Tuesday, April 30, 2019

Convexity Explains the High BitMEX ETH Funding Rate

BitMEX offers swaps that make it easy to lever a long or short bitcoin (BTC) and ether (ETH). The main reason it trades so much is that they are based outside of US or EU control in the little archipelago-nation of Seychelles, and also that it transacts only in Bitcoin. This combination makes it difficult for regulators to attack.

Their swap contracts are like futures contracts without expiry dates. A futures market trades off of a futures price, which is different than the spot price and its difference from the spot price is called the basis, where basis=spot - futures. The basis can be quite large in markets that are difficult to arbitrage, as sometimes the futures/forward prices are quite different than spot markets, and this difference can be explained as a risk premium, though this is just a catch-all term for anything outside of simple interest rates. A swap trading off of a spot price captures this basis via differential funding rates for the long and short. Thus, if the spot rate is $100 and the futures price in 1 month is $110, in a swap this would show up as a 10% monthly funding rate paid by the long to the short, which is implicitly what occurs in the futures market as a contract rolls towards expiration.

At BitMEX they call the basis the Funding Rate, and it is applied every 8 hours. Below are the annualized funding rates, by month, for the BTC and ETH swaps.

These rates imply that if you moved your BTC position into a long BTC swap at BitMEX, you would have added an extra 15% (annualized) to your return with little effort. Even better, the +45% ETH funding rate seems to imply arbitrage, wherein you could have hedged a long ETH position with a short position in the ETH swap, locking in a fat 45% annualized return while avoiding that entire Aug-present ETH price decline.

I asked people in Reddit their best explanation, and the answers were related to sentiment or simply a market imbalance. Yet there's a futures contract for the ETHBTC contract, and the basis there is quite small, about 5% annualized, which is inconsistent with this explanation. Further, ETH and BTC are highly correlated, about 90% using 2019 data, so it is implausible to think these two assets would have such radically different risk premia.

The trick is to adjust the funding rate for the BTC~ETH covariance in order to capture the convexity of USD returns. With this adjustment, the swap funding rates are about the same. This reminds me of the convexity adjustment in eurodollar futures is needed to compare eurodollar rates to interest rate swap rates due to the fact that eurodollar futures are settled daily while an interest rate swap is settled only at expiration. The convexity adjustment became well-publicized in the mid-90s, but for a good 5 years traders were oblivious, and while there was sufficient liquidity they did not make this convexity adjustment, generating arbitrage profits for a couple of savvy banks (you needed direct access to libor rates, so as an individual you could not arb this). In contrast, here it seems markets have always priced for this while it is still not widely understood or documented.

BTC Swap Payoff

Let us ignore the funding rate and look just at the profit generated by the asset price for a BTCUSD swap.  At BitMEX, the long profit is generated in BTC as follows for an entry price at time t, and  exit at t+1:

  • BTCswap Profit (in BTC)=Notional*[1/BTC(t) - 1/BTC(t+1)]
Note the seemingly backward ordering of the prices. This is because the BTC notional amount is in USD, and as we have a USD position, we have to ultimately translate that back into BTC.  Thus if you are long $1000 in the BTC swap, your USD profit would be

  • BTCswap profit (in USD)=$1000*[BTC(t+1)/BTC(t)-1]
As you are getting paid in BTC, however, you need to make the final adjustment of this into BTC, so you add this final price to payoff function:

  • BTCswap profit (in BTC)=$1000*[BTC(t+1)/BTC(t)-1]/BTC(t+1)
Applying some algebra gives the BitMEX payoff function:

  • BTCswap  profit (in BTC)=$1000*[1/BTC(t)-1/BTC(t+1)]
Thus, the BTC swap formula replicates the intuitive sense in which your notional amount is invested in BTC. The PNL is nonlinear in BTC, but linear in USD, as shown below:

ETH Swap Payoff

In contrast, the BitMEX ETH swap uses a different formula that BitMEX CEO Arthur Hayes notes is "attractive to speculators who wish to have exposure to a foreign asset, but without the corresponding exchange risk." This makes little sense because most investors care about their return in fiat, not BTC, and it is inconsistent with BitMEX's BTC swap, which generates a linear return in fiat currency. Most importantly, if you are trading on an asset's USD price the notional should be in USD; if the asset price is in BTC the notional should be in BTC. In this contract, the BitMEX ETH notional is basically in BTC but the asset price is in USD. The ETH perpetual swap is based on a profound misunderstanding of how a Quanto works.

