Volmex Implied Volatility Indices
Kadir Gökhan Babaoğlu, PhD
Volmex Labs
October 26, 2023
This paper develops a new methodology to calculate the implied volatility ofcrypto assets using option contracts from multiple sources. It describes how toconsolidate and filter the option data, calculate the implied variance and smooth
this variance in order to finalize the volatility calculation. At Volmex Labs, we
implemented this methodology to publish the Volmex implied volatility indiceswhich measure a 30-day implied volatility of crypto assets.11Introduction

Implied volatility (IV), believed to be the fear gauge2 of investors, simply refers
to the volatility that is extracted from (and implied by) options by matching theprice of these contracts. It is a single number that combines all the information
across available strikes for a given expiry, and measures overall expensiveness ofoption contracts.
The pricing of options depends not only on the price level of the underlying
asset but also on its volatility, which is actually a latent variable. This latentvariable (i.e., IV) is not observed directly but can be inferred from the pricesof option contracts, and is considered to be the expected volatility from now tothe expiration. Thus, IV reflects the option market participants’ expectations
about the price range of the underlying asset over the life of the contract andit is “forward-looking.”
IV calculations can be model-dependent or model-free. Model-dependent IVs
have restrictive assumptions on the underlying asset price and other state variable processes (stochastic volatility or jump intensity), whereas model-free IVs
work with minimal assumptions on the generating processes of state variables.
The Volmex IV indices use a model-free approach3 to calculate IV of crypto
assets, and reflect the forward-looking market view expressed as an index.
1 Currently available underlying crypto assets are ETH and BTC
2 See Whaley (2000).
3 See Demeterfi et al. (1999)

1

2Methodology

The Volmex IV calculation methodology does not depend on any model or any
data source. It consolidates option data from various option exchanges and creates a global option order book. It uses variance-swap replication method tocalculate implied variance in a model-free way and then uses the exponentiallyweighted moving average (EWMA) method to smooth the raw variance. Smoothing provides the advantage of de-noising the index without missing a trend. Itis parameterized in a balanced fashion so that it captures the emerging trends
but at the same time ignores noisy observations.
In order to infer the implied volatility from a set of option contracts, the Volmex
IV methodology follows the four steps below. Each of them is crucial for thecalculation of the final implied volatility value:
1. Data consolidation: The first step of this methodology is to gather all
the option and future contracts’ data from multiple sources and create a
global order book of options and risk-free curve.
2. Filtering: The second step filters the out-of-the-money (OTM) call andput option contracts to be used in the variance calculation.
3. Calculation: Raw implied variance calculation using variance swap replication is performed by computing the contributions of each of the selectedoptions, summing them up and making the adjustments to get the rawvalue, which is only based on the current selection of option contracts.
4. Smoothing: The last step calculates the final IV as the square rootof exponentially-weighted moving average (EWMA) of raw implied variances. The resulting smoothed IV reduces the noise and provides gradualupdating even when the raw values in the previous step become unstable.

3

Data consolidation

The Volmex IV methodology employs multiple data sources and each source
provides raw data in their own format. These data need to be extracted andstandardized in the same format before getting consolidated.

3.1

Data extraction

Source data should at least have bid/ask quotes and mark prices4 for each option
contract, together with a timestamp that indicates when the data are collected.
In addition, some providers have corresponding forward prices associated witheach option contract. These prices are mark prices and we use them when they
4 i.e., the reference price that providers use in P&L calculations2

are available. Section 3.3.1 describes the consolidation process of future prices
to create a yield curve that we use in the Volmex IV calculation.

3.2

Underlying asset price level

We assume that each exchange provides the current value of the underlying
asset price and its timestamp. Given the price and timestamp, we use the mostrecent price of all available prices as the global price and use it in Volmex IV
calculations.

