Perpetual futures: contract design and pricing
How a futures contract with no expiry stays near spot: forwards refresher, Shiller’s idea, index and mark prices, funding, no-arbitrage pricing and venue designs.
You’ll learn
- Explain why a dated future converges to spot and why a perpetual future has no such guarantee
- Calculate a funding rate from a premium index, including the clamp and its dead zone
- Distinguish last price, index price and mark price, and state what each one is used for
- Derive the futures price implied by the random-maturity results of Ackerer, Hugonnier and Jermann and of He, Manela, Ross and von Wachter
- Compare order-book and pool-based designs by asking who is the counterparty and who holds the residual risk
A contract that outlived its exchange
On May 13, 2016, the exchange BitMEX began trading XBTUSD, a bitcoin contract with no expiry date. It became the template for the most traded instrument in crypto. He, Manela, Ross and von Wachter report a median total daily perpetual volume of $101.9 billion across exchanges in 2022, about two to three times spot volume on those exchanges, using CoinGecko data that exclude venues known for misreported volume. BitMEX itself ceased trading on September 23, 2026, after announcing the closure on July 23, 2026. The contract design did not depend on the firm that popularized it.
Forwards, futures and why expiry matters
Forward contract: An agreement today to exchange an asset at a fixed future date for a price fixed today. No cash changes hands at the start, apart from any collateral. All gain or loss arrives at maturity.
Futures contract: A standardized, exchange-traded forward whose gains and losses are settled to margin accounts as the price moves, rather than all at maturity. This process is called marking to market.
Both contracts have a property that does most of the pricing work. At maturity, the contract and the asset are the same thing, so the two prices must match. That single pinned date lets an arbitrageur reason backward. Someone who holds the asset financed at rate r and sells the contract has locked in a known result. If the contract is too expensive relative to spot plus financing, the position earns more than the financing cost with no price risk, ignoring frictions.
F0 is the fair futures price today, S0 the spot price, r the continuously compounded financing rate, y the yield from holding the asset (staking income, for example, or a convenience yield), and T the time to maturity in years.
Example: cost of carry on a dated contract
- Example inputs: spot $2,000, financing 5% a year, zero asset yield, 90-day maturity, so T = 0.25.
- F0 = 2,000 × e0.05 × 0.25 = 2,000 × 1.012578 = $2,025.16.
- The $25.16 gap is the cost of financing the asset for a quarter. It must shrink to zero by expiry.
In crypto markets the gap often exceeds any plausible financing cost. Schmeling, Schrimpf and Todorov, in a Bank for International Settlements working paper, report that the gap between crypto futures and spot averages above 10% a year and can reach 60% a year. They attribute the variation mainly to a volatile convenience yield, linked to smaller investors seeking leveraged upside in booms and to scarce arbitrage capital. Lesson 14 returns to that trade and its risks.
Expiry has a cost for users. A trader who wants exposure past the delivery date must close one contract and open the next, paying fees and crossing the spread each time, and liquidity is split across maturities. Perpetuals remove both frictions. He et al. point out that a venue lists one perpetual for each underlying asset, so liquidity is not staggered across maturities. The price of this convenience is the loss of the pinned end date.
Shiller’s perpetual and the crypto version
The idea has an academic origin in a different problem. Robert Shiller’s 1993 paper in the Journal of Finance asked how to build derivatives on assets that are hard to price, such as houses. He proposed perpetual claims on an index of rents or dividends, with a contract settling in cash each day based on both the change in the futures price and that day’s dividend or rent index. The goal was price discovery. A market price for the perpetual claim would reveal a value that was otherwise difficult to observe directly.
He et al. note that the crypto version differs in purpose. A crypto asset generally has no rent stream to pass to the long side. Crypto spot prices are also easy to observe, because many venues trade them around the clock. So the job of the crypto perpetual is not to discover a hidden value. Its job is to offer leveraged long or short exposure, without a roll, that tracks an already visible price. The cash flow between the two sides was repurposed as well. In Shiller’s design it passes on a dividend. In the crypto design it pulls the contract toward spot.