The BitMEX ETH swap is a position defined by the number of contracts times a multiplier (1E-6). This is then multiplied by the difference in ETH's USD price to generate the position payoff. Abstracting from the funding rate, the payoff function for the ETH perpetual swap is:
  • ETHswap payoff (in BTC)=0.000001*#Contracts*[ETH(t+1) - ETH(t)]
To see what this implies, it is best to simply take the payoff function and turn it into USD by multiplying it by the USD BTC price at completion:
  • ETHswap payoff (in USD)=0.000001*#Contracts*[ETH(t+1) - ETH(t)]*BTC(t+1)
Given the ETH and BTC returns are about 90% correlated, this then generates the opposite return pattern to the BTC swap, where the USD return is convex, while the BTC return is linear. For example, if the ETH price rises 10% the BTC price will also probably rise, increasing returns for up movements. The same process works on downward movements to dampen losses. The result is the opposite of the BTC swap, a convex payoff in USD and a linear payoff in BTC.

The USD return can be derived by looking at the initial USD investment, which is the following at time t:
  • ETHswap USD investment=BTC(t)*0.000001*#Contracts*ETH(t)
As return=payoff/investment, the USD return on this ETH position is thus
  • ETHswap USD % return=BTC(t+1)/BTC(t)*[ETH(t+1) /ETH(t)-1]
  • ETHswap USD return=[1+ret(B)]*ret(E)
where ret(B) and ret(E) are the net returns for bitcoin and ether. The expected value of this is 

  • E{ETHswap USD return}=ret(E)+covariance(ret(E),ret(B))
as the covariance(a,b)=correl(a,b)*[stdev(a)*stdev(b)], this covariance term generates a boost to the USD payoff due to the significant ETH~BTC correlation. You can see why it is needed by looking at the convexity in the USD returns, an example of Jensen's inequality. Since June 2017, the volatility for BTC and ETH has been 90% and 118% respectively, with a correlation of 60%, generating a covariance adjustment term annualizes out to be about 64%. This is the return premium of the ETH swap compared to the ETH USD return over that period: 51% vs. 115% (all data reflecting arithmetically annualized returns). Thus if the average ETH return is boosted by 60% via the covariance adjustment, the 45% funding rate implies a comparable -15% funding rate for ETH, just as in the BTC. You can download an Excel sheet showing the math and data here.

A 15% return premium for going long at BitMEX is high, but not insanely so, and not without risk. The rate bounces around, and BitMEX could be hacked or shut down by regulators at any time, so I think the 15% funding credit longs get is fairly priced.

There are two qualifications. First, it neglects the fact you need to buy bitcoin to enter into them, and so you are essentially long the notional amount of bitcoin as well. Secondly, given a total return is essentially a geometric return, while an arithmetic return maintains a constant USD notional each period, one has to rebalance their position each period to generate the arithmetic return, so investors would need to adopt such an approach to generate the arithmetic return (see here for difference).


The BitMEX formula is poorly conceived: the swap has a BTC notional and references a USD price creating a convex USD payoff. Investors are forced to not only take a position on the future move in ETH, but also the ETH~BTC covariance. I understand why it is so popular, as it is very difficult to put on such a position hassle-free anywhere else, yet BitMEX should offer something with a linear fiat payoff like their BTC swap, as I think this would be even more popular. This could be done by having the BTC notional contract trade on the BTCETH price; alternatively, they could add a BTC return adjustment to make the USD return linear. Either approach would remove the importance of estimating the covariance term, which is a distraction for most investors, and also one that few really understand.