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Gatheral 60, September October 13, 2017 Some remarks on VIX futures and ETNs Marco Avellaneda Courant Institute, New York University Joint work with Andrew Papanicolaou, NYU-Tandon Engineering Outline VIX Time-Series: Stylized

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Some remarks on VIX futures and ETNs

Marco Avellaneda Courant Institute, New York University Joint work with Andrew Papanicolaou, NYU-Tandon Engineering

Gatheral 60, September October 13, 2017

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Outline

  • VIX Time-Series: Stylized facts/Statistics
  • VIX Futures: Stylized facts/Statistics
  • VIX ETNS (VXX, XIV) synthetic (futures, notes)
  • Modeling the VIX curve and implications to ETN trading/investing
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The CBOE S&P500 Implied Volatility Index (VIX)

πœπ‘ˆ

2 = 2π‘“π‘ π‘ˆ

π‘ˆ

∞

π‘ƒπ‘ˆπ‘(𝐿, π‘ˆ, 𝑇) 𝑒𝐿 𝐿2

  • Inspired by Variance Swap Volatility (Whaley, 90’s)
  • Here π‘ƒπ‘ˆπ‘(𝐿, π‘ˆ, 𝑇) 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 VSV 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* π‘Šπ½π‘Œ = π‘₯1

𝑗=1 π‘œ

π‘ƒπ‘ˆπ‘ 𝐿𝑗, π‘ˆ

1, 𝑇 βˆ†πΏ

𝐿𝑗

2 + π‘₯2 𝑗

π‘ƒπ‘ˆπ‘ 𝐿𝑗, π‘ˆ2, 𝑇 βˆ†πΏ 𝐿𝑗

2

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

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VIX: Jan 1990 to July 2017

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Lehman Bros Mode Mean

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VIX Descriptive Statistics

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

  • Definitely heavy tails
  • ``Vol risk premium theory’’ implies 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
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Is VIX a stationary process (mean-reverting)? Yes and no…

  • Augmented Dickey-Fuller test rejects unit root if we consider data since 1990.

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

  • Shorter time-windows, which don’t include 2008, do not reject unit root
  • Non-parametric approach (2-sample KS test) rejects unit root if 2008 is included.
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VIX Futures (symbol:VX)

  • Contract notional value = VX Γ— 1,000
  • Tick size= 0.05 (USD 50 dollars)
  • Settlement price = VIX Γ— 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)

VIX VX1 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.
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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

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

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

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Partial Backwardation: French election, 1st round

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Term-structures before & after French election

Before election (risk-on) After election (risk-off)

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VIX futures: Lehman week, and 2 months later

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A stylized description of the VIX futures cycle

  • Markets are ``quiet’’, volatility is low , VIX term structure is in contango (i.e. upward sloping)
  • Risk on: the possibility of market becoming more risky arises; 30-day S&P implied vols rise
  • 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)
  • Risk-off: uncertainty resolves itself, CMF drops and steepens
  • Most likely state (contango) is restored

Start here End here

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Statistics of VIX Futures Curves

  • Constant-maturity futures, π‘Šπœ, linearly interpolating quoted futures prices

π‘Š

𝑒 𝜐 = πœπ‘™+1 βˆ’ 𝜐

πœπ‘™+1 βˆ’ πœπ‘™ π‘Šπ‘Œπ‘™(𝑒) + 𝜐 βˆ’ πœπ‘™ πœπ‘™+1 βˆ’ πœπ‘™ π‘Šπ‘Œπ‘™+1(𝑒) π‘Šπ‘Œπ‘™(t)= kth futures price on date t, π‘Šπ‘Œ0= VIX, 𝜐0 = 0, πœπ‘™= tenor of kth futures

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1 M CMF ~ 65% 5M CMF ~ 35%

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PCA: fluctuations from average position

  • Select standard tenors πœπ‘™, 𝑙 = 0, 30, 60, 90, 120,150, 180, 210
  • Dates: Feb 8 2011 to Dec 15 2016
  • Slightly different from Alexander and Korovilas (2010) who did the PCA of 1-day log-returns.

π‘šπ‘œπ‘Š

𝑒𝑗 πœπ‘™ =

π‘šπ‘œπ‘Šπœπ‘™ +

π‘š=1 8

π‘π‘—π‘šΞ¨π‘š

𝑙

Eigenvalue % variance expl 1 72 2 18 3 6 4 1 5 to 8 <1

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Mode is negative

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13.5 % 18.7 %

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ETFs/ETNs based on futures

  • Funds track an ``investable index’’, corresponding to a rolling futures strategy
  • Fund invests in a basket of futures contracts
  • Normalization of weights for leverage:

𝑒𝐽 𝐽 = 𝑠 𝑒𝑒 +

𝑗=1 𝑂

𝑏𝑗 𝑒𝐺

𝑗

𝐺

𝑗 𝑗=1 𝑂

𝑏𝑗 = 𝛾, 𝛾 = leverage coefficient 𝑏𝑗 = fraction (%) of assets in ith future

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Average maturity of futures is fixed

  • Assume 𝛾 = 1, let 𝑐𝑗= fraction of total number of contracts invested in ith futures:
  • The average maturity πœ„ is typically fixed, resulting in a rolling strategy.

