Volatility Risk Premia in Crypto Options

Introduction

Options on bitcoin and ether have usually priced more volatility than followed. The gap between implied volatility and the realised volatility that comes after it is the simplest measure of the volatility risk premium. It is well documented in equity index options, and recent research finds it in bitcoin options. This paper measures it for BTC and ETH on Deribit, historically the main venue for crypto options, from April 2021 to October 2026.

The paper covers three things: the size of the gap, in volatility points and as a share of implied variance, set against the S&P 500; how the gap has varied with the market regime and over time; and how often, by how much and from what starting points realised volatility has run above implied. Implied volatility is Deribit's 30-day DVOL index. Realised volatility is measured over the following 30 days from Deribit's daily settlement prices. Each day from 1 April 2021 to 25 September 2026 starts one window, 2,004 per asset. All data are public and the analysis is reproducible.

These are market statistics, not the results of a strategy. The gap is not what an option seller receives. A seller's outcome depends on strikes and expiries, delta hedging, the path of prices, costs, margin and settlement currency, none of which is modelled here. No figure in this paper is the return of an investment strategy, actual or hypothetical, or of any Hedgehog Capital service. None of the statistics is an output of any Hedgehog model.

Hedgehog Capital is an AI-first global macro research-and-risk service for professional and high-net-worth investors. Clients keep capital on their own exchange accounts, where each client's dedicated Hedgehog AI trades. Hedgehog never holds client assets and is not a fund. This is one of five Hedgehog Capital white papers; the other four are in preparation.

Background and evidence

What the premium is. A variance swap exchanges realised variance over a period for a rate fixed at the start. Carr and Wu show that the 30-day variance swap rate can be synthesised from a strip of options, with an error that is third order in the size of any jump, and define the variance risk premium against that rate. In the 2004 working-paper version, on S&P 500 options from January 1996 to February 2003, mean realised volatility was 18.8, the synthetic variance swap rate 24.4 and the volatility swap rate, proxied by 30-day at-the-money implied volatility, 21.3. Market beta and the Fama-French factors explained little of the index premium.

Looking back or looking forward. Bollerslev, Tauchen and Zhou subtract the previous month's realised variance from the squared VIX, so their premium is known at the time. From January 1990 to December 2007, implied variance averaged 33.23 and realised variance 14.93 in monthly percentage-squared units, a premium of 18.30, more than half of implied variance. This paper uses the forward-looking gap: implied volatility against what followed.

What the premium pays for. Bollerslev and Todorov find that compensation for rare events accounts for a large fraction of the S&P 500 equity and variance premia. Bakshi and Kapadia read the premium from delta-hedged S&P 500 options and find it negative, more so when volatility is high, even after allowing for fear of jumps. Dew-Becker, Giglio, Le and Rodriguez find that in variance swaps from 1996 to 2014 only transitory, unexpected realised variance was priced, not news about variance one month to 14 years ahead. On that evidence the premium is compensation for realised variance over the contract, which suggests it sits at the short end, the horizon DVOL measures. Cheng finds that the premium in VIX futures falls or stays flat when risk rises, and that falling premiums have predicted rising risk. Carrying that result to options is an inference.

Crypto options. Alexander, Chen and Imeraj (2021, revised 2022) report that over 90% of exchange trading in crypto options had always been on Deribit, which then listed only inverse, coin-settled products. Deribit held nearly 90% of bitcoin options open interest when it launched DVOL on 31 March 2021 (The Block). Alexander, Deng, Feng and Wan find that bitcoin options trade far more short-dated contracts than S&P 500 options, with a call-put ratio of about 54:46 against about 4:6, and a much more symmetric smile that prices jumps in both directions. They argue that jump risk and thinner liquidity make market makers' inventory harder to hedge.

The closest precedent. Almeida, Grith, Miftachov and Wang study Deribit bitcoin options from July 2017 to December 2022 at a 27-day horizon, against backward-looking realised variance. Annualised implied variance averaged 0.72 and realised variance 0.58, a premium of 0.14. The authors compare this with about 0.02 for the S&P 500 in Bollerslev, Tauchen and Zhou's estimates: much larger in level, a smaller share of implied variance. In their high-volatility regime (649 days) the three figures were 0.86, 0.74 and 0.12; in their low-volatility regime (368 days), 0.46, 0.29 and 0.17. They conclude that bitcoin investors appear more concerned with variance and upside risk when volatility is low.

