Volatility Futures and ETNs: Statistics and Trading

Marco Avellaneda

NYU-Courant

Joint work with Andrew Papanicolaou, NYU-Tandon School of Engineering,

Xinyuan Zhang, NYU-Courant, Xinyu Fan, NYU-Courant

IMPA, Research In Options, November 27, 2017

Outline

• VIX Time-Series: Stylized facts/Statistics

• VIX Futures (CBOE)

• VIX ETNS (VXX, XIV, UVXY, SVXY, TVIX, EWZ)

• Modeling the VIX curve and applications to ETN trading/investing

• VSTOXX Futures and ETNs (Deutsche Borse/Eurex)

• Back-testing results

The CBOE S&P500 Implied Volatility Index (VIX)

𝜎𝑇2 =

2𝑒𝑟𝑇

𝑇 0

∞

𝑂𝑇𝑀(𝐾, 𝑇, 𝑆)𝑑𝐾

𝐾2

• Based on Implied Variance (Whaley, 90’s)

• Here 𝑂𝑇𝑀(𝐾, 𝑇, 𝑆) represents the value of the OTM (forward) option with strike K, or ATM if S=F.

• In 2000, CBOE created a discrete version of the IV in which the sum replaces the integral and the maturity is 30 days. Since there are no 30 day options, VIX uses first two maturities*

𝑉𝐼𝑋 = 𝑤1

𝑖=1

𝑛

𝑂𝑇𝑀 𝐾𝑖 , 𝑇1, 𝑆∆𝐾

𝐾𝑖2 + 𝑤2

𝑖

𝑂𝑇𝑀 𝐾𝑖 , 𝑇2, 𝑆∆𝐾

𝐾𝑖2

* My understanding is that recently they could have added more maturities using weekly options as well.

VIX time series- The Fear Gauge of the Market: Jan 1990 to July 2017

Lehman

Brexit

Lehman Bros

Mode

Mean

VIX statistics show heavy right-tail, skewness

VIX Descriptive Stats

Mean 19.51195

Standard Error 0.094278

Median 17.63

Mode 11.57

Standard Deviation 7.855663

Sample Variance 61.71144

Kurtosis 7.699637

Skewness 2.1027

Minimum 9.31

Maximum 80.86

• ``Vol risk premium theory’’: long-dated futures prices should be above the average VIX.

• This implies that the typical futures curve should be upward sloping (contango) since mode<average

Is VIX a stationary process? Yes and no…

• Augmented Dickey-Fuller test rejects unit root if we consider data from 1990 to 2017(Yahoo!Finance.com)

MATLAB adftest(): DFstat=-3.0357; critical value CV= -1.9416; p-value=0.0031.

• Shorter time-windows, data from 2009 to 2017 do not reject unit root

• Non-parametric approach (2-sample KS test) also rejects unit roots if we use 1990 to 2017

VIX Futures (CBOE, symbol:VX)• Contract notional value = VX × 1,000

• Tick size= 0.05 (USD 50 dollars)

• Final settlement price = Spot VIX × 1,000

• Monthly settlements, on Wednesday at 8AM, prior to the 3rd Friday (classical option expiration date)

• Exchange: Chicago Futures Exchange (CBOE)

• Cash-settled (obviously!)

VIXVX1 VX2 VX3 VX4 VX5 VX6

• Each VIX futures covers 30 days of volatility after the settlement date.• Settlement dates are 1 month apart.• Recently, weekly settlements have been added in the first two months.

