−5.1019
0.0138
IGa
IGa(6.0553, 88.2509)
−7.0553
0.0164
GGa
GGa(6.0715, 2.3493, 0.9052)
4.4959
0.0753
Ga
Ga(4.8790, 3.6151)
3.8790
0.0795
Type
Parameters
Front exp
Tail exp
KS test
Stable
S(1.2849, 1.0000, 3.9402, 13.7767)
−2.2849
0.0249
GB2
GB2(17.6690, 2.4125, 1.5731, 4.1218)
26.7951
−4.7951
0.0139
BP
BP(21.8899, 6.1274, 4.2232)
20.8899
−7.1274
0.0298
GIGa
GIGa(3.5363, 40.8574, 1.1920)
−5.2153
0.0134
IGa
IGa(4.8913, 70.5571)
−5.8913
0.0182
GGa
GGa(4.3867, 3.9442, 0.9731)
3.2687
0.0647
Ga
Ga(4.1254, 4.4086)
3.1254
0.0659
Type
Parameters
Front exp
Stable
S(1.2901, 1.0000, 3.3182, 15.9886)
Tail exp
KS test
−1.2901
0.0381
GB2
GB2(63.0279, 1.3326, 2.9404, 4.2422)
184.3272
−4.9184
0.0368
BP
BP(44.8997, 10.8471, 4.2232)
43.8997
−11.8471
0.0463
GIGa
GIGa(1.4520, 18.0537, 2.7628)
−5.0116
0.0375
IGa
IGa(8.9387, 152.9579)
−9.9387
0.0443
GGa
GGa(6.7354, 10.0120, 0.5250)
2.5360
0.0669
Ga
Ga(7.7138, 2.5092)
6.7138
0.0681
Type
Parameters
Front exp
Stable
S(1.3267, 1.0000, 3.8227, 16.0853)
GB2
GB2(53.3880, 2.6150, 1.7816, 3.0120)
94.1160
35.5647
Tail exp
KS test
−2.3267
0.0434
−5.6589
0.0391
BP
BP(36.5647, 8.8725, 4.2232)
−9.8725
0.0436
GIGa
GIGa(3.0721, 33.0522, 1.5909)
−5.8874
0.0423
IGa
IGa(7.2686, 123.1749)
−8.2686
0.0491
GGa
GGa(5.5107, 3.9062, 1.0601)
4.8419
0.0718
Ga
Ga(6.4518, 3.0512)
5.4518
0.0712
2594
MOGHADDAM ET AL.
101
2
2
0
0
-2
-4
-2
-4
-6
-6
-8
-8
90 92 95 97 00 02 05 07 10 12 15 17 20
90 92 95 97 00 02 05 07 10 12 15 17 20
0.16
0.16
data
N
S
GST
GCHU
0.14
0.12
0.12
0.1
PDF
0.08
0.08
0.06
0.06
0.04
0.04
0.02
0.02
0
data
N
S
GST
GCHU
0.14
0.1
PDF
101
4
VXO - RV
VIX - RV
4
0
-20
-10
0
10
20
-20
-10
0
10
20
VXO - RV
VIX - RV
PDF of VIX2 − RV2 (left) and VXO2 − RV2 from January 31, 1990, to December 29, 2017 (right) [Colour figure can be
viewed at wileyonlinelibrary.com]
FIGURE A11
TABLE A14
TABLE A15
MLE results for VIX-RV
MLE results for VXO-RV
Type
Parameters
KS test
Type
Parameters
KS test
Normal
N(0.4791, 3.8945)
0.0673
Normal
N(0.4412, 3.8358)
0.0618
Gen-Student's t
GST(0.9680, march 2,
1944.6948)
0.0447
Gen-Student's t
GST(0.8367, 3.2302,
2.8099)
0.0392
Tricomi
GCHU(3.1107, 2.1000,
3.0969, 0.9445)
0.0467
Tricomi
GCHU(2.8284, 2.1000,
3.2574, 0.8439)
0.0426
Stable
S(1.5807, −0.8377,
2.6247, 1.4591)
0.0135
Stable
S(1.6068, −0.7346,
2.6689, 1.2449)
0.0159
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