Crypto perpetuals come in three settlement types, classified by Ackerer, Hugonnier and Jermann. A linear contract is sized in the base asset and settled in the quote asset, such as 1 BTC settled in dollars or stablecoins. An inverse contract is sized in the quote asset and settled in the base asset, such as $10,000 of exposure settled in BTC. It let exchanges offer dollar-quoted exposure without holding dollars. A quanto contract settles in a third asset. The inverse payoff is not linear in price, which surprises many first-time readers.
Example: the inverse payoff is curved
- Example position: long an inverse contract with $10,000 face value at entry price $50,000. Profit in BTC = 10,000 × (1/entry − 1/exit).
- If the price falls 20% to $40,000: 10,000 × (1/50,000 − 1/40,000) = −0.05 BTC, worth −$2,000 at $40,000.
- If the price rises 20% to $60,000: 10,000 × (1/50,000 − 1/60,000) = +0.0333 BTC, worth +$2,000 at $60,000.
- The dollar result is symmetric, but the BTC result is not. The margin is held in BTC, so its dollar value falls in the same scenario that produces the loss.
Last price, index price and mark price
A perpetual venue needs reference prices for two different jobs: one to measure how far the contract has drifted from spot, and one to decide whether an account is solvent. Using the contract’s own last trade for either job would let one large order move the number that triggers other traders’ liquidations.
Index price (or oracle price): A composite of spot prices from several external venues. It is the venue’s estimate of the underlying asset’s value.
Mark price: The price a venue uses to value open positions for margin and liquidation. It blends the index with the contract’s own market, so that it follows real moves while resisting brief local distortions.
| Venue | Index or oracle | Mark price |
|---|---|---|
| Binance USDⓈ-M | Weighted average of spot prices across listed exchanges, with weights the venue can change | Median of three: index × (1 + last funding × time to next funding ÷ funding period); index + 30-second average of (mid − index); last price |
| Hyperliquid | Validators publish every 3 seconds a weighted median of spot mids from eight venues, then the stake-weighted median of those submissions is used | Median of three: oracle + 150-second exponential moving average of (mid − oracle); median of best bid, best ask and last trade on the venue; weighted median of perpetual mids on five other venues |
| dYdX v4 | Oracle price supplied at the protocol level | Margin requirements and account value are computed with the oracle price directly |
| GMX | Index price feed for each market | Liquidation checks use the minimum oracle price for longs and the maximum for shorts, plus the minimum collateral price |
Reference prices as documented by each venue, read November 2026. Parameters change, so treat this as a description of design choices and not as a permanent specification.
These are engineering compromises. A median of several inputs resists one bad input, but it cannot fix a case where the inputs fail together. If collateral is valued from a venue’s own thin market, a local dislocation can still trigger liquidations, which is the central dispute in the October 2025 case in Lesson 13.
The funding mechanism
Funding replaces convergence at expiry with a recurring cash flow between the two sides. If the contract trades above the index, longs pay shorts. If it trades below, shorts pay longs. Holding a long position costs more when the contract is expensive, and holding a short position earns more, so traders have a reason to take the side that pushes the price back. Funding is paid between positions. Binance and Hyperliquid both state that they collect no fee on it.
The standard formula, used with small variations by Binance and Hyperliquid, has two steps. First the venue measures how far the executable prices sit from the index. It uses impact prices, meaning the average fill for a set notional size on each side of the book, so that a few cents of quoted size at the top of the book cannot move the measure.
P is the premium index. It is zero when the index sits inside the impact bid and ask. Binance samples it every 5 seconds. Its impact size is 200 USDT divided by the initial margin rate at maximum leverage, so 4,000 USDT for a contract with a 20x maximum.
