Don't we want to base our trading decisions on current market conditions? Then why should we rely on time averages only because they are simple to comprehend? We can get current market volatility a lot faster by applying the ERGODICITY concept to financial markets. Ensemble averaging allows to measure market volatility quickly, based on only two points in time and is as relevant to volatility measurement as the traditional measures.

1. 1
Express measurement of market volatility using ergodicity concept
Jack Sarkissian
Managing Member, Algostox Trading LLC
email: jack@algostox.com
Abstract
We propose a number of volatility measures that are based on ensemble averaging instead of time
averaging. These measures allow fast measurement of current volatility without relying on series of past
data (realized volatility) of future expectations (implied volatility). The introduced quantities are tested on
a model market and are then related to actual market data. They display very adequate behavior and are
great complement to traditional volatility measures in analytics, securities valuation, risk management and
portfolio management.
Keywords: Volatility, ergodicity, quantitative finance, analytics, portfolio management, risk management
1. Introduction
Volatility is a measure of price variation of a financial instrument or portfolio over time. It is widely used
by finance professionals to price securities, measure risks, and make trading decisions. Traditionally, two
types of volatility are used: historical volatility, which is derived from time series of past market prices and
implied volatility, which is derived from the market price of a traded derivative (particularly options) [1].
Given time series of returns 𝑟𝑡, the historical volatility over time 𝑇 is defined as the standard deviation of
those returns:
Please cite as: J. Sarkissian, “Express measurement of market volatility using ergodicity concept”
(July 20, 2016). Available at SSRN: http://ssrn.com/abstract=2812353
© 2016 Algostox Trading. The enclosed materials are copyrighted materials. Federal law prohibits the
unauthorized reproduction, distribution or exhibition of the materials. Violations of copyright law will
be prosecuted.

2. 2
𝐻𝑉 = √
1
𝑇
∑(𝑟𝑡 − 𝑟̅)2
𝑇
𝑡=1
(1)
where 𝑟̅ is the average return. Historical volatility reflects the realized or past volatility. Implied volatility
𝐼𝑉 is defined as the volatility, with which the Black-Scholes option pricing model produces prices equal to
market prices:
𝑉(𝑠, 𝐾, 𝑇, 𝑰𝑽, 𝑟𝑓) = 𝑃 (2)
where 𝑠, 𝐾, 𝑇, 𝑟𝑓 and 𝑃 are the stock price, strike price, time to expiration, risk-free rate, and option market
price. Implied volatility reflects future volatility over time 𝑇. Both 𝐻𝑉 and 𝐼𝑉 have established themselves
as widely used volatility measures.
Interestingly, none of the two volatility measures describes the current, or instantaneous, volatility. In the
meantime, most of the time traders, asset managers and risk managers are concerned with the current value
of volatility – the instantaneous volatility. Due to unavailability of one, HV and VI and large variety of their
variations are used as a proxy to understand the current level of actual quantity [1-3].
To make matters worse, both historical and implied volatilities relate to extended time intervals. Historical
volatility uses series of ticks of backward data. Implied volatility is based on the time until the expiration
of the derivative, and as such represents the average over that period. Time averaging blurs the picture, and
slows down the reaction to market changes. Clearly, a better measure for volatility is necessary, one that
relies on current observables and does not suffer from these deficiencies.
In this paper we develop a view allowing to calculate market volatility based on only two points in time.
This is achieved by looking at each stock as a possible microstate of the entire system of stocks and applying
the ergodic assumption [4]. Under such assumption time averaging over a period can be replaced with
ensemble averaging related to current time. Ergodicity is a strong assumption, and more research has to be
done on ergodic properties of financial markets [5-9]. Despite that, volatility measures calculated in such
way are more relevant to the immediate volatility than the traditional measures, and can be used to
compliment them or even to replace them.
Here we focus only on volatility measurement and not forecasting. Firms and individuals adopting the
presented approach can solve the forecasting tasks as it applies to them.