Practitioner evidence. Block Scholes reported on Deribit Insights that BTC and ETH realised volatility hit all-time lows in 2023, and that the ratio of BTC at-the-money implied volatility to 90-day realised volatility fell below one for several months. Ahead of the spot ETF decision, a premium on later expiries put a kink in the term structure (Block Scholes, January 2024). Its weekly reports found BTC risk reversals skewed towards calls at all tenors on 7 January 2025 (Week 2) and a put premium at all tenors on 25 December 2025 (Week 52). The skew changed sign within one year.

What DVOL measures. Deribit's methodology selects the two expiries either side of 30 days, prices options from depth-weighted mid quotes, applies a variance-swap calculation and interpolates in time. It discards in-the-money options and options with a delta below 5%, computes a raw value every second and publishes an exponential moving average over 240 points. Option prices enter in coin terms, not US dollars. Dividing DVOL by the square root of 365 gives the implied one-standard-deviation daily move, which Deribit describes as an estimate of the expected daily move. Deribit may change the parameters at 48 hours' notice.

Three consequences follow. First, DVOL is a variance rate. Demeterfi, Derman, Kamal and Zou show that skew lifts the fair variance above at-the-money volatility, and Carr and Wu that the variance swap rate exceeds squared at-the-money implied volatility. A DVOL gap therefore overstates the gap at the money, and a variance swap also carries exposure to the wings. Second, DVOL omits the far wings, which on its own lowers it relative to a full-strip variance rate. Cboe's VIX methodology extends through put strikes until it meets two consecutive zero bids; DVOL stops at a 5% delta. The first two effects partly offset. Third, DVOL describes Deribit alone. Separately, because DVOL is the square root of a variance rate, it would exceed average realised volatility even if it forecast realised variance without bias (Jensen's inequality).

Data and method

Data. All data come from public endpoints, downloaded without API keys and stored unchanged.

• Deribit API: DVOL hourly and daily candles (get_volatility_index_data) from 24 March 2021; daily delivery prices (get_delivery_prices), the settlement prices for expiring futures and options, treated as 08:00 UTC values; hourly funding and index prices for BTC-PERPETUAL and ETH-PERPETUAL (get_funding_rate_history).
• Binance USD-M 8-hourly funding rates for BTCUSDT and ETHUSDT (Binance API).
• Coinbase Exchange daily candles for BTC-USD and ETH-USD (Coinbase Exchange API), as a check.
• FRED VIXCLS and SP500. FRED keeps ten years of daily S&P 500 history, which starts on 24 October 2016. The S&P 500 series is copyright S&P Dow Jones Indices and the VIX series copyright Cboe.

Definitions.

• Implied: DVOL at 08:00 UTC on the start day.
• Realised (primary): sqrt(365/30 x sum of the 30 squared daily log changes in the delivery price over the next 30 days) x 100. Zero-mean and in calendar days, as for a variance swap. Checks use the 720 hourly changes in the Deribit index and Coinbase daily closes.
• Gap: implied minus realised, in volatility points. A negative gap, where realised exceeded implied, is a shortfall; a run of consecutive shortfall start days is an episode.
• Variance-gap share: one minus mean realised variance over mean implied variance (DVOL squared). Unlike the gap in points, it is not raised by the difference between volatility and variance.
• Ex-ante spread: DVOL minus the previous 30 days' realised volatility.
• Funding regime: the trailing 7-day mean of the 8-hour funding rate at 08:00 UTC.
• Scaled move: the daily log change in the delivery price divided by DVOL/100/sqrt(365).
• S&P 500 gap: the VIX close minus realised volatility over the next 21 trading days, annualised with 252.

