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

𝐻𝑉 = √1

𝑇∑(𝑟𝑡 − �̅�)2

𝑇

𝑡=1

(1)

where �̅� 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

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 𝑤𝑖1 and acquires return 𝑟𝑖 at the end of the time step, the

resulting population distribution 𝑤𝑖(𝑟𝑖) is centered at

𝑅 = ∑ 𝑤𝑖𝑟𝑖

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%

Ca

pit

aliz

ati

on d

istr

ibu

tion

Return

0%

10%

20%

30%

40%

-2.0% -1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5% 2.0%

Ca

pit

aliz

ati

on d

istr

ibu

tion

Return

4

𝜔2 = ∑ 𝑤𝑖𝜉𝑖2

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∑ 𝑛𝑖𝑠𝑖

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= ∑ 𝑤𝑖𝑟𝑖

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= 𝜇𝑖𝑑𝑡 + 𝜎𝑖𝑑𝑧𝑖 (8)

where 𝜇𝑖 and 𝜎𝑖 are drift and volatility. Then we can write index return as

𝑑𝐼

𝐼= ∑ 𝑤𝑖𝜇𝑖𝑑𝑡 + ∑ 𝑤𝑖𝜎𝑖𝑑𝑧𝑖

𝑖𝑖

(9)

If 𝑧𝑖 are uncorrelated, its probability distribution has standard deviation

Ω2 = ∑ 𝑤𝑖2𝜎𝑖

2

𝑖

(10)

This Ω2 is a time average of the following ensemble average quantity

𝜔12 = ∑ 𝑤𝑖

2𝜉𝑖2

𝑖

(11)

so that

⟨𝜔12⟩𝑇 = Ω2 (12)

where ⟨… ⟩𝑇 stands for averaging over timeframe 𝑇.

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√∑ 𝑤𝑖𝜉𝑖2

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)𝑟𝛼𝑟𝑚𝑖𝑛

= 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

2𝜉𝑖2

𝑖

∑ 𝑤𝑖2

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)

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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) + √𝑓𝜎𝑓𝑑𝑧 (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

(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 = 0.1607xR² = 0.8675

0%

10%

20%

30%

0 0.5 1 1.5 2

eV

IX

Volatility level (f)

y = 0.2916x

R² = 0.6186

0%

20%

40%

60%

80%

100%

0 0.5 1 1.5 2

eV

aR

Volatility level (f)

y = 0.0181x

R² = 0.4719

0%

1%

2%

3%

0 0.5 1 1.5 2

avg

his

tori

c vo

lati

lity

Volatility level (f)

y = 0.1576x

R² = 0.7409

0%

5%

10%

15%

20%

25%

0 0.5 1 1.5 2

avg

eV

IX

Volatility level (f)

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

% e

Va

R

eVIX

y = 1.0061x

0%

10%

20%

30%

40%

0% 5% 10% 15% 20% 25% 30%

eV

IX2

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%

eVa

R

Index return

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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

(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%

eV

IX

eVaR

0%

10%

20%

30%

0% 10% 20% 30% 40%

VIX

eVIX

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%

eV

IX

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%

eVa

R

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

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%

eV

IX c

ha

nge

S&P 500 return

y = -18.003x + 0.0051

-90%

-60%

-30%

0%

30%

60%

90%

-3% -2% -1% 0% 1% 2% 3%

eV

aR

ch

ange

S&P 500 return

y = 7.4645x2 - 1.1185x - 0.0002

-4%

-2%

0%

2%

4%

-3% -2% -1% 0% 1% 2% 3%

VIX

ch

an

ge

S&P 500 return

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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%

eV

IX c

ha

nge

nex

t d

ay

eVIX change

y = -0.4674x + 0.0007

-100%

-80%

-60%

-40%

-20%

0%

20%

40%

60%

80%

100%

-100% -50% 0% 50% 100%

eV

aR

ch

ange

nex

t d

ayeVaR change

y = -0.1297x + 8E-05

-4%

-3%

-2%

-1%

0%

1%

2%

3%

4%

-4% -2% 0% 2% 4% 6%

VIX

ch

an

ge n

ext

day

VIX change

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

[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).

[4] L.D. Landau, E.M. Lifshitz, “Statistical Physics, Part 1”, Vol. 5, 3rd ed., (Butterworth-Heinemann,

1980)

[5] F. Lillo, R.N. Mantegna, “Variety and volatility in financial markets”, Phys. Rev. E 62, 6126–6134

(2000)

[6] F. Lillo, R.N. Mantegna, “Ensemble properties of securities traded in the NASDAQ market”, Physica

A, Volume 299, Issues 1-2, 161-167 (2001)

[7] G.F. Gu, W.X. Zhou, “Statistical properties of daily ensemble variables in the Chinese stock markets”,

Physica A, Volume 383, Issue 2, 497-506 (2007)

[8] O. Peters, “Optimal leverage from non-ergodicity”, Quantitative Finance, Volume 11, Issue 11, (2011)

[9] O. Peters, W. Klein, “Ergodicity breaking in geometric Brownian motion”, Phys. Rev. Lett. 110,

100603 (2013)