3. 3
2. Ensemble volatility measures: eVIX and eVaR
Volatility as a measure of time variation of price relies on time averaging. However, time averaging is not
the only option for statistical systems. If we look at the market as a statistical ensemble represented by
market components, we can characterize it through average values over that ensemble [5-7]. Such average
values must be capitalization weighted to adequately reflect each component’s contribution. We will see
that this point is not just a possible variation, but is required by calculations.
Let all market components start at zero time with zero return. Then, after a single time step, each component
will disperse, acquiring a return 𝑟𝑖. Capitalization distribution by return will take form shown in Fig. 1b.
Under the ergodicity assumption that distribution should repeat the shape of the probability distribution of
returns in time. If this is true, it can be used for ensemble averaging instead of time averaging as a way to
get fast measurement of volatility.
(a) (b)
Fig. 1. (a) Initial state of the market in return space: all stocks are at 𝑟𝑖 = 0% point; (b) final state of the
market in return space. Prices dispersed and return buckets are populated each according to the
probability of that return.
If i-th component has relative capitalization 𝑤𝑖
1 and acquires return 𝑟𝑖 at the end of the time step, the
resulting population distribution 𝑤𝑖(𝑟𝑖) is centered at
𝑅 = ∑ 𝑤𝑖 𝑟𝑖
𝑖
(3)
and has width
1
Defined as the share of i-th stock’s capitalization in the entire market.
0%
20%
40%
60%
80%
100%
-2.0% -1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0%
Capitalizationdistribution
Return
0%
10%
20%
30%
40%
-2.0% -1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0%
Capitalizationdistribution
Return

4. 4
𝜔2
= ∑ 𝑤𝑖 𝜉𝑖
2
𝑖
(4)
where
𝜉𝑖 = 𝑟𝑖 − 𝑅 (5)
If we wanted to relate these parameters to time-average based market quantities, we would compose a
capitalization weighted index, measuring average market performance:
𝐼 =
1
𝐼0
∑ 𝑛𝑖 𝑠𝑖
𝑖
(6)
Here 𝐼0 is the divisor, which is used in definition, but is unessential in this work. Return of the index is
equal
𝑑𝐼
𝐼
= ∑ 𝑤𝑖 𝑟𝑖
𝑖
(7)
Thus, index return is already described by ensemble average of Eq. (3). In order to get to its volatility, let
us assume that securities prices follow the traditional Wiener process:
𝑟𝑖 =
𝑑𝑠𝑖
𝑠𝑖
= 𝜇𝑖 𝑑𝑡 + 𝜎𝑖 𝑑𝑧𝑖 (8)
where 𝜇𝑖 and 𝜎𝑖 are drift and volatility. Then we can write index return as
𝑑𝐼
𝐼
= ∑ 𝑤𝑖 𝜇𝑖 𝑑𝑡 + ∑ 𝑤𝑖 𝜎𝑖 𝑑𝑧𝑖
𝑖𝑖
(9)
If 𝑧𝑖 are uncorrelated, its probability distribution has standard deviation
Ω2
= ∑ 𝑤𝑖
2
𝜎𝑖
2
𝑖
(10)
This Ω2
is a time average of the following ensemble average quantity
𝜔1
2
= ∑ 𝑤𝑖
2
𝜉𝑖
2
𝑖
(11)
so that
〈𝜔1
2〉 𝑇 = Ω2 (12)
where 〈… 〉 𝑇 stands for averaging over timeframe 𝑇.

5. 5
Before we proceed, we must note that the real dependence of index volatility is more complex, but (a) it is
still proportional to overall component volatility: Ω~𝜎, and (b) in this section we are only considering a
model in order define ensemble-average volatility measures and study how they reflect time average
quantities. Relation of ensemble quantities to real market data will be considered in Section 4.
Based on these considerations, let us introduce the following volatility measures:
1. The eVIX index (stands for ensemble VIX)
𝑒𝑉𝐼𝑋 = 𝐷√∑ 𝑤𝑖 𝜉𝑖
2
𝑖
(13)
where 𝐷 = √
1 𝑦𝑒𝑎𝑟
𝑇
is the annualizing multiplier and is equal 16 = √256 if return data is daily.
2. 𝛼-percentile eVaR (standing for ensemble Value-at-Risk)
𝛼-quantile 𝑉𝑎𝑅 = −𝐷 𝑟𝛼,
where 𝑟𝛼: ∑ 𝑤(𝑟𝑖)
𝑟 𝛼
𝑟 𝑚𝑖𝑛
= 𝛼, and summation is performed in the order of increasing 𝑟𝑖.
(14)
Unlike eVIX, which only measures the dispersion and excludes the average return, eVaR includes the return
and is a better measure of risk of loss.
3. The eVIX2 index for convenience in dealing with equity indexes
𝑒𝑉𝐼𝑋2 = 𝐷√
∑ 𝑤𝑖
2
𝜉𝑖
2
𝑖
∑ 𝑤𝑖
2
𝑖
(15)
Before relating these volatility measures to market data let us test them on a model market. With the model
we have control of input parameters and can characterize outcome vs. input.
3. Modeling results
In this section we consider a model market with 100 components. Its weight composition is random and
annualized volatilities are random too, ranging from 8% to 24% under normal market conditions.
Volatilities in Eq. (8) are constant. Here we make them time-dependent by scaling by a common factor, so
that
𝜎𝑖(𝑡) = 𝑓(𝑡)𝜎𝑖 (16)