Sample and statistics. Start days run from 1 April 2021, the day after DVOL's launch, to 25 September 2026: 2,004 per asset. The rank regime compares each day's DVOL with up to 364 previous daily readings, needs at least 179, and covers 1,833 start days from 19 September 2021. Overlapping windows share 29 of their 30 days, so means carry Newey-West standard errors (Bartlett kernel, 30 lags) and confidence intervals come from a moving-block bootstrap (30-day blocks, 2,000 draws). A Bartlett kernel gives little weight to the longest overlaps, so we also report t-statistics on non-overlapping windows. Taking every 30th start day, at each of 30 offsets, gives 66 or 67 non-overlapping windows. The sample therefore holds about 67 non-overlapping windows per asset, about 12 a year. Volatility clusters, so even these are not fully independent.

Tercile and decile cut-offs use the full sample and so involve hindsight; the rank against previous readings does not. Splits by the price change over a window condition on the outcome. Event splits compare windows ending before a date with windows starting on or after it. Every statistic is a market statistic, and none is converted into the outcome of trading options.

Findings: the size of the gap

Over 2,004 start days, BTC's DVOL averaged 59.5 and subsequent realised volatility 50.7: a mean gap of 8.8 points and a median of 10.6, with implied above realised on 71.9% of days. ETH's DVOL averaged 73.5 against 67.6: a mean gap of 5.9, a median of 8.0 and 66.0% of days (Table 1). In variance terms, mean realised variance was 25.4% below mean implied variance for BTC and 12.8% below for ETH.

The gap is clear for BTC, marginal for ETH and noisy for both. Its 5th and 95th percentiles were -17.9 and 34.6 points for BTC and -32.4 and 39.8 for ETH. The Newey-West t-statistics were 6.0 and 2.9. On non-overlapping windows they range from 3.8 to 5.5 for BTC and from 1.9 to 2.7 for ETH across the 30 start offsets, so the ETH mean is only marginally significant. The 95% bootstrap intervals run from 6.0 to 11.8 points for BTC and 2.2 to 10.0 for ETH. Across the 30 sets of non-overlapping windows, BTC's mean gap had a median of 9.0 (range 7.4 to 10.2) and ETH's 5.8 (4.7 to 7.3).

Part of the gap in points is a matter of units. Because DVOL is the square root of a variance rate, it would exceed average realised volatility even if it forecast realised variance without bias (Jensen's inequality). Measured over the full sample, this effect could account for up to 3.1 points of BTC's 8.8-point mean gap and 4.2 of ETH's 5.9. Measured within DVOL deciles, it is about 1.8 and 2.6 points. The same effect tends to raise the share of days with implied above realised. The variance-gap share does not have this bias.

The estimator matters most in the tails (Table 2). Hourly index data give mean gaps of 7.5 for BTC and 4.9 for ETH, and Coinbase closes 8.1 and 4.3. Daily and hourly realised volatility correlate at 0.90 and 0.92. In crash windows with intraday reversals the hourly measure is far higher: ETH's worst gap is -54.0 on settlement prices and -102.5 on hourly data, both in late April 2021.

Bigger in points, smaller as a share over the same period. Over the same period the VIX exceeded subsequent S&P 500 realised volatility by 4.0 points on average, on 85.9% of start days (Table 3). The crypto gaps were larger in points. As a share of implied variance they were smaller: 29.4% for the S&P 500, 25.4% for BTC and 12.8% for ETH. Realised ran more than 25% above implied on 4.5% of S&P 500 start days, against 11.7% for BTC and 16.0% for ETH.

The comparison depends on the period. From October 2016, a sample that includes the 2020 sell-off, the S&P 500 variance-gap share was 17.7%, below BTC's, and its worst gap -72.0. Equity gaps are also lumpier, with skewness of -2.29 over the common period against 0.04 for BTC. Over the period shared with Almeida and co-authors, start days from 1 April 2021 to 31 December 2022, BTC implied variance averaged 0.68 and realised 0.47, a gap of 0.21 and a share of 30.8%. ETH's figures were 1.02, 0.81, 0.21 and 20.8%. The constructions differ, so the comparison is indicative.