𝑐𝑗 =

π‘œπ‘— π‘œπ‘˜ = 𝐽 𝑏𝑗 𝐺𝑗 . πœ„ =

𝑗=1 𝑂

𝑐𝑗 π‘ˆπ‘— βˆ’ 𝑒 =

𝑗=1 𝑂

π‘π‘—πœπ‘—

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

𝑒𝐽 𝐽 = 𝑠𝑒𝑒 + 𝑐 𝑒 𝑒𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝑒𝐺 2

𝑐 𝑒 𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝐺 2

Weights are based on 1-M CMF , no leverage 𝑐 𝑒 = π‘ˆ2 βˆ’ 𝑒 βˆ’ πœ„ π‘ˆ2 βˆ’ π‘ˆ

1

πœ„ = 1 month = 30/360 Notice that since we have Hence π‘’π‘Š

𝑒 πœ„ = 𝑐 𝑒 𝑒𝐺 1 + (1 βˆ’ 𝑐 𝑒 )𝑒𝐺 2+ 𝑐′ 𝑒 𝐺 1 βˆ’ 𝑐′ 𝑒 𝐺 2

π‘’π‘Š

𝑒 πœ„

π‘Š

𝑒 πœ„ = 𝑐 𝑒 𝑒𝐺 1 + (1 βˆ’ 𝑐 𝑒 )𝑒𝐺 2

𝑐 𝑒 𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝐺 2

+ 𝐺

2 βˆ’ 𝐺 1

𝑐 𝑒 𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝐺 2

𝑒𝑒 π‘ˆ2 βˆ’ π‘ˆ

1

π‘Š

𝑒 πœ„ = 𝑐 𝑒 𝐺 1 + (1 βˆ’ 𝑐 𝑒 )𝐺 2

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

𝑒𝐽 𝐽 = 𝑠𝑒𝑒 + 𝑐 𝑒 𝑒𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝑒𝐺 2

𝑐 𝑒 𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝐺 2

= 𝑠 𝑒𝑒 + π‘’π‘Š

𝑒 πœ„

π‘Š

𝑒 πœ„ βˆ’

𝐺

2 βˆ’ 𝐺 1

𝑐 𝑒 𝐺

1 + (1 βˆ’ 𝑐 𝑒 )𝐺 2

𝑒𝑒 π‘ˆ2 βˆ’ π‘ˆ

1

𝑒𝐽 𝐽 = 𝑠 𝑒𝑒 + π‘’π‘Š

𝑒 πœ„

π‘Š

𝑒 πœ„ βˆ’

πœ– ln π‘Š

𝑒 𝜐

πœ– 𝜐

𝜐=πœ„

𝑒𝑒

Slope of the CMF is the relative drift between index and CMF

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Example 2 : XIV, Short 1-M rolling futures

𝑒𝐾 𝐾 = 𝑠 𝑒𝑒 βˆ’ π‘’π‘Š

𝑒 πœ„

π‘Š

𝑒 πœ„ +

πœ– ln π‘Š

𝑒 𝜐

πœ– 𝜐

𝜐=πœ„

𝑒𝑒

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

πœ„ = 1 month = 30/360

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

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Stationarity/ergodicity of CMF and consequences

π‘Šπ‘Œπ‘Œ0 βˆ’ π‘“βˆ’ 𝑠 𝑒 π‘Šπ‘Œπ‘Œπ‘’ = π‘Šπ‘Œπ‘Œ0 1 βˆ’

π‘Š

𝑒 𝜐

π‘Š

𝜐 π‘“π‘¦π‘ž βˆ’

𝑒 πœ– ln π‘Š

𝑑 πœ„π‘’π‘‘

πœ–πœ

π‘“βˆ’ 𝑠 𝑒 π‘Œπ½π‘Š

𝑒 βˆ’ π‘Œπ½π‘Š 0 = π‘Œπ½π‘Š π‘Š

𝜐

π‘Š

𝑒 𝜐 π‘“π‘¦π‘ž

𝑒 πœ– ln π‘Š

𝑑 πœ„π‘’π‘‘

πœ–πœ

βˆ’ 1 Proposition: If VIX is stationary and ergodic, and 𝐹

πœ– ln π‘Š

𝑑 πœ„

πœ–πœ

> 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):

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Feb 2009 All data, split adjusted VXX underwent five 4:1 reverse splits since inception Huge volume Flash crash US Gov downgrade

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Taking a closer look, last 2 1/2 years

yuan devaluation brexit trump korea ukraine war Note: borrowing costs for VXX are approximately 3% per annum This means that we still have profitability for shorts after borrowing Costs.