F is the funding rate for an 8-hour interval, P̄ the average premium over that interval and i the interest component, 0.01% per 8 hours by default. clamp(x, a, b) limits x to the range a to b. Payment = position notional × F.
The clamp has a practical consequence. When the average premium lies between −0.04% and +0.06%, the clamp does not bind, the two P̄ terms cancel, and funding equals exactly the interest component. So a printed funding rate of 0.01% per 8 hours says only that the premium was inside a band. It is not a measurement of demand. Beyond the band, funding follows the premium one for one, offset by 0.05%.
Two funding calculations from venue documentation
- Binance’s own worked case: P̄ = 0.0429%, i = 0.01%. Then i − P̄ = −0.0329%, inside the clamp. F = 0.0429% − 0.0329% = 0.0100%.
- Hyperliquid’s own worked case: impact bid $10,100 and oracle $10,000, so the premium is 1%. Then i − P̄ = 0.01% − 1% = −0.99%, clamped to −0.05%. F = 1% − 0.05% = 0.95%.
- Example payment: a long position with $10,000 notional at F = 0.01% pays 10,000 × 0.0001 = $1 to shorts.
Venues differ on the details. Binance pays every 8 hours by default and may shorten the interval in extreme volatility, and it weights later samples more heavily in its 8-hour average. Hyperliquid computes an 8-hour rate but pays one eighth of it each hour, on a notional converted at the oracle price, not the mark. dYdX pays hourly, divides the premium by 8 and adds a default interest component of 0% in cross markets. GMX, a pool-based venue, uses a different mechanism altogether: the side with larger open interest pays the smaller side, at a rate that adapts over time with the imbalance. A number with the label “funding” therefore does not mean one thing.
This script was run on November 5, 2026. The premium it rebuilt from impact prices matched the venue’s reported premium for all three assets (−0.5, −0.42 and −2.73 basis points, where a basis point is 0.01%). Hourly funding on BTC and ETH was 0.00125%, the default rate, while SOL had dropped slightly below it at 0.001%. Those figures describe that moment only.
What the theory says the price should be
Without an expiry date, the usual backward reasoning has no starting point. Two lines of research solved this in different ways. Their answers agree on the central idea: the funding stream acts like a random maturity.
Ackerer, Hugonnier and Jermann write the funding at date t+1 as κ(ft − xt) + ιxt, where f is the futures price, x the spot price, κ the premium rate and ι the interest factor. Because entering a future costs nothing, no-arbitrage requires the expected cash flow under a pricing measure to be zero. Rearranging gives a recursion, shown here in the constant-parameter case:
E is the expectation under the risk-neutral measure. Iterating forward under a no-bubble condition removes the f term.
The perpetual is priced as the spot price at a random future date. A larger κ means stronger anchoring and a shorter effective maturity.
Example: effective maturity and the premium it implies
- Example inputs: κ = 1/3 per day, so the mean of θ is 3 days. Suppose spot is expected to grow at g = 0.03% a day under the pricing measure, and ι = 0.
- The sum of κ(1 + κ)−(n+1)(1 + g)n over n is a geometric series that equals κ / (κ − g).
- f / x = 0.33333 / (0.33333 − 0.0003) = 1.0009, a premium of about 9 basis points.
- If g = 0, then f = x. With strong anchoring and no drift under the pricing measure, the contract price equals spot.
He et al. take a different route that deals directly with the clamp. They define random-maturity arbitrage: a strategy that may be unwound at a bounded stopping time depending on prices, instead of on a fixed date. For the Binance specification with a premium scale κ, clamp width γ and interest component ι, and a constant financing rate r per funding period, they show that absence of such arbitrage requires:
sgn(·) returns +1 or −1. For Binance they use κ = 1, γ = 0.05% and ι = 0.01%. When r = ι the price lies in an interval, not at a single point.
Example: the benchmark on either side of the interest component
- With 1,095 funding periods a year, a financing rate of 5% a year is r = 0.05 / 1095 = 0.00457% per period. That is below ι = 0.01%, so the sign is +1.