Settlement dates:

Sep 20, 2017

Oct 18, 2017

Nov 17, 2017

Dec 19, 2017

Jan 16, 2018

Feb 13, 2018

Mar 20, 2018

April 17, 2018

VIX futures 6:30 PM Thursday Sep 14, 2017

Note: Recently introduced weeklies are illiquid and should not be used to build CMF curve

InterConstant maturity futures (x-axis: days to maturity)

Partial Backwardation: French election, 1st round

Term-structures before & after French election

Before election(risk-on)

After election(risk-off)

VIX futures: Lehman week, and 2 months later

Extreme backwardation and high volatility

The VIX futures cycle (a.k.a. Risk-on/Risk-off)

• Market is ``trending/quiet” (risk-on), volatility is low , VIX term structure is in contango (i.e. upward sloping)

• The possibility of market becoming more risky arises; 30-day S&P implied vols rise

• Risk-off: VIX spikes, CMF flattens in the front , then `curls up’, eventually going into backwardation

• Backwardation is usually partial (CMF decreases only for short maturities), but can be total in extreme cases (2008)

• Uncertainty eventually resolves itself, CMF curve drops and steepens, risk-on comes back

• Most likely state (contango) is restored

1. Start here

2. End here

3. Repeat…..

Statistics of VIX Futures are studied with CMF curves

• Constant-maturity futures, 𝑉𝜏, linearly interpolating quoted futures prices

𝑉𝑡𝜏 =

𝜏𝑘+1 − 𝜏

𝜏𝑘+1 − 𝜏𝑘𝑉𝑋𝑘(𝑡) +

𝜏 − 𝜏𝑘

𝜏𝑘+1 − 𝜏𝑘𝑉𝑋𝑘+1(𝑡)

𝑉𝑋𝑘(t)= kth futures price on date t, 𝑉𝑋0= VIX, 𝜏0 = 0, 𝜏𝑘= tenor of kth futures

PCA: quantify CMF fluctuations

• Select standard tenors 𝜏𝑘, 𝑘 = 0, 30, 60, 90, 120,150, 180, 210

• Data: Feb 8 2011 to Dec 15 2016

• Estimate:

• Slightly different from Alexander and Korovilas (2010) who did the PCA of 1-day log-returns.

𝑙𝑛𝑉𝑡𝑖𝜏𝑘 = 𝑙𝑛𝑉

𝜏𝑘 +

𝑙=1

8

𝑎𝑖𝑙Ψ𝑙𝑘

Eigenvalue % variance expl

1 72

2 18

3 6

4 1

5 to 8 <1

Mode is negative

Mode is zero

13.5 %

18.7 %

The most likely curvehas VIX at 13.5

It is highly concave,

15%: 0 to 1 month7%: 1 to 2 months

(Data: Feb 8 2011 to Dec 15 2016)

ETFs/ETNs based on futures: the `equitization’ of VIX

• Funds track an ``investable index’’, corresponding to a rolling futures strategy

• Invest in a basket of futures contracts

• Normalization of weights:

𝑑𝐼

𝐼= 𝑟 𝑑𝑡 +

𝑖=1

𝑁

𝑎𝑖

𝑑𝐹𝑖

𝐹𝑖

𝑖=1

𝑁

𝑎𝑖 = 𝛽, 𝛽 = leverage coefficient

𝑎𝑖 = fraction (%) of assets in ith future

ETFs/ETNs have a target average maturity

• Assume 𝛽 = 1, let 𝑏𝑖= fraction of total number of contracts invested in ith futures:

• The average maturity 𝜃 is typically fixed, resulting in a rolling strategy.

𝑏𝑖 =𝑛𝑖

𝑛𝑗=

𝐼 𝑎𝑖

𝐹𝑖.

𝜃 =

𝑖=1

𝑁

𝑏𝑖 𝑇𝑖 − 𝑡 =

𝑖=1

𝑁

𝑏𝑖𝜏𝑖

Example 1: VXX (maturity = 1M, long VIX futures, daily rolling)

𝑑𝐼

𝐼= 𝑟𝑑𝑡 +

𝑏 𝑡 𝑑𝐹1 + (1 − 𝑏 𝑡 )𝑑𝐹2

𝑏 𝑡 𝐹1 + (1 − 𝑏 𝑡 )𝐹2

Weights are based on 1-MCMF , no leverage

𝑏 𝑡 =𝑇2 − 𝑡 − 𝜃

𝑇2 − 𝑇1

𝜃 = 1 month = 30/360

Weights correspondto CMF (linear int.)

