Convergence of Perpetual Futures

We recommend those not familiar with perpetual futures and funding rates start with this article to gain an intuitive overview of how they work.

This piece answers the question of why perpetual futures prices track their underlying indices. Readers will see that the funding rate is an incentive mechanism the encourages convergence in many ways beyond just arbitrage opportunities.

Perpetual futures (perps) are financial instruments whose price tracks its underlying index.  For example, the price of an ETH-USD PERP tracks the price of ETH in USD.  The funding rate is what makes this happen, but you already knew that.  This piece is going to examine exactly how and why this happens in the real world.

The funding rate adjusts the payoff of holding an open perp position.  It does this in response to the difference between the perp market price and the underlying price.  Trades that reduce the difference between the price of the perp and the underlying index become more profitable and vice versa.  The wider the differential between the prices, the more convergence trades are incentivised.

When the price of a perpetual future is above its underlying index the funding rate is positive and long positions pay funding to short positions.  This is a slight oversimplification as we’ll see later in the series, but directionally correct.

Traditional, fixed term futures are financial contracts that obligate parties to transact at a predetermined price and and date. They are guaranteed to converge to their underlying index at a specific time.  Perpetual futures have a weaker notion of random-maturity convergence.  This means a holder of a perp is guaranteed to receive the same payoff as holding the underlying index at some point in the future (see below), they just don’t know when that point will be.  

Funding rates introduce an element of active position management that does not exist with fixed term futures.  With fixed term futures a holder can trade a future, knowing that over a specific period their return will match the underlying (excluding a relatively stable cost of carry).  

This is not the case with perps.  Variable funding rates make the payoff from holding perps path dependent.  If a trader holds a long position in a perp that consistently trades above its underlying index the passive trader will lose out on funding and underperform the index.  Until we get funding rate premium derivatives, there is no passive way to manage these risks.  

In the absence of frictions traders can avoid funding by selling their position, at a greater profit than the underlying index implies, and buying back almost immediately over the funding payment.  Should enough traders pursue a version of this and markets naturally converge (see below).

The consequence of this path dependency is that perpetual futures encourage active trading.  At least a portion of participants need to optimise for funding in some way.

Funding Rates in Action

These are all abstract concepts.  It’s time to look at some examples of different types of traders strategies being affected by funding.   

Consider the case where trader Bob goes long 1 BTC Perp when the market price and the index price are both the same, $100.  Then imagine the index price stays the same, but for whatever reason, the price on the perpetual exchange rises to $110.  Now Bob is going to be paying $10 every 8 hours in funding to BTC perp shorts.  This is not very appealing to him so he rather closes his position out for $10 profit before the end of the funding period.  Others thinking along the same lines as Bob do so too.  This selling moves the market back to the $100 index price.

Of course, if it was Bob’s intention all along to have long exposure to BTC he can always re-enter his position again either after the funding period or if expected funding reduces to a tolerable level.  

Enter Alice, the arbitrage bot.  Alice notices BTC perps trading at $110 on an exchange when they are trading at $100 elsewhere.  She does not want any exposure to BTC and instead goes short BTC on the perp exchange at $110 and long the same amount of BTC elsewhere (either on another perp exchange, an expiry futures exchange or just buying BTC spot).  Alice has deep pockets and continues making this trade until her trades have enough impact to converge the prices.  

Now Charlie enters with his sophisticated market making bot.  Again the price on the perp exchange is $110 per BTC while it trades at $100 elsewhere.  Charlie does not have the deep pockets of Alice as he provides his service on many different exchanges and needs to carefully manage his collateral.  Charlie makes sure to show aggressive offers on the perp exchange and aggressive bids elsewhere (to hedge his inventory).  This has the effect of gradually exhausting net buying pressure on the perp exchange without diverging prices further and without putting himself into a position with significant BTC exposure.  Charlie is earning positive funding rates while his counterparties are earning negative funding.  Charlie’s actions provide resistance to the price divergence and eventually his counterparties will begin to act like Bob above and the prices should converge.