- F / S = 1 / (1 + 0.0005 − 0.0000457) = 0.99955, which is 4.5 basis points below spot.
- A financing rate of 15% a year gives r = 0.0137%, above ι, so the sign is −1.
- F / S = 1 / (1 − 0.0005 − 0.000137) = 1.00064, which is 6.4 basis points above spot.
Angeris, Chitra, Evans and Lorig approach the problem as contract design, asking what funding formula makes a given payoff replicable in a continuous-time market with no transaction costs. He et al. criticize that framework for assuming the perpetual’s payoff is a fixed function of spot, which rules out studying deviations. These are three different questions, not three competing estimates of one number.
What the data show
The theory gives a benchmark. The prices frequently sit far from it. Using hourly Binance data for BTC, ETH, BNB, DOGE and ADA from early 2020 to March 11, 2024, He et al. find a mean absolute deviation of about 52% to 90% a year across assets, measured in annualized terms. That is much larger than the deviations documented in traditional currency markets. The deviations move together across assets and declined by about 22 percentage points a year. Their size is comparable to a no-arbitrage bound calibrated to actual trading fees.
Their explanation centers on limits to arbitrage. Binance’s April 2022 introduction of portfolio margining let arbitrageurs net collateral across spot and futures positions, and deviations compressed significantly afterward. Order flow also had a larger price impact when initial deviations were already large. And past spot returns positively predicted the deviation, which they read as extrapolative demand for leverage.
Does the perpetual lead spot, or merely track it?
Alexander, Choi, Park and Sohn (2020), using minute-by-minute data, find that BitMEX derivatives led prices on major bitcoin spot exchanges, were more informationally efficient than spot and served as effective hedges. That supports a view of the perpetual as the main venue of price formation in that period. The opposite caution is also serious. That study covers one exchange in one period, the leading venues have changed since, and lead-lag results depend on the data frequency, on trading fees and on which spot venues form the comparison. A contract with high leverage may also react first to the same news without adding information. The honest position is that lead-lag findings are conditional on the sample, and that a price which moves first has not thereby been shown to cause anything.
Is the anchor actually an anchor?
One view holds that funding is a weak mechanism. Deviations of tens of percent a year persisted for years, and an arbitrageur can lose money for a long time before convergence. He et al. themselves note that funding-rate arbitrage is not risk-free because no date forces the trade to close at a profit. The other view holds that the anchor works well enough for its purpose. Those annualized numbers correspond to small gaps at any moment, since a 50% annualized funding gap is only about 0.05% per 8-hour period, and they have narrowed as arbitrage capital grew. Both can be true, because “close to spot” depends on the holding period under discussion.
Order books and pools
Venue design answers one question above all others: who stands on the other side of a position, and who absorbs the loss if that position cannot pay? The two broad families answer differently. The table describes documentation as read in November 2026.
| Feature | Order book (Binance, Hyperliquid, dYdX) | Pool (GMX) |
|---|---|---|
| Counterparty | Another trader, matched by price and time | The liquidity pool. GMX states that the pool is the counterparty to traders |
| Execution price | Set by resting orders, so a large order walks the book | Set by an oracle feed, plus a price-impact charge or credit, applied at close or decrease, based on the change in the long-short imbalance |
| Balancing force | Premium-based funding | Imbalance-based funding, a borrow fee paid by the larger side, and price impact |
| Who supplies liquidity | Market makers, and on Hyperliquid also a community-owned protocol vault | Depositors, who earn fees and bear trader profit and loss |
| Where state lives | Binance: private ledger. Hyperliquid: matching state inside chain state. dYdX: book in memory on each node, with positions and fills on chain | On chain, in smart contracts |
| Stress failure mode | Thin books, liquidation slippage and bad debt | Depositors take trader gains, so the pool caps pending profit relative to pool value and may deleverage winners |
Sources: GMX, Hyperliquid and dYdX documentation. Each venue has further parameters not shown.