We have

Hence

𝑑𝑉𝑡𝜃 = 𝑏 𝑡 𝑑𝐹1 + (1 − 𝑏 𝑡 )𝑑𝐹2+ 𝑏′ 𝑡 𝐹1 − 𝑏′ 𝑡 𝐹2

𝑑𝑉𝑡𝜃

𝑉𝑡𝜃

=𝑏 𝑡 𝑑𝐹1 + (1 − 𝑏 𝑡 )𝑑𝐹2

𝑏 𝑡 𝐹1 + (1 − 𝑏 𝑡 )𝐹2+

𝐹2 − 𝐹1

𝑏 𝑡 𝐹1 + (1 − 𝑏 𝑡 )𝐹2

𝑑𝑡

𝑇2 − 𝑇1

𝑉𝑡𝜃 = 𝑏 𝑡 𝐹1 + (1 − 𝑏 𝑡 )𝐹2

Dynamic link between Index and CMF equations (long 1M CMF, daily rolling)

𝑑𝐼

𝐼= 𝑟𝑑𝑡 +

𝑏 𝑡 𝑑𝐹1 + (1 − 𝑏 𝑡 )𝑑𝐹2

𝑏 𝑡 𝐹1 + (1 − 𝑏 𝑡 )𝐹2

= 𝑟 𝑑𝑡 +𝑑𝑉𝑡

𝜃

𝑉𝑡𝜃

−𝐹2 − 𝐹1

𝑏 𝑡 𝐹1 + (1 − 𝑏 𝑡 )𝐹2

𝑑𝑡

𝑇2 − 𝑇1

𝑑𝐼

𝐼= 𝑟 𝑑𝑡 +

𝑑𝑉𝑡𝜃

𝑉𝑡𝜃

− 𝜕 ln𝑉𝑡

𝜏

𝜕 𝜏𝜏=𝜃

𝑑𝑡Slope of the CMF isthe relative drift between index and CMF

Example 2 : XIV, Short 1-M rolling futures

𝑑𝐽

𝐽= 𝑟 𝑑𝑡 −

𝑑𝑉𝑡𝜃

𝑉𝑡𝜃

+ 𝜕 ln𝑉𝑡

𝜏

𝜕 𝜏𝜏=𝜃

𝑑𝑡

This is a fund that follows a DAILY rolling strategy, sells futures, targets 1-month maturity

𝜃 = 1 month = 30/360

In order to maintain average maturities/leverage, funds must ``reload’’ on futures contracts which tend to spot VIX and then expire. Hence, under contango, long ETNs decay, short ETNs increase.

Stationarity/ergodicity of CMF and consequences

𝑉𝑋𝑋0 − 𝑒− 𝑟 𝑡 𝑉𝑋𝑋𝑡 = 𝑉𝑋𝑋0 1 −𝑉𝑡

𝜏

𝑉0𝜏 𝑒𝑥𝑝 − 0

𝑡 𝜕 ln 𝑉𝑠𝜃𝑑𝑠

𝜕𝜏

𝑒− 𝑟 𝑡 𝑋𝐼𝑉𝑡 − 𝑋𝐼𝑉0 = 𝑋𝐼𝑉0𝑉0

𝜏

𝑉𝑡𝜏 𝑒𝑥𝑝 0

𝑡 𝜕 ln 𝑉𝑠𝜃𝑑𝑠

𝜕𝜏− 1

If VIX is stationary and ergodic, and 𝐸𝜕 ln 𝑉𝑠

𝜃

𝜕𝜏> 0, static buy-and-hold XIV or

short-and-hold VXX produce sure profits in the long run, with probability 1.