6. 6
Factor 𝑓 ≈ 1 under normal market conditions and can go up to 2.5 in extremes. This factor follows mean
reverting process
𝑑𝑓 = 𝜃(1 − 𝑓) + √𝑓𝜎𝑓 𝑑𝑧 (17)
where 𝜃 is the reversion speed (here 𝜃 = 0.01) and 𝜎𝑓 is the factor volatility (here 𝜎𝑓 = 0.05). Time
evolution of the resulting volatility measures is shown in Fig. 2. Correlation of eVIX and eVaR with 𝑓 is
better seen in Figs. 3a-b, where they are plotted directly against factor 𝑓. For comparison historical volatility
(30-day average) has the lowest 𝑅2
, while even the eVIX averaged over the same time period performs
better, Fig. 3c-d. This indicates that ensemble volatility measures react faster and to the right proportion to
changes in volatility levels.
Fig. 2. Time evolution of volatility factor (represented by its
1
10
-th) and the corresponding eVIX, and
95%-eVaR
0%
20%
40%
60%
80%
100%
0 200 400 600 800 1000
eVIX 95%-eVAR f/10

7. 7
(a) (b)
Fig. 3. (a) eVIX vs. volatility level 𝑓, and (b) 95%-eVaR vs. volatility level 𝑓. Both have a distinct linear
relationship with 𝑓.
(c) (d)
Fig. 3. (c) Average historical volatility vs. volatility level 𝑓, and (d) average eVIX vs. volatility level 𝑓.
Despite being averaged over the same time as historical volatility, the eVIX has bigger 𝑅2
.
As expected, the ensemble volatility measures are interrelated, as shown in Figs. 4a-b. The eVaR shows
linear relationship with eVIX, but has enough independence to be considered as a separate measure. The
eVIX2 measure to a good degree repeats eVIX, so we will limit usage to eVIX alone.
y = 0.1607x
R² = 0.8675
0%
10%
20%
30%
0 0.5 1 1.5 2
eVIX
Volatilitylevel (f)
y = 0.2916x
R² = 0.6186
0%
20%
40%
60%
80%
100%
0 0.5 1 1.5 2
eVaR
Volatilitylevel (f)
y = 0.0181x
R² = 0.4719
0%
1%
2%
3%
0 0.5 1 1.5 2
avghistoricvolatility
Volatilitylevel (f)
y = 0.1576x
R² = 0.7409
0%
5%
10%
15%
20%
25%
0 0.5 1 1.5 2
avgeVIX
Volatilitylevel (f)

8. 8
(a) (b)
Fig. 4. (a) 95%-eVaR vs. eVIX. The eVaR shows linear relationship with eVIX, but has enough
independence to be considered as a separate measure, (b) eVIX2 vs eVIX the two quantities appear to be
redundant.
Scatterplot in Fig. 5a shows that relationship between eVIX and index returns is quite arbitrary, though
larger returns are associated with larger eVIX, no matter what sign the return has. This symmetry is
expected since eVIX only measures the dispersion of population distribution while excluding index return.
The eVaR does not exclude it, and therefore negative returns tend to associate with larger eVaR as shown
in Fig. 5b.
(a) (b)
Fig. 5. (a) eVIX vs. index returns and (b) 95%-eVaR vs. index returns. Because eVIX does not include
index return in its definition and eVaR does, their sensitivity to return sign is different.
y = 1.8173x
0%
20%
40%
60%
80%
100%
0% 5% 10% 15% 20% 25% 30%
95%eVaR
eVIX
y = 1.0061x
0%
10%
20%
30%
40%
0% 5% 10% 15% 20% 25% 30%
eVIX2
eVIX
y = 8839.5x2
+ 1.24x + 0.1227
0%
5%
10%
15%
20%
25%
30%
-0.6% -0.4% -0.2% 0.0% 0.2% 0.4% 0.6%
eVIX
Index return
y = 17997x2 - 23.155x + 0.2203
0%
20%
40%
60%
80%
100%
-0.6% -0.4% -0.2% 0.0% 0.2% 0.4% 0.6%
eVaR
Index return