An order book does not need a price feed to find a trade price, but it needs a depth of willing market makers. A pool offers a quote for any size up to its limits with no resting orders, but it depends entirely on its oracle, and depositors are short the traders’ winners. Hyperliquid shows these categories blur. It runs an order book and also has HLP, a protocol vault that provides liquidity and performs liquidations, with depositors sharing its profit and loss. There is no consensus on which design is safer, and each has had public failures covered in Lessons 12, 13 and 15.
Common mistakes
- Treating funding as a fee to the exchange. On the documented venues here it passes between positions.
- Assuming a default funding print means the market is balanced. The clamp makes a wide band of premiums produce the same number.
- Using last price to judge whether a position was at risk. Liquidations use the mark price, which can differ from the last trade.
- Calling a perpetual a chain of 8-hour futures. He et al. note it cannot be replicated that way, because an 8-hour future converges to spot at each roll and the perpetual does not.
- Reading no-arbitrage as risk-free. The random-maturity argument allows for arbitrarily long interim losses that a margin account may not survive.
- Comparing funding across venues without checking the interval, the interest component and which price converts notional.
Problem set
- A contract has an average premium of 0.03% over an 8-hour interval, with i = 0.01% and a ±0.05% clamp. Find the funding rate. Then repeat for a premium of 0.20% and for −0.20%.
Show solution
For 0.03%: i − P̄ = −0.02%, inside the clamp, so F = 0.03% − 0.02% = 0.01%. For 0.20%: i − P̄ = −0.19%, clamped to −0.05%, so F = 0.15%. For −0.20%: i − P̄ = +0.21%, clamped to +0.05%, so F = −0.15%, and shorts pay longs. - Show the band of premiums within which funding equals i exactly, for i = 0.01% and a clamp of ±0.05%.
Show solution
F = P̄ + (i − P̄) = i whenever |i − P̄| ≤ 0.05%. That means P̄ lies between i − 0.05% = −0.04% and i + 0.05% = +0.06%. Outside this band, F = P̄ − 0.05% above it and F = P̄ + 0.05% below it. - An example long inverse contract has a $20,000 face value and entry at $40,000. The exit price is $50,000. Compute the profit in BTC and its dollar value at exit. Then explain why it differs in BTC terms from a 25% fall to $30,000.
Show solution
Profit = 20,000 × (1/40,000 − 1/50,000) = 20,000 × 0.000005 = 0.1 BTC, worth $5,000 at $50,000. For a fall to $30,000: 20,000 × (1/40,000 − 1/30,000) = −0.1667 BTC, worth −$5,000 at $30,000. The dollar amounts are symmetric for those exits, but the BTC amounts are not, because profit is a function of 1/price. - In the Ackerer, Hugonnier and Jermann setup with ι = 0, compare κ = 1 per 8-hour period with κ = 1/8 per 8-hour period. Find the mean of θ in hours for each case, and say which contract should track spot more tightly.
Show solution
The mean of θ is 1/κ periods. With κ = 1, the mean is 1 period, or 8 hours. With κ = 1/8, it is 8 periods, or 64 hours. The first has the stronger anchor and shorter effective maturity, so it should track spot more tightly, other things equal. - Use the He et al. formula for κ = 1, γ = 0.05% and ι = 0.01% to find F/S for a financing rate of 9% a year, using 1,095 periods a year. Is the result above or below spot? What changes at 12% a year?
Show solution
At 9%: r = 0.09/1095 = 0.00822%, below ι, so the sign is +1 and F/S = 1/(1 + 0.0005 − 0.0000822) = 0.99958, about 4.2 basis points below spot. At 12%: r = 0.01096%, above ι, so the sign is −1 and F/S = 1/(1 − 0.0005 − 0.0001096) = 1.00061, about 6.1 basis points above spot. The sign flips because the financing cost moved above the interest component. - Why might an order-book venue and a pool-based venue show different “funding” for the same asset at the same moment, even if both markets are fair?
