Can choice merchants take a sq. root? Surprisingly, possibly not. This publish exhibits that VIX futures costs exhibit important deviations from their option-implied higher bounds—the sq. root of variance swap ahead charges—thus violating the legislation of 1 value, a elementary idea in economics and finance. The deviations widen during times of market stress and predict the returns of VIX futures. Simply because the inventory market struggles with multiplication, the fairness volatility market seems unable to take a sq. root at instances.
The Legislation of One Worth and Anomalies
The legislation of 1 value (LOOP) states that belongings with similar payoffs will need to have the identical value. If belongings with similar payoffs have totally different costs, aggressive merchants will exploit the deviations to make arbitrage earnings, thereby eliminating LOOP violations.
Monetary markets are a chief testing floor for LOOP. With low transaction prices and the flexibility to purchase and promote related securities, conventional theories predict that LOOP ought to maintain broadly throughout monetary market settings. A rising literature, nonetheless, challenges this view. There at the moment are well-documented examples of LOOP violations in equity, fixed income, credit, and currency markets.
Documenting LOOP violations improves our understanding of monetary markets. When belongings with carefully associated payoffs commerce at considerably totally different costs for extended durations of time, there have to be market frictions or inefficiencies that impede arbitrage exercise, even for stylish merchants and monetary establishments. By uncovering these frictions, we will higher interpret market costs and developments.
Fairness Volatility No-Arbitrage Deviation Measure
This publish and the related staff report research relative pricing in fairness volatility markets by evaluating the costs of VIX futures to variance swap ahead charges implied by S&P 500 index choices (“artificial variance swap forwards”). The VIX futures and index choices markets present a pure setting to check LOOP as a result of they provide practically redundant securities upon which arbitrage pricing locations tight restrictions.
Specifically, VIX futures pay the distinction between the futures value and the VIX index at maturity. The VIX index is defined because the sq. root of a one-month artificial variance swap price. The swap price is “artificial” as a result of it’s computed from a portfolio of S&P 500 index choices. This relationship binds collectively the pricing of VIX futures and index choices.
It follows that VIX futures costs are bounded above by the sq. root of variance swap ahead charges. This motivates defining the no-arbitrage deviation measure,
The deviation measure is the distinction between the n month futures value and its higher certain, the one-month variance swap ahead price beginning on the futures expiration date, expressed in annualized volatility items.
The deviation measure has a simple interpretation. When the measure is optimistic, the futures value is above its higher certain. This means a LOOP violation wherein a dealer can promote futures and pay mounted in forwards to lock in a riskless revenue. When the deviation measure is damaging and huge in magnitude, the futures contract is reasonable relative to the ahead price.
The chart beneath stories the deviation measure averaged throughout the entrance six futures contracts from March 26, 2004 to June 30, 2020. The deviation measure reveals substantial variability over time. Optimistic deviations reveal the presence of LOOP violations. Destructive deviations are sometimes giant in magnitude, typically greater than 2-3 p.c in dimension even after averaging throughout contracts.
To place the deviation measure in context, VIX futures bid-ask spreads have averaged round 0.05 p.c in recent times. For the damaging deviations, estimates of the distinction between VIX futures higher and decrease bounds are round 1.7 p.c on common in response to a dynamic term-structure model, indicating that most of the damaging deviations are LOOP violations as effectively.
Along with their giant dimension, the chart reveals that the no-arbitrage deviations have a tendency to say no round durations of elevated systematic danger, such because the monetary disaster and COVID-19 disaster, in addition to excessive frequency occasions just like the fairness flash crash and VIX futures spike in February 2018. When danger will increase, the decline within the deviation measure signifies that VIX futures costs enhance by lower than variance swap ahead charges implied by index choices.