Integrating the I-equation for VXX and the corresponding J-equation for XIV (inverse):

Feb 2009

All data since inception, split adjusted.VXX underwent five 4:1 reverse splits since inception

Since 2016Flash crash

US Gov downgrade

A closer look shows self-similar pattern (last 2 1/2 years)

yuan devaluation

brexit

trump

korea

ukraine war

Note: borrowing costs for VXX are approximately 3% per annumThis means that we still have profitability for shorting after borrowingCosts.

china

brexit

trump

le pen

korea

Modeling the CMF curve dynamics

• VIX ETNs are exposed to (i) volatility of VIX (ii) slope of the CMF curve

• 1-factor model is not sufficient to capture observed ``partial backwardation’’ and ``bursts’’ of volatility

• Parsimony suggests a 2-factor model

• Assume mean-reversion to investigate the stationarity assumptions

• Sacrifice other ``stylized facts’’ (fancy vol-of-vol) to obtain analytically tractable formulas.

`Classic’ log-normal 2-factor model for VIX

𝑉𝐼𝑋𝑡 = 𝑒𝑥𝑝 𝑋1 𝑡 + 𝑋2 𝑡

𝑑𝑋1 = 𝜎1𝑑𝑊1 + 𝑘1 𝜇1 − 𝑋1 𝑑𝑡

𝑑𝑋2 = 𝜎2𝑑𝑊2 + 𝑘2 𝜇2 − 𝑋2 𝑑𝑡

𝑑𝑊1 𝑑𝑊2 = 𝜌 𝑑𝑡

𝑋1 = factor driving mostly VIX or short-term futures fluctuations (slower)

𝑋2 = factor driving mostly CMF slope fluctuations (faster)

These factors should be positively correlated.

Use Q-measure to model CMFs

𝑉𝜏 = 𝐸𝑄 𝑉𝐼𝑋𝜏 = 𝐸𝑄 𝑒𝑥𝑝 𝑋1 𝜏 + 𝑋2 𝜏

Ensuring no-arbitrage betweenFutures, Q = ``pricing measure’’ withMPR

𝑉𝜏 = 𝑉∞ 𝑒𝑥𝑝 𝑒− 𝑘1𝜏 𝑋1 − 𝜇1 + 𝑒− 𝑘2𝜏 𝑋2 − 𝜇2 −1

2 𝑗𝑖=1

2 𝑒− 𝑘𝑖𝜏𝑒−𝑘𝑗𝜏

𝑘𝑖+ 𝑘𝑗𝜎𝑖𝜎𝑗𝜌𝑖𝑗

`Overline parameters’ correspond to Q-measure. Assuming a linear market price of risk, therisk factors X are distributed like OU processes with ``renormalized’’ parameters under Q.

Estimating the model means: find 𝑘1, 𝜇1, 𝑘2, 𝜇2, 𝑘1, 𝜇1, 𝑘2, 𝜇2, 𝜎1, 𝜎2, 𝜌, 𝑉∞ using historical data (P measure)

Estimating the model, 2011-2016 (post 2008)

• Kalman filtering approach

Estimating the model, 2007 to 2016 (contains 2008)

Stochastic differential equations for ETNs (e.g. VXX)

𝑑𝐼

𝐼= 𝑟 𝑑𝑡 +

𝑑𝑉𝑡𝜃

𝑉𝑡𝜃

− 𝜕 ln𝑉𝑡

𝜏

𝜕 𝜏𝜏=𝜃

𝑑𝑡

𝑑𝐼

𝐼= 𝑟 𝑑𝑡 +

𝑖=1

2

𝑒− 𝑘𝑖𝜃𝜎𝑖𝑑𝑊𝑖 +

𝑖=1

2

𝑒− 𝑘𝑖𝜃 𝑘𝑖 − 𝑘𝑖 𝑋𝑖 + 𝑘𝑖𝜇𝑖 − 𝑘𝑖 𝜇𝑖 𝑑𝑡

Substituting closed-form solutionin the ETN index equation we get:

Equilibrium local drift =

𝑖=1

2

𝑒− 𝑘𝑖𝜃 𝑘𝑖 𝜇𝑖 − 𝜇𝑖 + 𝑟 𝜎𝐼2 = 𝑗𝑖=1

2 𝑒− 𝑘𝑖𝜏𝑒 −𝑘𝑗𝜏 𝜎𝑖𝜎𝑗𝜌𝑖𝑗

Results of the Numerical Estimation for VIX ETNsForecast profitability for short VXX/long XIV, in equilibrium

Notes :

(1) For shorting VXX one should reduce the ``excess return’’ by the average borrowing cost which is 3%.It is therefore better to be long XIV (note however that XIV is less liquid, but trading volumes in XIV are

increasing.