Show solution
They measure different things. An order-book venue typically derives funding from the premium of executable prices over an index. GMX derives it from the long-short imbalance on its own pool, with a rate that adapts over time, and it also charges a separate borrow fee. Even two premium-based venues can differ on interval, interest component and clamp. So the same label covers different mechanisms, and the difference need not reveal any mispricing.
References
- Shiller, R. J. (1993). Measuring Asset Values for Cash Settlement in Derivative Markets: Hedonic Repeated Measures Indices and Perpetual Futures. Journal of Finance 48(3), 911-931. Source
- He, S., Manela, A., Ross, O., and von Wachter, V. (2022, this draft September 2026). Fundamentals of Perpetual Futures. arXiv:2212.06888. Source
- Ackerer, D., Hugonnier, J., and Jermann, U. (2024). Perpetual Futures Pricing. NBER Working Paper 32936; published in Mathematical Finance 36(3), 481-499 (2026). Source
- Angeris, G., Chitra, T., Evans, A., and Lorig, M. (2022). A primer on perpetuals. arXiv:2209.03307. Source
- Schmeling, M., Schrimpf, A., and Todorov, K. (2023). Crypto carry. BIS Working Paper 1087. Source
- Alexander, C., Choi, J., Park, H., and Sohn, S. (2020). BitMEX bitcoin derivatives: Price discovery, informational efficiency, and hedging effectiveness. Journal of Futures Markets 40(1), 23-43. Source
- Binance. Introduction to Binance Futures Funding Rates (documentation, read November 2026). Source
- Binance. Mark price documentation for USDⓈ-M futures (read November 2026). Source
- Hyperliquid. Funding, oracle and robust price indices documentation (read November 2026). Source
- dYdX. Funding documentation (read November 2026). Source
- GMX. Trading fees documentation and providing-liquidity documentation (read November 2026). Source
- Redman, J. (July 23, 2026). BitMEX Closes Doors as the Perpetual Swap It Invented Rules Rivals. Bitcoin.com News. Secondary report of the closure notice. Source
- CoinDesk (July 23, 2026). Report of BitMEX notifying users that it would end operations by September 23, 2026. Source
Exercise
- Run the script in this lesson, or send the same POST request with any HTTP tool. No account or key is needed, though some venues restrict access by region.
- For three assets, rebuild the premium from impactPxs and oraclePx and confirm it matches the reported premium.
- Apply the funding formula to the reported premium, divide by 8 and compare with the reported hourly funding. The reported premium is one reading and funding uses an average, so expect small gaps. Note where the clamp produces the default rate.
- Choose an asset whose funding differs from the default rate, and explain the difference using the premium band from this lesson.
- Write down one thing this snapshot cannot tell you. For example, it does not show who holds the positions.
Check yourself
1. Why does a dated future have a simpler pricing argument than a perpetual?
2. The premium averaged 0.05% over an 8-hour interval on a contract with i = 0.01% and a ±0.05% clamp. What does a funding print of 0.01% show?
3. Why do venues liquidate using a mark price instead of the last trade?
4. In the geometric-maturity result, what happens to the effective maturity of the contract as κ rises?
5. He et al. find that deviations compressed after Binance introduced portfolio margining. What does this finding most directly support?
6. A trader is long an inverse BTC contract with BTC margin. The price falls by 30%. Which statement is correct?
7. On a pool-based venue, what is the most direct consequence of a large gain by traders?
8. A paper reports a Sharpe ratio above 3 for a funding-based arbitrage strategy from 2020 to 2024. Which is the most defensible reading?
Describes past on-chain activity and how our data is built. Not investment advice. Method details: data methodology. Live panels: tracked whale wallets, updated 2026-11-05 15:41 UTC.
