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.

Wednesday, March 13, 2019

Why Taleb's Antifragile Book is a Fraud

In Nassim Taleb’ book Antifragile he emphasizes that ‘if you see a fraud and do not say fraud, you are a fraud,’ I am thus compelled to note that Antifragile is a fraud because its theme is based on intentional misdirection. The most conspicuous and popular examples he presents are also explicitly mentioned as not the essence of antifragility. Indeed, incoherence is Taleb’s explicit strategy, as the Wikipedia entry on Antifragility notes Taleb presents his book in a way to make it difficult to criticize. He tried to squeeze a weltanschauung onto the Procrustean bed of his Black Swan and generated a synecdoche that confuses the part with the whole.

I bring this up because last month I was listening to a Joe Rogan podcast where they mentioned hormesis, the concept that small amounts of a toxin or stressor strengthen an organism. The guest noted hormesis was discovered in the 1950s when researchers noticed a little bit of herbicide paradoxically makes plants stronger. Actually, this phenomenon was found back in 1888 concerning yeast, though the basic idea is probably timeless, in that everyone understands exercise strengthens muscles while immobilizing a limb after an injury leads to atrophy, and too much stress causes problems to muscles and limbs. A glass of wine a day is a tonic, too much leads to cirrhosis. Even concerning poison, there's the term Mithridatism, which comes from King Mithridates (160 BC) self-administering small amounts of a toxin to build up his immunity.

tweeted that Taleb thinks he invented hormesis, whereupon Taleb’s sock-puppet quickly noted that antifragility is not hormesis, and Antifragile explicitly mentions hormesis and its 1888 discovery, as well as Mithridatism. A snippet of his Twitter rebuttal is here. Tweets are not the place for snarky subtlety. My point was that far as antifragility works, it's hormesis, in spite of Taleb's qualification that "hormesis is a metaphor" for antifragility. This got me wondering how such a contradiction happened.

Taleb states his neologism antifragility is "beyond resilience or robustness." He defines antifragility more precisely as "a convex response to a stressor or source of harm, leading to a positive sensitivity to increase in volatility." Thus hormesis is not an example of antifragility, because, in the parlance of finance, hormesis is like increasing the value of a portfolio by reducing one’s beta, while antifragility is increasing the value of a portfolio by increasing its gamma. Gamma is a measure of convexity, the signature feature of a put or call option, and in uncertain environments, a higher gamma leads to higher value.

The common takeaway of Antifragile, however, is simple resilience. For example, in Jonathan Haidt's book the Coddling of the American Mind he credits Taleb’s concept of antifragility for arguing that protecting students from ideas they find offensive leads to them becoming more fragile, anxious, and easily discouraged. Actively confronting ideas we don't like makes us tougher and smarter, in that as JS Mill wrote, ‘he who knows only his own side of the case knows little of that.’ Haidt's book mentions unstructured play for children, the immune system, and exposure to peanuts as examples of Taleb's concept of antifragility. Wikipedia’s entry on antifragile gives as its primary example that of bone density being strengthened by exposure to stress. These are all examples of hormesis.

Convex Payoff
If you own an option you have positive convexity and benefit from higher volatility; you are long volatility (aka long vega). It has long been known that financial options, especially out-of-the-money put and call options, have poor average returns. A good example is provided by VXX ETF (reissued recently as the VXXB), which is long vega and loses money with about the same Sharpe ratio as the SP500 index. Being long vega is like shorting the market, good in bad times, but over the long run a bad investment.

Taleb is aware of this and states that good antifragile things are not financial options because they "are sold by someone," but rather real options, which he thinks are free: "we don't pay for options given to us by nature and technical innovation." His prominent example here is Thales of Miletus. Aristotle gave us the story that Thales secured the rights to wine presses at a relatively low rate, which is an option: he had the right, not the obligation to use the wine presses. When the harvest proved to be bountiful, and so demand for the presses was high, Thales charged a high price for their use and reaped a considerable profit. Taleb states the key to Thales's fortune was his awareness of his 'lack of knowledge,' in that as he owned an option he enabled himself to benefit from uncertainty.

I do not have data on wine presses circa 600 BC, but currently, such options are priced. For example, in futures markets, there is a thing called the basis, the difference between the future and cash price of a good reflecting the yield on the asset vs. the opportunity cost of money (ie, the interest rate). One of the components of that basis is the convenience yield, in that if there is a shortage, having the actual commodity will have a great value. For goods subject to shortages (eg, wine presses), this increases the cash price over the future price because having the good on hand can be very valuable in a crisis. William Easterly has argued that Western countries dumping grain on African countries during shortages deprives farmers of the essential revenue for many farmers, the profit that pays for merely breaking even during normal times, discouraging endogenous markets. Spikes in demand are a large part of any asset owner's income. Simple awareness of a 'lack of knowledge' about future demand is not helpful, because it presumes the seller assumes the convenience yield or option value is zero, and that's unlikely. 