Findings: compression over time

The gap has narrowed (Tables 4 and 5). BTC's mean gap fell from 18.5 points for 2021 start days to 4.5 in 2025 and 3.0 in 2026 to 25 September, and its variance-gap share from 34.7% to 7.7%. ETH's mean gap fell from 17.6 to 4.6 and was negative in 2025, at -3.6, with implied above realised on only 47.7% of days, even though ETH's DVOL was higher in 2025 than in 2023 or 2024.

Part of this reflects lower implied levels. BTC's mean DVOL fell from 91.5 for 2021 start days to 43.0 for 2026 start days to 25 September (Table 4), with a lowest 2025 reading at 08:00 UTC of 33.6 on 19 September, and the gap scales with the implied level, as the regime findings show. ETH in 2025 does not fit that pattern. The yearly evidence is thin. BTC's Newey-West t-statistic fell from 4.7 in 2021 to 0.9 in 2026; each full year rests on about 12 non-overlapping windows, and 2021 and 2026 on about 9. Hourly data point the same way, with BTC mean gaps of 15.6, 12.2, 6.9, 6.0, 3.6 and 0.8 from 2021 to 2026.

Market structure changed too. The SEC approved the listing and trading of spot bitcoin ETPs on 10 January 2024 (SEC). On 23 May 2024 it approved exchange rule changes to list and trade eight spot ether ETPs (Release No. 34-100224). Nasdaq listed options on BlackRock's iShares Bitcoin Trust (IBIT) on 19 November 2024 (Markets Media, citing Nasdaq). Table 6 compares windows before and after each date.

Every split shows a narrower gap afterwards, but DVOL was lower in each later period, so the splits mix the event with the level effect. The compression coincided with lower DVOL, spot ETPs and listed IBIT options. These data cannot separate the causes.

Findings: regime-conditional magnitude

The gap scaled with implied volatility. In BTC's top DVOL decile, 87.7 to 143.7, the mean gap was 27.6 points and implied exceeded realised on 98.0% of start days (Table 7). All 201 of those start days fell between April 2021 and November 2022, about seven non-overlapping windows from a single period, and the decile boundaries use the full sample. In the bottom decile, 32.2 to 38.3, the mean gap was -1.8 and the share 46.8%.

A linear fit gives 0.38 points of extra BTC gap per point of DVOL (t-statistic 6.3, R-squared 0.21) and 0.35 for ETH (t-statistic 4.7, R-squared 0.14). Realised volatility rose by only 0.62 and 0.65 points per point of DVOL, so DVOL overstated subsequent realised volatility most at high levels. It still explained 41% of the variation in subsequent BTC realised volatility and 37% for ETH, against 22% and 29% for trailing realised volatility.

ETH's decile means were 2.0, 3.0, 1.4, -2.1, 0.2, 2.2, 6.7, 2.5, 16.4 and 27.1 points: thin below a DVOL of about 93 and concentrated in the top two deciles. By full-sample DVOL tercile the mean gap was 2.7, 7.4 and 16.2 points for BTC and 1.7, 1.5 and 14.6 for ETH. Terciles of trailing 30-day realised volatility, also on full-sample cut-offs, give the same ordering: mean gaps of 3.3, 5.9 and 17.0 points for BTC and 1.8, 5.3 and 10.6 for ETH.

The ordering holds on a measure observable at the time. Table 8 ranks each day's DVOL against up to 364 previous daily readings, as an observer could at the time. This describes the sample. It is not a tested trading rule, and start days within a regime overlap heavily. In BTC's top third the mean gap was 14.7, implied exceeded realised on 89.0% of days and the worst gap was -19.1. In the bottom third the figures were 4.3, 61.7% and -32.3. BTC spent 966 of 1,833 start days in the bottom third of its own range, a sign of DVOL's falling trend.

Calm regimes held the deepest BTC shortfalls. The low full-sample DVOL tercile contained the worst BTC gap and the highest share of deep shortfalls: gaps below -10 on 20.4% of days, against 10.2% in the high tercile. In the real-time rank regime the figures were 20.5% and 3.6%. This differs from Almeida and co-authors, who found a higher premium in their low-volatility regime over 2017 to 2022. Their regimes cluster option-implied densities and their realised variance looks backward; ours sort on DVOL and look forward over 2021 to 2026. Neither result is general. Both show that the premium depends on the regime.