9. 9
4. Relation of ensemble indexes to VIX and equity returns
Let us now describe the relation of ensemble volatility measures to real market data. The most obvious
choice is the S&P500 and its CBOE Volatility Index – the VIX. The VIX index for year 2012 along with
corresponding daily eVIX and 95%-eVaR are shown in Fig. 6. Ensemble measures are choppier than VIX,
since they are related to current market conditions, while VIX is related to time average of expected
volatility. At times eVaR takes negative values. This happens when index return is positive and so large
that it offsets the return dispersion.
Fig. 6. Time evolution of VIX, and daily eVIX, and 95%-eVaR in 2012. The ensemble measures are
choppier than the VIX, since they are related to current market conditions, while VIX is related to time
average of expected volatility.
Relation between volatility indexes
Connection of ensemble volatility measures to VIX is easier seen if eVaR is taken as the argument. Linear
relation between both VIX and eVIX with eVaR can be traced and is shown in Figs. 7a-b. This contrasts
with relation between VIX and eVIX, which is not so apparent in Fig. 7c.
-20%
0%
20%
40%
60%
80%
100%
VIX eVIX eVaR

10. 10
(a) (b)
Fig. 7. Comparison of volatility measures (a) VIX vs. 95%-eVaR, and (b) eVIX vs. 95%-eVaR.
Fig. 7c. Comparison of volatility measures VIX vs. eVIX.
Relation between volatility indexes and the underlying index returns
Similar to Figs. 5a-b, Figs. 8a-b show eVIX and eVaR plotted against S&P 500 index returns. Again, the
relation of eVIX is symmetric, so large returns imply bigger eVIX no matter if they are positive or negative.
That component is also present in eVaR, but in addition to that eVaR has a much stronger negative tilt.
Once again, we recall that eVaR is measured from zero return, and not from the index return, so negative
returns increase eVaR. The VIX index appears to combine both the quadratic component and the tilt as
shown in Fig. 8c. In that sense, it behaves more like eVaR than eVIX.
y = 0.0479x + 0.163
0%
10%
20%
30%
-20% 0% 20% 40% 60% 80% 100%
VIX
eVaR
y = 0.0541x + 0.1725
0%
10%
20%
30%
40%
-20% 0% 20% 40% 60% 80% 100%
eVIX
eVaR
0%
10%
20%
30%
0% 10% 20% 30% 40%
VIX
eVIX

11. 11
(a) (b)
Fig. 8. (a) eVIX vs. S&P500 returns, and (b) 95% eVaR vs. S&P500 returns. eVIX does not have the tilt.
Fig. 8c. VIX vs. S&P500 returns. The VIX index behaves more like eVaR than eVIX.
Relation between volatility change and underlying index returns
Next, we show how volatility indexes change with S&P 500 returns. Change in absolute value is discussed,
not a relative change. Once again, the eVIX changes are quite arbitrary, but symmetric, as shown in Fig. 9a.
Changes of eVaR are negatively correlated with index returns, Fig. 9b. And once again, VIX resembles
eVaR more than eVIX, Fig. 9c.
y = 112.33x2 + 0.7164x + 0.1794
0%
10%
20%
30%
40%
-3% -2% -1% 0% 1% 2% 3%
eVIX
S&P 500 return
y = 188.86x2
- 17.787x + 0.2644
-20%
0%
20%
40%
60%
80%
100%
-3% -2% -1% 0% 1% 2% 3%
eVaR
S&P 500 return
y = 55.673x2 - 0.9101x + 0.1726
0%
5%
10%
15%
20%
25%
30%
-3% -2% -1% 0% 1% 2% 3%
VIX
S&P 500 return