3.3

Future contracts: Interest rate term structure

Option exchanges offer futures trading as well, and these data could be retrieved
when we receive option data.
If data is attached to the option data, i.e., mark prices of corresponding futures or forward prices, then we do not retrieve futures quotes to keep the time
discrepancy at minimum between option and future, and build the interest rate
term structure (risk-free curve) as in Section 3.3.1.
However, if data is not available in the option data set, we retrieve futures
quotes and process as described in Section 3.3.2.
3.3.1

Using mark prices

We calculate the interest rate using the global price level mentioned in Section
3.2 using the formula below.
F = S × er×t

(1)

where F is the forward price, S is the underlying price level, t is year-to-maturity(YTM) and r is the interest rate. Thus, the implied interest rate isrimp =ln F − ln S.
t(2)Once interest rates are calculated for all the future contracts, we calculate theaverage interest rate for each expiry. This gives us the yield curve for theunderlying asset.
3.3.2

Using bid-ask quotes

If future contracts data comes separately, we use bid and ask quotes to consolidate the option data and then build the yield curve based on the formula inequation 2.
To consolidate the data, we collect bid/ask quotes and pick the maximum bid3

and minimum ask for each maturity. We compute mid-price (which is the average of bid and ask prices) as the forward price. After that, using equation 2, wecalculate the interest rates for each maturity therefore finding the yield curve
needed for further steps of Volmex IV calculation.

3.4

Option contracts

Processing the options data that we retrieve from various exchanges5 has twosteps. First, we extract the information from each of the exchange, and then
combine them as a single data set of options. Details of these two steps are
discussed in the following sections.
3.4.1

Source data

1. Select near and next-term options:• Near-term: Options with longest maturity that is less than or equal
to index maturity• Next-term: Options with shortest maturity that is more than indexmaturity
2. Eliminate invalid quotes under the following scenarios:• Negative bid/ask spread6• Mark price is out of bid/ask range7• Mark price is not positive.
We perform these operations for the data of each exchange, and next sectionuses them to create a consolidated option data set to use in the remaining steps.
3.4.2

Consolidate data

1. Combine all data in one set. Since the option data are normalized in theprevious step, we do not have any issues to concatenate these data sets.
2. Select near and next-term options: Providers may have different dates asnear and/or next-term expiry dates. In this case, we choose the expiry
that is listed on the most exchanges.
3. Consolidate bid and ask quotes for each option contract with same type8 ,strike and expiry:• Select quotes: maximum of bids and minimum of asks available.
5 Currently these exchanges are Deribit and OKX
6 Ask price is less than bid price
7 Mark price is less than bid price or greater than ask price
8 i.e., call or put4• Select mark prices: mark price of the option with smallest bid/askspread.
4. Eliminate invalid quotes under the following scenarios:• Negative bid/ask spread.9• Mark price is not positive.
5. Eliminate the quotes with large spreads:• Calculate bid spread as the difference between mark price and bidprice. Set the bid spread to zero if negative.
• Calculate ask spread as the difference between ask price and markprice. Set the ask spread to zero if negative.
• Calculate spread as the sum of bid and ask spreads.
• Calculate the maximum allowed spread (MAS) as the minimum ofbid and ask spreads, multiplied by SPREAD MULTIPLIER10 .
• Calculate the global maximum spread (GMS) as SPREAD MIN11 multiplied by SPREAD MULTIPLIER• Remove the quote if its spread is greater than both GMS and MAS.
6. Calculate mid-price as the average of bid and ask prices.
7. Calculate interest rate12 using yield curve introduced in Section 3.3.
Once all data are consolidated in a single data set, we move to the next stage
where we filter out-of-the-money (OTM) option contracts.