Frequency of LOOP Violations
The desk beneath presents the frequency of LOOP violations by contract for various thresholds. Panel A exhibits that higher certain violations are most frequent for the entrance three contracts with LOOP violations on greater than 10 p.c of days. Panel B exhibits that decrease certain violations are most frequent for longer-dated contracts and happen for round 25 p.c of observations. The decrease certain is estimated because the option-implied higher certain minus the distinction in higher and decrease bounds from a term-structure mannequin.
The outcomes point out that VIX futures exhibit frequent LOOP violations relative to their no-arbitrage bounds that persist even for big thresholds. The workers report offers additional proof that the deviation measure is figuring out an arbitrage unfold by exhibiting that the deviation measure robustly predicts the returns of VIX futures in extra of variance swap forwards.
Deviation Measure versus Danger and Demand
What drives the deviation measure and legislation of 1 value violations? If arbitrageurs are danger averse or have restricted capital, demand shocks can push costs away from elementary values. As well as, when danger will increase, financing constraints might bind extra tightly and drive arbitrageurs to exit positions.
The plot beneath investigates how the deviation measure pertains to danger and demand components. It plots the typical deviation towards the VIX Index and supplier web place in VIX futures from the Commodity Futures Buying and selling Fee’s Dedication of Merchants Report (CoT Report) normalized by open curiosity. Whereas the VIX index is a well known danger issue, earlier research spotlight supplier web positions in VIX futures as an essential demand variable that impacts futures prices and risk premia, and probably serves as a veil for retail hedging demand for VIX Change-Traded-Merchandise (ETPs).
The plots recommend that the deviation measure is reducing in systematic danger and rising in demand to purchase VIX futures. Because the deviation measure tracks the distinction in costs for practically similar claims throughout two markets, it’s stunning to discover a damaging relationship with danger, as danger ought to play the same function in each markets. One speculation for the outcomes is that hedgers might take revenue on lengthy positions in VIX futures when danger will increase, resulting in demand shocks which are correlated with will increase in systematic danger. The workers report offers vector autoregression and panel regression evaluation that finds assist for this speculation.
Deviation Measure throughout COVID-19 Pandemic
How did the deviation measure react throughout the COVID-19 pandemic? The determine beneath plots the deviation measure averaged throughout contracts towards a one-week forward forecast from a vector autoregression with the VIX index and supplier web place in VIX futures as variables. The deviation measure exhibited a muted response throughout the early phases of the pandemic, much like the discovering from a associated article exhibiting that VIX futures didn’t anticipate or present early warnings indicators for a way a lot the VIX index would enhance. Even because the VIX elevated from about 15 on February 18 to 47 on March 10, the deviation measure remained according to the forecast. It was not till the VIX index reached all-time highs in mid-March that the deviation measure declined considerably and produced a big forecast error. After March, the deviation measure remained low all through April and early Could and tended to shock to the draw back in comparison with the forecast, indicating even bigger dislocations than regular within the relative pricing of VIX futures and index choices.
In abstract, VIX futures exhibit systematic LOOP deviations from index choices. This result’s stunning as a result of the no-arbitrage relationship behind the deviation measure is well-known to choice merchants. The proof means that the deviations are associated to systematic danger and demand stress. Throughout the COVID-19 disaster, the deviations widened to all-time lows and revealed the presence of enormous dislocations within the relative costs of VIX futures and index choices that persevered into June 2020.
Peter Van Tassel is an economist within the Federal Reserve Financial institution of New York’s Analysis and Statistics Group.
Charles Smith is a senior analysis analyst within the Financial institution’s Analysis and Statistics Group.
How one can cite this publish:
Peter Van Tassel and Charles Smith, “The Legislation of One Worth in Fairness Volatility Markets,” Federal Reserve Financial institution of New York Liberty Avenue Economics, February 1, 2021, https://libertystreeteconomics.newyorkfed.org/2021/02/the-law-of-one-price-in-equity-volatility-markets.html.
The views expressed on this publish are these of the authors and don’t essentially mirror the place of the Federal Reserve Financial institution of New York or the Federal Reserve System. Any errors or omissions are the duty of the authors.