Volatility Futures and ETNs: Statistics and Trading

Transcription

1 IMPA, Research In Options, November 27, 2017 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

2 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

3 The CBOE S&P500 Implied Volatility Index (VIX) Based on Implied Variance (Whaley, 90 s) σ 2 T = 2erT OTM(K, T, S) dk T 0 K 2 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 = w 1 n i=1 OTM K i, T 1, S K K i 2 + w 2 i OTM K i, T 2, S K K i 2 * My understanding is that recently they could have added more maturities using weekly options as well.

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

5 Mode Mean Lehman Bros

6 VIX statistics show heavy right-tail, skewness VIX Descriptive Stats Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Minimum 9.31 Maximum ``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

7 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= ; critical value CV= ; p-value= 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

8 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 3 rd 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.

9 VIX futures 6:30 PM Thursday Sep 14, 2017 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

10 Inter Constant maturity futures (x-axis: days to maturity) Note: Recently introduced weeklies are illiquid and should not be used to build CMF curve

11 Partial Backwardation: French election, 1 st round

12 Term-structures before & after French election Before election (risk-on) After election (risk-off)

13 VIX futures: Lehman week, and 2 months later Extreme backwardation and high volatility

14 The VIX futures cycle (a.k.a. Risk-on/Risk-off) 1. Start here 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 2. End here 3. Repeat..

15 Statistics of VIX Futures are studied with CMF curves Constant-maturity futures, V τ, linearly interpolating quoted futures prices V t τ = τ k+1 τ τ k+1 τ k VX k (t) + τ τ k τ k+1 τ k VX k+1 (t) VX k (t)= kth futures price on date t, VX 0 = VIX, τ 0 = 0, τ k = tenor of kth futures

16 PCA: quantify CMF fluctuations Select standard tenors τ k, k = 0, 30, 60, 90, 120,150, 180, 210 Data: Feb to Dec Estimate: lnv ti τ k = lnv τ k + 8 l=1 a il Ψ l k Slightly different from Alexander and Korovilas (2010) who did the PCA of 1-day log-returns. Eigenvalue % variance expl to 8 <1

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18 Mode is negative

19 Mode is zero

20 (Data: Feb to Dec ) 18.7 % The most likely curve has VIX at 13.5 It is highly concave, 13.5 % 15%: 0 to 1 month 7%: 1 to 2 months

21 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 N di I = r dt + df i a i F i i=1 a i = fraction (%) of assets in ith future Normalization of weights: N i=1 a i = β, β = leverage coefficient

22 ETFs/ETNs have a target average maturity Assume β = 1, let b i = fraction of total number of contracts invested in i th futures: b i = n i n j = I a i F i. The average maturity θ is typically fixed, resulting in a rolling strategy. θ = N i=1 b i T i t = N i=1 b i τ i

23 Example 1: VXX (maturity = 1M, long VIX futures, daily rolling) di I = rdt + b t df 1 + (1 b t )df 2 b t F 1 + (1 b t )F 2 Weights are based on 1-M CMF, no leverage b t = T 2 t θ T 2 T 1 θ = 1 month = 30/360 Weights correspond to CMF (linear int.) V t θ = b t F 1 + (1 b t )F 2 We have dv t θ = b t df 1 + (1 b t )df 2 + b t F 1 b t F 2 Hence dv t θ V t θ = b t df 1 + (1 b t )df 2 b t F 1 + (1 b t )F 2 + F 2 F 1 b t F 1 + (1 b t )F 2 dt T 2 T 1

24 Dynamic link between Index and CMF equations (long 1M CMF, daily rolling) di I = rdt + b t df 1 + (1 b t )df 2 b t F 1 + (1 b t )F 2 = r dt + dv t θ V t θ F 2 F 1 b t F 1 + (1 b t )F 2 dt T 2 T 1 θ di I = r dt + dv t θ V t ln V t τ τ τ=θ dt Slope of the CMF is the relative drift between index and CMF

25 Example 2 : XIV, Short 1-M rolling futures This is a fund that follows a DAILY rolling strategy, sells futures, targets 1-month maturity θ dj J = r dt dv t θ V + t ln V t τ τ τ=θ dt 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.