(2) Realized Sharpe ratios are higher. For instance the Sharpe ratio for Short VXX (with 3% borrow) from Feb 11To May 2017 is 0.90. This can be explained by low realized volatility in VIX and the fact that the model predictssignificant fluctuations in P/L over finite time-windows.

Jul 07 to Jul 16 Jul 07 to Jul 16 Feb 11 to Dec 16 Feb 11 to Jul 16

VIX, CMF 1M to 6M VIX, 1M, 6M VIX, CMF 1M to 7M VIX, 3M, 6M

Excess Return 0.30 0.32 0.56 0.53

Volatility 1.00 0.65 0.82 0.77

Sharpe ratio (short trade) 0.29 0.50 0.68 0.68

Variability of rolling futures strategies predicted by model (static ETN strategies).

Black=actual historical, colored= simulated paths

Passing from OU factors to CMF curve shapes

Solve for X in the first equation, substitute in second equation.

Expressing VXX dynamics in terms of the shape of the CMF curve as an SDE

• This is an actionable result for quantitative trading.

• Negative drift for high volatility and/or high contango

“Phase Diagram”: X=ln V(30), Y=d lnV(30)/d𝜏

High vol, backwardation

Low vol, contango

𝝁 < 𝟎 (𝒔𝒉𝒐𝒓𝒕 𝑽𝑿𝑿)

𝝁 > 𝟎 (𝒍𝒐𝒏𝒈 𝑽𝑿𝑿)

Dynamic trading of VIX ETNs can produceSharpe ratios as highas 1.7 (not shown)

Low vol, partial backwardation

High vol, contango (very unlikely)

Epilogue: European Volatility Trading

• VSTOXX is the European counterpart of VIX.

• VSTOXX is based on the implied variance of the Eurostoxx 50 Index

• VSTOXX futures have similar terms as VIX contracts (Eurex, symbol= V2TX)

• ETNs and ETFs referencing VSTOXX are much less liquid than U.S. counterparts

VSTOXX remains higher than VIX since inception

European Volatility ETNs: EXIV and EVIX

EVIX tracks a 30-day constant-maturity rolling longfutures strategy and EXIV tracks a 30-day shortfutures strategy. (Launched May 2017)

ETN weights are similar to the US counterparts

EVIX (leverage=1)

a = number of days until FVS1 expiresb = number of days until FVS2 expires

\Weights:

EXIV (leverage=-1)

Weights:

Simulated short EVIX strategy since Sep 2013

Unlike VXX, shorting EVIXloses money from June2014 to June 2016

Brexit

Measuring contango with constant-maturity futures

Measuring V2TX/VSTOXX contango (30-day slope from spot)

“Thresholding‘’ with the S-indicator

- Xinyuan Zhang

ETN

Trading strategy: short VSTOXX ETN only when S is ``large enough’’.

[1] The CBOE Volatility Index- VIX, White Paper, Chicago Board of Options Exchange, http://www.cboe.com/micro/vix/vixwhite.pdf

[2] Alexander, C. and Korovilas, Understanding ETNs on VIX Futures(SSRN, 2012)

[3] Avellaneda, M., and Papanicolaou, A., Statistics of VIX Futures and Applications to Trading Volatility ETNs (SSRN, Sep 2017)

[4] http://www.eurexchange.com/exchange-en/products/vol/vstoxx/vstoxx--futures-and-options .[5] https://www.investing.com/indices/stoxx-50-volatility-3stoxx-eur-historical-data

A few references…

1 M CMF ~ 65%

5M CMF ~ 35%