The backward-looking spread said little. DVOL minus trailing 30-day realised volatility, an ex-ante measure in the spirit of Bollerslev, Tauchen and Zhou, averaged 8.3 points for BTC and was positive on 78.5% of days. Its correlation with the gap that followed was 0.10 for BTC and 0.21 for ETH, with regression R-squared values of 0.01 and 0.04. The level of DVOL was the better guide.

Negative funding has coincided with wider gaps. Binance funding was positive on 90.4% of BTC and 88.3% of ETH start days. On the 192 BTC days with negative trailing funding, the mean gap was 14.7 points and implied exceeded realised on 88.0% of days, against 8.1 points and 70.2% otherwise (Table 9). Deribit's perpetual funding gives the same ordering.

Negative funding coincided with higher DVOL: 62.4 against 59.2 for BTC, and 85.8 against 71.9 for ETH. For BTC the pattern survives within DVOL terciles. In the low tercile the mean gap was 10.3 points with negative funding (60 days) and 2.0 with positive funding (608 days). ETH is mixed: in its middle tercile, negative funding coincided with a gap of -3.4 (76 days). Funding levels correlated at only -0.05 (BTC) and -0.13 (ETH) with the subsequent gap, and the negative-funding samples are small. In a BIS working paper, Schmeling, Schrimpf and Todorov find that carry in crypto futures, the futures-spot basis, is large and time-varying, that it is driven by trend-chasing smaller investors and scarce arbitrage capital, and that high carry has predicted price crashes. Perpetual funding, used here, is a related but different measure.

Findings: tail risk

Shortfalls were frequent and clustered. Realised exceeded implied on 28.1% of BTC and 34.0% of ETH start days. BTC had 42 shortfall episodes, with a median length of 10 start days and a maximum of 34. ETH had 52, the longest running 59 start days from January to March 2025. Table 10 lists the largest.

Calm markets hid the worst BTC shortfall. The window starting 29 January 2026 opened with DVOL at 38.0, in BTC's bottom decile. Realised volatility over the next 30 days was 70.2 as the settlement price fell 27.5%. BTC settled at 88,204 on 29 January and 65,018 on 6 February, when DVOL stood at 71.9 at 08:00 UTC, its highest 08:00 reading of 2026, having touched 90.3 intraday on 5 February. BTC's lowest February settlement price was 63,127 on 24 February. ETH fell from 2,957 to 1,903 between 29 January and 6 February. In four of the five worst BTC episodes, DVOL was below 50 on the worst start day.

Shortfalls came in both directions. Of BTC start days with realised above implied, 57.4% saw the price fall and 42.6% saw it rise; ETH's split was the same. The window from 26 February 2024 produced a BTC gap of -19.1 points as the price rose 35.9%. Table 11 shows the gap widest when prices moved least. It was negative after falls of more than 20% and thin after rallies of more than 20%, at 4.0 points for BTC and 3.3 for ETH. The gap correlated at -0.36 (BTC) and -0.39 (ETH) with the absolute price change, and only 0.16 and 0.19 with the signed change. This fits the more symmetric bitcoin smile: upside jumps mattered as well as downside ones.

DVOL overstated typical daily moves and understated rare ones. Dividing each day's price change by the daily move DVOL implied gives a scaled move. If DVOL were accurate and moves normal, its standard deviation would be one. It was 0.889 for BTC and 0.955 for ETH, and 80.1% of BTC days fell within one implied standard deviation, against 68.3% under a normal distribution. The tails were far heavier. BTC moved more than four implied standard deviations on 8 days, against 0.1 expected, and ETH more than five on 6 days (Table 12).

The largest scaled moves ran both ways, often from low implied levels. BTC rose 9.8% on the day starting 20 August 2026, with DVOL at 36.1, a move of 4.95 standard deviations. It fell 8.6% on the day starting 10 October 2025 (-4.31), in a liquidation cascade that followed a US announcement of 100% tariffs on Chinese imports; CoinShares cites an officially reported $19 billion of liquidations and says the true figure is likely higher. ETH rose 17.2% on the day starting 19 August 2026 and fell 18.7% on the day starting 4 August 2024, each 6.54 standard deviations. Of BTC's 22 days beyond three standard deviations, 19 came in 2023 or later, when DVOL was lower; for ETH, 16 of 19.

