VIX futures term structure and the expectations hypothesis

Abstract

Tests of the expectations hypothesis reveal that the slope of the VIX futures term structure predicts the direction but not the magnitude of the evolution of the short-end of the curve, but predicts neither the direction nor the magnitude of short-term changes in the long-end of the curve. Relative value seeking spread trades, constructed to exploit such violations, deliver excess returns with annualized Sharpe ratios equal or greater than those of volatility-writing strategies deployed by VIX ETN's for a majority of the 32 spread trade combinations tested. I demonstrate that profits from beta-neutral variations of the spread trades, which are not compensation for taking on equity downside risk by design, are propagated by inflows of capital into VIX futures markets, after controlling for factors that measure changes in the availability of hedge fund capital, risk appetite, and momentum. At the heart of profits, and by extension the term structure anomalies, is a disproportionally elevated basis propagated by long VIX demand that enters the futures market through ETN channels.

Keywords:

• Financial futures
• Implied volatilities
• Quantitative trading strategies
• Term structure
• Expectations hypothesis
• Role of derivatives securities mis-pricing

JEL Codes:

• B23
• G13
• G14

Acknowledgements

I am grateful for invaluable comments and suggestions from Michael Hutchison, Joshua Aizenman, Carl Walsh, Chris Limnios, Jeffrey Pontiff, Eric Fischer, Vlad Sushko, Garrett Lee, Joe Chan, Ludwig Chincarini, Roger Peng, Julian Sebastian, anonymous referees, and conference/seminar participants from University of California, Santa Cruz, Louisiana State University, University of Victoria, University of San Francisco, HSBC, ABF, Bank of International Settlements, Cornerstone Research, Vega Economics, Google, Guggenheim Partners, Goldman Sachs. Research support from the Cota-Robles Foundation and the Sury Initiative for Global Finance and International Risk Management (SIGFIRM) is also gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 The VIX index is commonly referred to as the market's ‘fear gauge’.

2 The co-movement is imperfect for reasons highlighted in Section 3.

3 One reason is that the VIX is constructed using mid-prices on SPX options. Transactions generally do not occur at mid- prices, as mid-prices represent an average of the prevailing bid and ask prices. Investors buy options at the offer and sell at the bid.

4 Alexander and Korovilas (2012) furnish a thorough background

5 The VIX index is an expectation of realized volatility in the S&P 500 index over the next 30 days, in other words, a 30-day variance swap. The VIX term structure, or long dated variance swap, is a projection of realized volatility beyond 30 days.

6 Vega calculation: Number of contracts × futures prices ×$1000.

7 The one and two-factor dynamic equilibrium models of Eraker and Wu (2017), for instance, do well at explaining the near and next-term futures returns, however, explanatory power falls with tenor, and the analysis does not go beyond the fifth tenor.

8 Note, there are some papers in the literature that use the term ‘expectations hypothesis’ when describing tests that explore the existence of risk premia. However, I define the expectations hypothesis as a test of the extent to which the term structure describes the evolution of the VIX itself, as in Johnson (2016).

9 See Poon and Granger (2003) for a thorough survey of volatility forecasting, based on 93 peer-reviewed research articles.

10 The term forecast bias is used in the literature to denote a deviation from EH, or it is used to convey the existence of risk premia (to the extend implied volatility is viewed as a forecast of future realized volatility, under the risk-neutral measure).

11 Maturities occur monthly. In eight out of twelve months in the year, VIX futures settle on the third Wednesday of each month, in the other four months, the futures expire on the fourth Wednesday of the month.

12 Accessible at www.cboe.com/micro/vix/vixwhite.pdf.

13 See for instance, Fama and French (1998), to appreciate the contrast. Also a number of researches have proposed alternative pricing approaches. Zhang and Zhu (2006), for instance, use the Heston stochastic volatility model to price VIX futures. Zhu and Zhang (2007) value VIX futures by applying a stochastic variance model to the evolution of the VIX itself and to deriving the term structure of forward variance. Lin (2007) uses an affine jump-diffusion model with jumps in both index and volatility processes to arrive at VIX futures prices. Sepp (2008) applies a similar framework for calibration of both VIX futures and options on the VIX. Zhang and Huang (2010) highlight the importance of dynamics assumptions and parameter estimation by contrasting the results of a number of different approaches.

14 Note ${\mathcal{F}}_t^1$ references the futures contract with the nearest expiry to t. As time goes by, ${\mathcal{F}}_t^1$ will naturally reference different individual futures contracts as contracts expire and roll down.

15 Other factors that would contribute to flood insurance premiums are cost of capital, interest rates, the general health of the insurance industry, disruption by insurtech players, etc.

16 $\left(\frac{1}{12}\right){0.14}^2+\left(\frac{1}{12}\right){0.1777}^2=\left(\frac{2}{12}\right){0.16}^2$

17 This results in a futures term structure of j = km, where m is generally 1 month.

18 See Blume (1974). Arithmetic returns from high-frequency data (i.e. daily) may overstate the performance of buy and hold strategies, while geometric return calculations tend to understate the actual performance.

19 Calculation: $(1-10\mathrm{\%})\times (1+10\mathrm{\%})=-1.0\mathrm{\%}.$

20 For instance, the cost of a six-month at-the-money option is not two-times the cost of the three-month at-the-money option, both at time t, all else equal. Assuming a flat Black Scholes implied volatility term structure, it is less than two-times, or specifically $\sqrt{\frac{1}{2}}<(2\times \sqrt{\frac{1}{4}})$.

21 Explicitly, gamma is the rate of change of delta, the sensitivity of the option price to changes in spot rates.

22 See Chapters 30–33 in Schwager (1984) for background. Also, Szymanowska et al. (2014) implement this strategy in their analysis of futures risk premia.

23 Assuming a VIX futures price of 22 (see table 1, panel 1), investor incurs bid-ask spread transaction costs of 0.05 on 22.00, or 0.23%, when rebalancing the trade each month. Since the typical month has 21 trading days, this amounts to approximately 0.01% in costs per day.

24 Examples: S&P 500 index, EUR/USD exchange rate, gold.

25 Consumption commodities such as oil and orange juice would also fall in this category, as would bitcoin.

26 For press coverage: https://www.cnbc.com/2018/02/06/the-obscure-volatility-security-thats-become-the-focus-of-this-sell-off-is-halted-after-an-80-percent-plunge.html

27 See Fama and French (1993).

28 It is important to note that changes in open interest is a more direct measure of the flow of funds for VIX ETN's than the market capitalization data of the ETN's. Market capitalization may increase when investment performance is positive, or when new money enters the strategy. Similarly, it is reduced by negative investment performance, as well as redemptions or withdrawals, including fund closures and client defections. Thus, changes in investment performance have the potential to influence market capitalization, such that the true flow of funds is best captured by futures open interest.

29 Alexander and Korovilas (2012) describe this in greater detail. Also for an example of the construction of VIX ETN's, refer to https://www.sec.gov/Archives/edgar/data/312070/000119312512118832/d317408d424b3.htm.

30 Published by Hedge Fund Research, Inc., the HFRXRVA Index tracks the net-asset value managed by relative value investment managers who maintain positions in which the investment thesis is predicated on realization of a valuation discrepancy in the relationship between multiple securities.

31 A dollar invested in the VXX ETN (one of the largest) on its inception date of 1/29/2009 would be worth less than a penny less than 6 years later, as of 12/30/2014.