12. 12
Fig. 9. (a) eVIX change vs. S&P500 returns, and (b) eVaR change vs. S&P500 returns. eVIX change does
not have the tilt.
Fig. 9c. VIX change vs. S&P 500 returns.
Autocorrelation of volatility measure changes
Daily ensemble measures have substantial negative autocorrelation for time shift up to 1 day as shown in
Table 1. In other words, an eVIX or eVaR increase is usually followed by a decrease the next day, and vice
versa. Unlike them the degree of autocorrelation in VIX is much smaller. Corresponding scatterplots are
with time shift of 1 day are provided in Figs. 10a-c.
y = 96.626x2 + 0.6633x - 0.0075
-20%
-10%
0%
10%
20%
-3% -2% -1% 0% 1% 2% 3%
eVIXchange
S&P 500 return
y = -18.003x + 0.0051
-90%
-60%
-30%
0%
30%
60%
90%
-3% -2% -1% 0% 1% 2% 3%
eVaRchange
S&P 500 return
y = 7.4645x2
- 1.1185x - 0.0002
-4%
-2%
0%
2%
4%
-3% -2% -1% 0% 1% 2% 3%
VIXchange
S&P 500 return

13. 13
Table. 1. Autocorrelation of volatility indexes.
Time shift 1d 2d 3d
VIX -13% 2.1% -14.9%
eVIX -44% -3.9% -7.5%
eVaR -47% -3.6% 1.2%
(a) (b)
Fig. 10. (a) eVIX change vs. itself shifted by 1 day, and (b) same for 95% eVaR.
Fig. 10c. VIX change vs. itself shifted by 1 day
These results show that ensemble-average based volatility measures are:
1. Adequate measures of market volatility
y = -0.4438x - 0.0011
-20%
-15%
-10%
-5%
0%
5%
10%
15%
20%
-20% -10% 0% 10% 20%
eVIXchangenextday
eVIXchange
y = -0.4674x + 0.0007
-100%
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
-100% -50% 0% 50% 100%
eVaRchangenextday
eVaR change
y = -0.1297x + 8E-05
-4%
-3%
-2%
-1%
0%
1%
2%
3%
4%
-4% -2% 0% 2% 4% 6%
VIXchangenextday
VIXchange

14. 14
2. More relevant to current volatility than the traditional measures
3. Faster reacting to market changes than the traditional measures
5. Conclusions
By switching from time averaging to ensemble averaging we were able to introduce market volatility
measures that rely on data only between two points in time and are fast to react to changing market
conditions. Testing them on a model market we showed that they are adequate in describing the true degree
of market fluctuations. Using them on real market data – the S&P 500 – allowed to reveal their relation
with VIX and index returns. We demonstrated that out of the two measures eVaR is more tightly related to
VIX than eVIX. Yet, they are distinct measures and should be regarded as independent and complimentary.
Unlike VIX, both eVIX and eVaR displayed strong autocorrelation.
Validity of switching between the averaging types is related (but not limited) to the ergodic properties of
the system. Ergodicity means that given enough time a statistical system will run through its entire phase
space of microstates [4]. As a result, the entire ensemble will be represented over time, and ensemble
averaging will converge with time averaging. The biggest destructing factor for ergodicity is correlation,
which we know is present in financial markets. Although ergodic properties of financial markets remain to
be explored more thoroughly and some research has already been done [5-9], the obtained results already
show that ergodic principle (hypothesis) is applicable and useful.
The eVIX and eVaR measures can be applied to market sectors as well as the whole market. They can be
used as risk measures for diversified portfolios, alongside the traditional risk measures. Due to their speed,
the ensemble measures are useful as early indicators of financial processes in the markets, signaling
opportunities, risks, instabilities, pinpointing their sources, and allowing faster reaction. Firms and trading
desks can make use of eVIX and eVaR in analytics, securities valuation, portfolio management, and risk
management.
6. References
[1] S.T. Rachev, S.V. Stoyanov, F.J. Fabozzi, “Advanced Stochastic Models, Risk Assessment, and
Portfolio Optimization”, (Wiley 2008)
[2] J. London, “Modeling Derivatives in C++”, 1st
edition (Wiley 2004)

15. 15
[3] D.G. Goldstein, N.N. Taleb, "We don't quite know what we are talking about when we talk about
volatility", Journal of Portfolio Management, 33 (4), 84-86 (2007).
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1980)
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(2000)
[6] F. Lillo, R.N. Mantegna, “Ensemble properties of securities traded in the NASDAQ market”, Physica
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100603 (2013)