4

Filtering
1. Index maturity is set at 30 days and year convention is 365 days, which
gives index YTM as 30/365 ≈ 0.082192 years.
2. Calculate the implied forward price of the strike that has minimum absolute mid-price difference between call and put options, for near andnext-term options: Fimp = K + F × (C − P ) where F is the forward price,
C is the call option price, P is put option price, and both options arequoted in the amounts of underlying.
3. Set the largest strike that is less than the implied forward Fimp as ATM
strike KAT M for near and next-term options.
9 Ask price is less than bid price

10 Currently set at 10.0
11 Equal to tick size; currently set at 0.0005 for BTC and ETH
12 Use linear piece-wise interpolation and extrapolate further tenors at the last observed
values5

4. Select OTM options with respect to KAT M for each set of near and nextterm options and set their price as mid-price. If both call and put options
are selected for the same strike (i.e., KAT M ), then take the average ofthem.
5. Select the options with strikes greater than Kmin and less than Kmax ,Kmin = Fimp /RANGE MULT
Kmax = Fimp × RANGE MULT
where Fimp ensures that the range of strikes change dynamically. Currently RANGE MULT is set to 2.5. For instance, if Fimp = 56, 000, thenstrikes will be selected between 22, 400 and 140, 000.
6. Sort the options on strikes and eliminate the options after observing fiveconsecutive bid prices that are equal or less than minimum bid threshold.13

5Raw implied variance calculation

Once filtering is done, we first make sure the strike range is always from Kminto Kmax by log-linearly extrapolating the strikes using the slopes between top
point (ATM strike and option price) and last points available on each side (lowest and highest valid strikes and option prices).
Secondly, we extend the number of strikes and option prices using log-linear
piece-wise interpolation in order for the variance formula (equation 3) to be
more stable when the market becomes less liquid.
Near and next-term option prices are then plugged in the implied variance calculation that replicates variance swaps. The implied variance formula14 for nearand next-term options is,2X
1Fi2
2σi,t=wi,j V (Kj ) −−1 (3)Ti
Ki,ATMj∆Kj, i ∈ {NEAR, NEXT}, Fi is the implied forward price,
Kj2
Ki,AT M is the ATM strike level, V (K) is the price15 of OTM option with strikeK, and Ti is years to expiry.
where wi,j = eri Ti

22Once σNEAR,tand σNEXT,tare calculated following the formula above, implied
13 Equal to tick size
14 See Demeterfi et al. (1999)
15 i.e., mid-price6

variance at the index maturity is interpolated using the weighting scheme below:(TNEXT − TINDEX )/TINDEX(TNEXT − TNEAR )/TNEAR(TINDEX − TNEAR )/TINDEXωNEXT,t =(TNEXT − TNEAR )/TNEXTωNEAR,t =(4)(5)

where TINDEX has been set as 30/365.
Thus, the raw value of implied variance at the index maturity is,222σRAW,t= ωNEAR,t σNEAR,t+ ωNEXT,t σNEXT,twhich is calculated continuously at a frequency that we acquire new data.

6Smoothing

Raw implied variance could suffer from noise since it is calculated when a newdata set is available16 and it might be over-sensitive to sudden changes.

6.1

Exponentially-weighted moving average (EWMA)

To eliminate the noise in raw implied variance and capture the fundamental
trends in the market, an EWMA of raw implied variance is employed as shown
below:
222σSMOOTH,t= λσSMOOTH,t−1+ (1 − λ)σRAW,t(6)2where σSMOOTH,t−1
is the previous value of the smoothed implied variance andits weight is the smoothing parameter λ.

6.2

Half-life: Choosing the smoothing parameter

The degree of smoothing is adjusted by the smoothing parameter λ. Equation
6 could be defined recursively, as below:
22σSMOOTH,t= λτ σSMOOTH,t−τ+ (1 − λ)

τ−1X2λτ σRAW,t−ii=0

and we can find the smoothing parameter value that implies a specific half-life17
τ using the derivation below,λτ =1⇒ λ = e− ln(2)/τ .
216 Currently it is every second
17 Currently it is set as 1 minute between 8:30am to 7:30am UTC, and 2 minutes between
7:30am and 8:30am UTC7

Above equation gives λ = 0.97716 when we set the half-life to 30 seconds andimplied variance calculation is performed every second,
0.9771630sec×1calculation/sec = 0.9771630 ≈ 0.5
which simply means that the weight of current smooth variance will be halved
every 30 seconds (i.e., 30 calculations given that calculations are performed
every second) if we choose λ as 0.97716.