On most days DVOL priced more movement than occurred: that is the premium. On a few days it priced far less: that is the tail. In this sample, a risk measure calibrated to the typical day would have understated the rare one.

Findings: persistence, correlation and the calendar

The gap does not carry over. The autocorrelation of the daily BTC gap falls from 0.96 at one day, an artefact of overlapping windows, to -0.07 at 30 days; ETH's falls to 0.03. Across non-overlapping windows the lag-one autocorrelation had a median of -0.07 for BTC and 0.04 for ETH. DVOL itself is persistent, with 30-day autocorrelations of 0.83 and 0.78. Last month's gap said little about this month's. The level of DVOL said more.

BTC and ETH moved together. Their daily gaps correlated at 0.81 and their DVOL levels at 0.90. When BTC's realised volatility exceeded implied, ETH's did too on 83.1% of days. ETH's DVOL averaged 14.0 points above BTC's, with the yearly difference ranging from 2.2 in 2023 to 23.2 in 2025. Links to equities were loose: BTC's DVOL correlated with the VIX at 0.34 in levels, and BTC's gap with the S&P 500 gap at 0.33.

Weekends are quieter, and DVOL appears to price that. Weekends were 28.6% of hours but carried 16.2% of BTC and 18.4% of ETH realised variance, from hourly index data. The BTC share fell from 22.4% in 2021 to between 12.0% and 13.6% in 2024 to 2026. Kaiko reported in June 2024 that the weekend share of BTC trade volume had fallen to a record low of 16% so far that year, from 28% in 2019. A 30-day window that starts on a Friday or Saturday holds roughly one more weekend day than the window a day earlier; one that starts on a Sunday or Monday holds one fewer. From January 2023 to October 2026, BTC's DVOL fell on average into Friday and Saturday (0.43 and 0.67 points; t-statistics -2.15 and -4.48) and rose into Sunday and Monday (0.60 and 0.83). ETH moved the same way, though its fall into Friday was smaller and not statistically significant. That fits options pricing weekends as quieter, but it is an inference, not a test.

From April 2021 to December 2022 the fall into Friday was much larger: 1.72 points for BTC and 2.00 for ETH (t-statistics -3.18). On-chain option vaults then sold options in weekly auctions, mostly on Fridays. Analysis by OrBit Markets, published by Paradigm, found one-week ETH at-the-money implied volatility around Friday auctions about 4 vols below the previous 24 hours, with traders anticipating the flow. The calendar effect alone predicts some fall into Friday, so these data cannot attribute the larger fall to the vaults, but the timing matches.

DVOL describes one venue. A MerQube and Kaiko report, which Kaiko states was paid for by Bitvavo, finds that IBIT options open interest passed $10bn within three weeks of launch and peaked at $53.3bn on 9 October 2025. Bloomberg reported on 29 September 2025 that IBIT options open interest, at nearly $38bn, had passed Deribit's $32bn in bitcoin options (Bitcoin Magazine, citing Bloomberg); Glassnode, on different figures, describes the lead as brief. On Kaiko's skew measure, IBIT sat below Deribit's BTC options on 3 March 2025 (13.6% against 23.5%) and well above them on 1 July 2025 (33.4% against 18.1%), and the two were level by late September 2025 (17.5% against 17.6%). Kaiko notes that the inversion coincided with the SEC raising IBIT position limits from 25,000 to 250,000 contracts and with the introduction of cash-settled FLEX options. A premium measured on Deribit need not hold elsewhere.

Implications for allocators

These points describe what the market record implies for a risk framework applied to this premium. They do not describe any implementation, including Hedgehog's, and are not advice or a recommendation to any investor.

Definitions change the answer. In points the crypto gap exceeded the S&P 500's. As a share of implied variance it was smaller over 2021 to 2026, though BTC's share exceeded the S&P 500's ten-year figure of 17.7%. A variance-swap index overstates the gap at the money, a gap in volatility points carries a convexity effect that a variance measure does not, and a backward-looking spread answers a different question from a forward-looking gap. Estimates of the premium are comparable only once their definition, horizon and realised estimator are stated.