The three caricatures above describe how funding rates incentivise convergence for three different types of trader on the perpetuals market.  The first and last do not involve arbitrage at all and are one of the reasons perpetual markets work better for lower liquidity pairs.

Expectations of Convergence Leading to Convergence

In practice it is a combination of the traders described above in the market that keep prices in line.

Markets are adaptive.  Expected prices quickly become actual prices. If enough traders expect prices to converge, the expected convergence becomes part of their strategies and this belief does most of the heavy lifting keeping prices in line.

Looking at another example, Mallory goes long BTC-USD PERP at $100 when the USD price is $100.  Near the end of the first funding period the PERP price has risen to $110 while the index price is still $100.  Mallory decides to sell and buy back the PERP to avoid the funding payment.  As she tries to do this however the market price begins to drop almost as though she is being front-run.  This predictable phenomenon is a result of others trying to avoid the funding rate payment, market makers expecting this, and other second order effects.

The next iteration of expectations resulting in reality is traders expecting this end of period convergence a bit earlier and prices converging a little earlier in the funding period.  This happens until prices never get too out of line as there is an expectation of them converging soon after and there is always someone to take advantage of that.

Replicating the Payoff of an Underlying Index

What if the prices do not converge?  How can traders be sure that they will replicate the payoff of the underlying index at some point in the future?  The funding rate solves this problem.  

Dave White of Paradigm explained three mental models of funding rates mimicking price moves brilliantly in the cartoon guide to perps.  The original is definitely well worth a read, here is a succinct summary.

Perps as a PNL Loan

Imagine a deal where a trader goes long an index and their PNL is periodically calculated and owed to them by their counterparty, e.g. they go long one contract at $100 and if the index goes to $110 they are owed $10 at the end of the period.  The interest rate on any PNL is 100% per day.

In the above example the perp market price move acts like payment of PNL owed as the trader could realise their PNL by closing their position.  The funding rate acts like the interest rate on any mismatch in PNL payment.

This is how funding rates act as a PNL loan.  The market as a rational actor is incentivised to repay this loan (converge to the index price) within a reasonable time frame as it is cheaper than the 100% per day interest rate.  

Funding rates are not 100% per day.  The way they function as a P&L loan acts as a 100% per day rate because of the 1:1 linear funding rate formula.  Either a trader makes 100% of the profit/loss as the markets stay in line, or the funding rate makes up for any mismatch.  In this case the funding rate is paying 100% of any price mismatch per day.

Perps as a Loan Swap

This time imagine two parties entering into an agreement to borrow one side of a trading pair from the other.  For example if the price of BTC is $100, Alice borrows one BTC from Bob and Bob borrows $100 from Alice.  Both loans are at 100% interest rates, but because their value is the same, the interest cancels out.  

Whenever the price of BTC changes the values of the loans will no longer be equal and one party will need to pay interest each funding period until they cash settle their loan differential.  

Again the perp price converging to its index is equivalent to repayment in this example.  Again the market is incentivised to settle this differential (converge to index) as it is cheaper.

Again the 100% rate is to make sure that payoffs align with perp payoffs and is not due to perps being expensive in any way.

Perps as One Day Futures

This provides a strategy to shift a perpetual position into its underlying index using the funding rate.  A trader could do this to gain direct exposure and therefore match the payoff of the underlying.

At any time an unleveraged trader could sell their perp and buy the underlying provided the perp price is in line with or higher than the underlying price.  If the perp price is below the underlying price the trader could instead of selling their full amount in one go, could sell 1/24th of their position every hour.  The funding payment would exactly offset their shortfall each hour and over a few days they would gradually shift their entire position to the underlying.


At their core perpetual futures are a price oracle and a funding rate.  The oracle specifies the index to track and the funding rate incentivises market participants to take actions that keep prices close to this index.

This incentive goes beyond simply bringing arbitrageurs into the market although this is the simplest mechanism to understand.  Traders and market makers’ strategies are also affected by the funding rate.  Perpetual futures markets can exist whether arbitrage is possible or not, allowing long tail derivative markets.

Even in the case where prices never converge the payoff of the underlying is replicated at some point in the future.  This point is unknown, hence the term random-maturity arbitrage of perpetual futures.