26 Stationarity/ergodicity of CMF and consequences Integrating the I-equation for VXX and the corresponding J-equation for XIV (inverse): VXX 0 e r t VXX t = VXX 0 1 V t τ t ln V Vτ exp s 0 0 τ θ ds e r t XIV t XIV 0 = XIV 0 V 0 τ t ln V Vτ exp s t 0 τ θ ds 1 If VIX is stationary and ergodic, and E ln V s θ τ > 0, static buy-and-hold XIV or short-and-hold VXX produce sure profits in the long run, with probability 1.

27 All data since inception, split adjusted. VXX underwent five 4:1 reverse splits since inception Flash crash Since 2016 Feb 2009 US Gov downgrade

28 A closer look shows self-similar pattern (last 2 1/2 years) ukraine war Note: borrowing costs for VXX are approximately 3% per annum This means that we still have profitability for shorting after borrowing Costs. yuan devaluation brexit trump korea

29 korea le pen china brexit trump

30 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.

31 `Classic log-normal 2-factor model for VIX VIX t = exp X 1 t + X 2 t dx 1 = σ 1 dw 1 + k 1 μ 1 X 1 dt dx 2 = σ 2 dw 2 + k 2 μ 2 X 2 dt dw 1 dw 2 = ρ dt X 1 = factor driving mostly VIX or short-term futures fluctuations (slower) X 2 = factor driving mostly CMF slope fluctuations (faster) These factors should be positively correlated.

32 Use Q-measure to model CMFs V τ = E Q VIX τ = E Q exp X 1 τ + X 2 τ Futures, Q = ``pricing measure with MPR Ensuring no-arbitrage between V τ = V exp e k 1 τ X 1 μ 1 + e k 2 τ X 2 μ e k i τ e k j τ 2 ji=1 σ k i + k i σ j ρ ij j `Overline parameters correspond to Q-measure. Assuming a linear market price of risk, the risk factors X are distributed like OU processes with ``renormalized parameters under Q. Estimating the model means: find k 1, μ 1, k 2, μ 2, k 1, μ 1, k 2, μ 2, σ 1, σ 2, ρ, V using historical data (P measure)

33 Estimating the model, (post 2008) Kalman filtering approach

34 Estimating the model, 2007 to 2016 (contains 2008)

35 Stochastic differential equations for ETNs (e.g. VXX) θ di I = r dt + dv t θ V t ln V t τ τ τ=θ dt Substituting closed-form solution in the ETN index equation we get: 2 di I = r dt + e k i θ σ i dw i + i=1 2 i=1 e k i θ k i k i X i + k i μ i k i μ i dt Equilibrium local drift = 2 e k i θ k i μ i μ i + r σ 2 2 I = ji=1 e k i τ e k jτ σ i σ j ρ ij i=1

36 Results of the Numerical Estimation for VIX ETNs Forecast profitability for short VXX/long XIV, in equilibrium 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 Volatility Sharpe ratio (short trade) 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 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.

37 Variability of rolling futures strategies predicted by model (static ETN strategies). Black=actual historical, colored= simulated paths

38 Passing from OU factors to CMF curve shapes Solve for X in the first equation, substitute in second equation.

39 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

40 Phase Diagram : X=ln V(30), Y=d lnv(30)/dτ Low vol, contango High vol, contango (very unlikely) μ < 0 (short VXX) Dynamic trading of VIX ETNs can produce Sharpe ratios as high as 1.7 (not shown) μ > 0 (long VXX) Low vol, partial backwardation High vol, backwardation

41 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

42 VSTOXX remains higher than VIX since inception

43 European Volatility ETNs: EXIV and EVIX EVIX tracks a 30-day constant-maturity rolling long futures strategy and EXIV tracks a 30-day short futures strategy. (Launched May 2017)

44 ETN weights are similar to the US counterparts EVIX (leverage=1) a = number of days until FVS1 expires b = number of days until FVS2 expires \Weights: EXIV (leverage=-1) Weights:

45 Simulated short EVIX strategy since Sep 2013 Unlike VXX, shorting EVIX loses money from June 2014 to June 2016 Brexit

46 Measuring contango with constant-maturity futures

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

48 Thresholding with the S-indicator Trading strategy: short VSTOXX ETN only when S is ``large enough. ETN - Xinyuan Zhang

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50 A few references [1] The CBOE Volatility Index- VIX, White Paper, Chicago Board of Options Exchange, [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] -futures-and-options. [5]

51 1 M CMF ~ 65% 5M CMF ~ 35%