The regime changed the gap. The gap was widest when implied volatility was high, absolutely and against its own history, and thinnest when it was low. Low-DVOL periods held the worst BTC shortfall and the highest share of deep shortfalls, so treating low implied volatility as low risk would have misread this sample. DVOL level and rank, trailing realised volatility and funding were all observable at the time, but none was tested here as a trading rule. Regime identification is the subject of a companion white paper in preparation, Regime Detection in Digital Asset Markets.

The tail was heavier than implied. Under normally distributed moves at the implied level, 0.1 BTC days beyond four standard deviations were expected; there were 8. Tail estimates scaled from implied volatility under a normal distribution would have understated this. The empirical distribution and the episodes in Table 10 give a fuller picture.

BTC and ETH moved as one exposure. Daily gaps correlated at 0.81, and when BTC's realised volatility exceeded implied, ETH's did too on 83.1% of days. Risk measured on each asset alone would have understated the combined exposure.

Both tails were live. Of BTC shortfall windows, 42.6% ended with a higher price. The smile prices jumps both ways, and the sign of the skew changed within 2025. A short call is exposed to rallies as a short put is to falls. Israelov and Nielsen show that a covered call also carries an uncompensated exposure that behaves like a reversal strategy unless it is hedged.

Last month was a poor guide. The gap did not persist from one window to the next, and the backward-looking spread said little about what followed.

The index, the venue and the settlement currency matter. DVOL omits the far wings, is smoothed and covers Deribit only, and listed IBIT options priced skew differently for much of 2025. Until August 2025, Deribit's BTC and ETH options were inverse, settled in the coin itself. A short put margined in coin loses on the option as the price falls while its collateral loses dollar value. Deribit added USDC-settled BTC and ETH options from 19 August 2025 (Deribit). They trade alongside the inverse contracts, which remain listed.

Predictable flow has amplified stress. On 5 February 2018 the S&P 500 fell 4% and the VIX rose 20 points, its largest daily rise since the 1987 crash. The BIS found that leveraged long and inverse volatility ETPs, whose assets had reached about $4 billion at the end of 2017, both had to buy VIX futures just before 16:15, when they published their daily net asset values. At 16:08, 115,862 contracts, roughly a quarter of the market, traded within one minute. The inverse product XIV fell 84% and was terminated. Augustin, Cheng and Van den Bergen use the episode to illustrate the risks of hedge and leverage rebalancing when markets are highly concentrated and volatile. A CFA Institute summary of their paper concludes that risk managers and regulators should monitor such rebalancing. In crypto, weekly vault auctions created predictable supply of a similar kind.

The rest of the book matters. Shortfalls came with large price moves, so they can coincide with stress elsewhere in a crypto book. Book-level limits are the subject of a companion white paper in preparation, Multi-Strategy Portfolio Construction Under EVAR Constraints.