6.3

Volmex implied volatility

The Volmex IV of a crypto asset x, abbreviated as xVIV,18 is the squared rootof the smoothed implied variance that has been calculated, multiplied by 100,since it is expressed as percentage points:
q2xVIVt = 100 × σSMOOTH,t

7

Conclusion

This paper introduces a new methodology to calculate the implied volatilityof crypto assets at a given expiry/tenor, and Volmex uses this methodology
at 30 days maturity to calculate each IV index. It has the novel features of:i) consolidating option and futures data across multiple exchanges to create
a global option book, and ii) smoothing using exponentially-weighted moving
averaging, which takes every observation into account while keeping the finalindex moving smoothly.

References
K. Demeterfi, E. Derman, M. Kamal, and J. Zou. A guide to volatility andvariance swaps. The Journal of Derivatives, 6(4):9–32, 1999.
R. E. Whaley. The investor fear gauge. The Journal of Portfolio Management,
26(3):12–17, 2000.

18 e.g. EVIV is for Ethereum (ETH) Volmex IV, and BVIV is Bitcoin (BTC) Volmex IV

8Appendix A

Exceptions

The Volmex IV calculation depends on the health of the data and some exchanges might experience difficulties around 8 a.m. UTC for several reasons.
In addition, option market makers may not always quickly react to the suddenchanges in the market and pull their quotes for a small amount of time for revaluation and re-quoting purposes.
These scenarios may cause significant reduction in the amount of available option quotes in the exchanges and may affect the calculation of raw implied
variance.

A.1

Scenarios

Two scenarios are considered as exceptions:
1. Failure in calculating the raw implied variance numerically
2. Raw implied variance less than Black-Scholes at-the-money (ATM) implied variance (BSIV)In the above scenarios, we use BSIV and Volatility Tail Index (VTI) to calculate
the raw implied variance.

A.2

Variables

Exceptions could happen to three variables:
1. Near-term raw implied variance
2. Next-term raw implied variance
3. Raw implied variance (30-day)
Each of these variables have their own BSIV and VTIs.

A.3

Black-Scholes ATM Implied Volatility (BSIV)

When we calculate the reference ATM strike, KAT M for near and next-term
option contracts, we select the five19 contracts closest to KAT M and calculate
Black-Scholes implied volatility (BSIV) of these contracts, select two smallest of
the available BSIVs, and then calculate the average of them. This gives us nearand next-term BSIVs and we use the weighting scheme in Equations 4 and 5 tocalculate the 30-day BSIV. All BSIVs are also expressed in percentage points.
19 If all of the selected quotes are invalid, try 10 first and then 15.

9A.4

Volatility Tail Index (VTI)VTI measures the percentage difference between the Volmex IV and BSIV. Since
the Volmex IV uses OTM options and BSIV is only informative about ATM
options, the ratio of these two would indicate how tails are priced differently
than central strike region (i.e., ATM). We calculate raw VTIs for near-term,
next-term and 30-day expiries, and then smooth them using the same value of
λ that we use in Equation 6.
VTIraw,t = 100 × (xVIVt /BSIVt − 1)
VTIsmooth,t = λ × VTIsmooth,t−1 + (1 − λ) × VTIraw,twhere xVIV refers to the Volmex IV of crypto asset x.

A.5

Calculation

In case of any exception in the calculation of raw implied variance (of near-term,next-term, or 30-day), we use BSIV and VTI values to calculate raw varianceas shown below:
2
VTIsmooth,t−1BSIVt
2× 1+σRAW,t =100
10010