Taleb gives other examples, for instance: avoiding doctors, having different alternatives on vacation or for dinner, the ability to switch jobs, a rent-controlled apartment, or being married to an accountant but an occasional fling with a rock star. To the extent these options are desirable, they are not underpriced, and certainly not free. People realize this, which is why they are rarely mentioned outside the book. The bottom line is that things with convexity are too costly in general because people love lottery tickets; convexity is not the essence of any antifragile example because it is generally too costly. 

Another misleading application is in biology or economics, where populations or markets that have had to withstand more competition and external variability dominate those with benign environments. A bacteria population in the lab loses its ability to withstand stressors, an industry protected from new entrants loses its ability to compete when technology changes the game, or an industry protected against failure becomes bloated and less robust. Systems that allow or even encourage failure thrive relative to those that protect its members from failure. In economics, the basic idea of exiting losing businesses is perhaps the most crucial advantage of the free market over socialism. Failure strengthens the herd, whether animals or firms

Taleb states that his notion of antifragility is behind the following: "evolution, culture, ideas, revolutions, political systems, technological innovation, cultural and economic success, corporate survival, good recipes, the rise of cities, cultures, legal systems, equatorial forests, bacterial resistance." Success within these domains comes from looking at the competitive success of groups benefiting from hormesis vs. those insulated from it. Competition leads to the resiliency and efficiency needed to survive.

Back to his definition of antifragile, it is not that the prospering agents are convex to stress, instead that as survivors their progeny takes over the extinct's lebensraum; the dynamic effect of robustness in a system based on survival of the fittest looks like convexity.  Indeed, Taleb notes the property applies to the group, not the individuals: "the surviving cohort is stronger than the initial one—but not quite the individuals since the weaker ones died." So here, like with hormesis, he notes it is a metaphor, but really not. He is quite aware that such systems are not direct examples of antifragility because the agents that generate convexity at the higher level are merely robust, via hormesis.

A robust business has to innovate because every business model changes over time. Jeff Bezos notes success takes someone with a stubborn vision yet flexible on details, because without a strong vision one's strategy overreacts to current failure or success, while without flexibility one cannot adapt when things do not work precisely as planned, as they always do. Note here we see a classic example of 'moderation in all things,' where the optimum lies between an excess and deficit.  In contrast, Taleb describes a caricature of vision via his "teleological fallacy" which is the "illusion that you know exactly where you are going." One could go on all day about the inadequacies of strawmen, as Taleb does.

An industry protected from failure or change via union work rules or bailouts removes the micro-instability needed at all levels of a company to develop innovation, robustness, and a healthy familiarity with failure. While some rules and regulations are good, most are merely a pretext for barriers to entry protecting current workers and firms. This is just about allowing hormesis to do its thing.

Becoming excellent first requires a lot of domain-specific hard work, with a focus that enlarges some things while excluding others. Jordan Peterson argues that flow comes from operating at the edge of our competence, with enough mastery to generate satisfaction yet enough novelty to be challenging. To an outsider, such explorations can seem like random tinkering, but for an expert, it is a variation on their unusual intuition. Suggesting that the general strategy of accumulating convex exposures is the key to success is a profound error, as in the difference between the benefit of anger, and anger directed at the right person at the right time and in the right way.

An essential attribute of someone who innovates is their ability to embrace failure. Adversity is a great teacher, why mother giraffes knock their newborns down just after first learning how to stand;  they have to learn quickly on the African grasslands. Embracing failure is easy to say but hard to do, which Taleb acknowledges. Actually, he doesn't explicitly acknowledge this, but as he never mentions why he worked for several banks while he was a trader (blow up?) or the fact that every close friend he mentions in Antifragile is highly successful, suggests he sees failure as a characteristic of unremarkable losers.

Failure will always be costly, and due to moral hazard, no one will sell you a put option on your failures.  In addition to acclimating ourselves to failure, just as useful are the Christian virtues of faith, hope, and love. If you have faith in what you hope for regardless of your economic success, and love someone who loves you for who you are, failures under the sun are not so terrible. This allows you to explore more virgin territory so that when the unexpected happens, you might be in the right place at the right time.

There are two ways to generate an option payoff. One is to buy an option; another is via dynamic replication, which involves doubling down a position as it becomes more in-the-money. The outsized success of winners over losers in dynamic systems generates large convexities, but to be a winner, the keys are not buying options, but rather, via resilience acquired through hormesis, survive long enough to achieve success indirectly via a combination of vision, excellence, and flexibility (obliquity). To describe the essence of this as creating option payoffs focuses people on explicit optionality, as opposed to the optionality that comes via hormesis. Resilience plus what is often called common sense, generates outsized winners in dynamic zero-sum competition. This is why everyone mentions examples of hormesis, waves their hands, and hopes no one notices the bait-and-switch.

Promoting the new idea that acquiring options on the next Black Swan is the basis of "our own existence as a species on this planet" is the sort of hyperbole you hear at TED talks. It is the sort of thing bureaucrats love because they are generally too high up to have much domain-specific expertise, and the incoherent but plausible-sounding theme allows one to talk about strategy without actually knowing anything specific. Then you give examples of your great idea that are really something else entirely, and fade to black...