Risks and limitations

• Short history. About 67 non-overlapping 30-day windows per asset, about 12 a year and about 9 in each of the partial years 2021 and 2026. Volatility clusters, so even non-overlapping windows are not fully independent.
• Overlap and hindsight. Newey-West errors, the block bootstrap and non-overlapping checks correct the statistics but cannot add independent information. The Bartlett kernel gives higher t-statistics than non-overlapping windows. Tercile and decile cut-offs use the full sample.
• DVOL is a variance rate. It overstates the gap at the money, is smoothed, excludes options with a delta below 5%, and its parameters can change at 48 hours' notice. Because it is the square root of a variance rate, it would exceed average realised volatility even if it forecast realised variance without bias (Jensen's inequality). This effect could account for up to 3.1 points of BTC's 8.8-point mean gap and 4.2 of ETH's 5.9. The variance-gap share does not have this bias.
• One venue. DVOL and the settlement prices are Deribit's. Deribit was chosen because it has historically been the main venue for crypto options, which carries a risk of survivorship bias. Options on other venues, including CME and listed IBIT options, are not measured, and IBIT options have priced skew differently. We do not track changes in the exchanges that make up Deribit's index.
• Estimator choice. Daily settlement prices miss intraday reversals. ETH's worst gap is -54.0 on daily data and -102.5 on hourly data.
• Settlement prices. We treat Deribit's published delivery prices as 08:00 UTC values. Almeida and co-authors describe Deribit's bitcoin settlement price as a weighted average across major exchanges over the 30 minutes before 08:00 UTC. We have not confirmed the current construction with Deribit.
• Confounding. Event splits mix the event with the fall in DVOL. Price-change splits condition on the outcome. Negative-funding subsamples are small.
• Conventions. The S&P 500 measure uses trading days and 252; the crypto measures use calendar days and 365.
• Not a strategy. The gap is the difference between two market measures. DVOL is built as a variance-swap rate, but the gap is not the return of any position and should not be read as, or converted into, a return or a risk-adjusted ratio. It ignores strike selection, delta hedging, the path of prices, spreads, fees, margin calls, liquidation and settlement currency. Each can dominate the outcome of a real position, above all in the tail windows documented here.
• Structural change. The market changed materially over the sample. These patterns may not persist, and past market behaviour does not indicate future behaviour.
• Inference. The weekend and Friday interpretations, the link to vault auctions and the parallel with Cheng's VIX futures results are inferences rather than tests.

Conclusion

From April 2021 to October 2026, DVOL exceeded the next 30 days' realised volatility on 71.9% of BTC and 66.0% of ETH start days, by 8.8 and 5.9 points on average; part of that difference may reflect the difference between volatility and variance. Over the same period, the gap was larger than the S&P 500's in points and smaller as a share of implied variance; against the S&P 500's 2016 to 2026 record, BTC's share was larger. The gap narrowed over the sample, largely as DVOL fell.

Its size depended on the regime. It scaled with implied volatility, on full-sample cut-offs and on a rank observable at the time, and was thinnest in calm markets, which held the worst BTC shortfall. Shortfalls ran in both directions, hit both assets together and did not persist. DVOL overstated most daily moves and understated a few by a wide margin.

For risk frameworks, the record points to precise definitions, conditioning on the regime, sizing for the tail and treating BTC and ETH as one exposure. These are market statistics, not strategy results. This paper is for professional and high-net-worth investors only. It is not intended for retail investors or for US citizens or residents, and it is not investment, tax or legal advice.

About the data

Data: Deribit public API v2 (DVOL hourly and daily candles for BTC and ETH from 24 March 2021; daily delivery prices; hourly funding and index prices for BTC-PERPETUAL and ETH-PERPETUAL), Binance USD-M 8-hourly funding rates, Coinbase Exchange daily candles, and FRED series VIXCLS (copyright Cboe) and SP500 (copyright S&P Dow Jones Indices), all from public endpoints and stored unchanged. Period: data through 25 October 2026 (UTC); 30-day windows start each day from 1 April 2021 to 25 September 2026 (2,004 per asset); the equity comparison uses 1,375 start days from 1 April 2021 to 22 September 2026 and 2,491 from 24 October 2016. Method: implied volatility is DVOL at 08:00 UTC on the start day; realised volatility is the zero-mean, 365-day annualised volatility of the next 30 daily log changes in the Deribit delivery price, with hourly Deribit index and Coinbase daily estimators as checks; the gap is implied minus realised in volatility points, and the variance-gap share is one minus mean realised variance over mean DVOL squared; the S&P 500 gap is the VIX close minus realised volatility over the next 21 trading days, annualised with 252. Means carry Newey-West standard errors (Bartlett kernel, 30 lags; 21 for the S&P 500); confidence intervals use a moving-block bootstrap (30-day blocks, 2,000 draws, fixed seed); non-overlapping checks use every 30th start day at each of 30 offsets and report the range of t-statistics. Tercile and decile cut-offs use the full sample; the DVOL rank uses only prior data (up to 364 previous daily readings). All figures are market statistics; none is the return of a strategy, actual or hypothetical, and none is an output of any Hedgehog model. The scripts and stored raw data reproduce every figure.