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china brexit trump le pen korea

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Modeling CMF curve dynamics

  • VIX ETNs are exposed to (i) volatility of VIX (ii) slope of the CMF curve
  • We propose a stochastic model and estimate it.
  • 1-factor model is not sufficient to capture observed ``partial backwardation’’ and ``bursts’’
  • f 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.
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`Classic’ log-normal 2-factor model for VIX

π‘Šπ½π‘Œπ‘’ = π‘“π‘¦π‘ž π‘Œ1 𝑒 + π‘Œ2 𝑒 π‘’π‘Œ1 = 𝜏1𝑒𝑋

1 + 𝑙1 𝜈1 βˆ’ π‘Œ1 𝑒𝑒

π‘’π‘Œ2 = 𝜏2𝑒𝑋

2 + 𝑙2 𝜈2 βˆ’ π‘Œ2 𝑒𝑒

𝑒𝑋

1 𝑒𝑋 2 = 𝜍 𝑒𝑒

π‘Œ1 = factor driving mostly VIX or short-term futures fluctuations (slow) π‘Œ2 = factor driving mostly CMF slope fluctuations (fast) These factors should be positively correlated.

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Constant Maturity Futures

π‘Šπœ = 𝐹𝑅 π‘Šπ½π‘Œπœ = 𝐹𝑅 π‘“π‘¦π‘ž π‘Œ1 𝜐 + π‘Œ2 𝜐

Ensuring no-arbitrage between Futures, Q = ``pricing measure’’ with MPR

π‘Šπœ = π‘Šβˆž π‘“π‘¦π‘ž π‘“βˆ’

𝑙1𝜐 π‘Œ1 βˆ’

𝜈1 + π‘“βˆ’

𝑙2𝜐 π‘Œ2 βˆ’

𝜈2 βˆ’

1 2 π‘˜π‘—=1 2 π‘“βˆ’

π‘™π‘—πœπ‘“βˆ’π‘™π‘˜πœ

𝑙𝑗+ π‘™π‘˜

πœπ‘—πœ

π‘˜πœπ‘—π‘˜

`Overline parameters’ correspond to assuming a linear market price of risk, which makes the risk factors X distributed like OU processes under Q, with ``renormalized’’ parameters. Estimating the model means finding 𝑙1, 𝜈1, 𝑙2, 𝜈2, 𝑙1, 𝜈1, 𝑙2, 𝜈2, 𝜏1, 𝜏2, 𝜍, π‘Šβˆž using historical data

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Estimating the model, 2011-2016 (post 2008)

  • Kalman filtering approach
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Estimating the model, 2007 to 2016 (contains 2008)

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

𝑒𝐽 𝐽 = 𝑠 𝑒𝑒 + π‘’π‘Š

𝑒 πœ„

π‘Š

𝑒 πœ„ βˆ’

πœ– ln π‘Š

𝑒 𝜐

πœ– 𝜐

𝜐=πœ„

𝑒𝑒 𝑒𝐽 𝐽 = 𝑠 𝑒𝑒 +

𝑗=1 2

π‘“βˆ’

π‘™π‘—πœ„πœπ‘—π‘’π‘‹ 𝑗 + 𝑗=1 2

π‘“βˆ’

π‘™π‘—πœ„

𝑙𝑗 βˆ’ 𝑙𝑗 π‘Œπ‘— + π‘™π‘—πœˆπ‘— βˆ’ 𝑙𝑗 πœˆπ‘— 𝑒𝑒

Substituting closed-form solution in the ETN index equation we get:

Equilibrium local drift =

𝑗=1 2

π‘“βˆ’

π‘™π‘—πœ„

𝑙𝑗 πœˆπ‘— βˆ’ πœˆπ‘— + 𝑠 𝜏𝐽

2 = π‘˜π‘—=1 2

π‘“βˆ’

π‘™π‘—πœπ‘“ βˆ’π‘™π‘˜πœ πœπ‘—πœ π‘˜πœπ‘—π‘˜

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Results of the Numerical Estimation for VIX ETNs:

Model’s prediction of 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 11 To May 2017 is 0.90. This can be explained by low realized volatility in VIX and the fact that the model predicts significant 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

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

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Model also applies to dynamic asset- allocation

  • Assuming HARA (power-law), Merton’s problem reduces to solving

Linear-quadratic Hamilton Jacobi Bellman equation, which has an explicit solution.

  • We find that optimal investment in VIX ETPs, then looks like

πœ„(π‘Œ1, π‘Œ2) = π‘‘π‘π‘œπ‘‘π‘’. 𝜏𝐽

2

𝜏𝐽

2

2 βˆ’ πœ–π‘šπ‘œπ‘Šπœ ð𝜐 + π΅π‘Œ1 + πΆπ‘Œ2 =

1 𝜏𝐽

2 𝑏0 + 𝑏1

πœ–π‘šπ‘œπ‘Šπœ ð𝜐

+ 𝑏2 π‘šπ‘œπ‘Šπœ

β€˜myopic’ drift HJB term

Conclusion : Trading strategies should be `learnt’ from the (i) slope of the curve AND (ii) the VIX level.

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Happy birthday Jim!