Saturday, May 25, 2024

Guillaume Lambert’s Panoptic


Panoptic CEO Giaullome Lambert and COO Jesper Kristensen

Panoptic promotes a vision where users can create virtually any standard combination of puts and calls on the blockchain without an oracle, auction, or limit order books. This would be very useful because options are difficult for market makers because of their convexity and leverage. An option's lambda captures leverage:

Lambda = option%change/underlying%change

A lambda of 10 implies a 1% price change generates a 10% option price change. Option lambdas range from 2 to 100 depending on volatility, maturity, etc., but generally are around five to 15. This is one of the primary reasons options are popular: option buyers can get 10x leverage without worrying about margin. Options often expire worthless, so they aren't less risky than leverage, but removing the liquidation scenario simplifies it.

With only leverage, this would not be a problem for market makers, as it would just be the difference between a market maker trading 10x more size than another. However, options generally have a high convexity, as measured by its gamma.

Gamma = change in delta/change in underlying price

With the delta changing, market makers are not just trading an asset 10x the size of its underlying but must hedge this convexity, which is costly. This volatility increases the spread on options relative to stocks due to the gamma alone. The three orders of magnitude greater latency in the blockchain exposes option market makers to insurmountable adverse selection.

My favorite solution would be to create an option contract that traded on the implied volatility, not the price. As one wag once said, “Implied volatility is the wrong volatility in the wrong model to get the right price.” All Black-Scholes inadequacies can be rectified via a suitable skew adjustment to the implied volatilities for various maturities and moneyness. Implied vols fluctuate much less than the price, generating an order of magnitude less option price volatility than actual option prices. One could trade on implied vol and then use a forward-starting underlying price, such as the price read from a liquid AMM's contract, and apply the Black-Scholes model to calculate the price. I used to work for an options market maker, and implied vols are how traders intuit prices anyway, as multiple rolling maturities and strikes make option prices an afterthought.

A New Way?

The basic stochastic process used in financial derivative theory started in biology. In 1827, Robert Brown described the path of pollen grains moving in water in what was to be called Brownian motion, the forerunner of the random walk applied to stock prices. Thus, it makes sense that Guillaume Lambert, a Cornell biophysicist, was well-equipped to analyze option-like properties of constant product automated market makers like Uniswap. After writing a few pieces and getting a Uniswap grant, he eventually secured funding to start Panoptic, a protocol for buying and selling option positions based on AMM liquidity provider (LP) positions.

His fundamental insight is that since Uniswapv3 LP positions have negative gamma, with ranges defined very granularly (every ten basis points), one can use these as the base for replicating any option. The basic idea comes from the Black-Scholes partial differential equation, which shows that an option premium equals the expected convexity cost, the expected value over all expected price paths, covering different gammas and stock prices.

If the option seller delta hedges continuously, his cost will be based on a similar function, the only difference being that volatility will potentially change over time (this was rebranded as loss versus rebalancing, but it's the same thing).

Thus, the PnL for the hedged seller is just the difference between the realized and implied variances. In equilibrium this is zero, implying the expected convexity costs (hedged or unhedged) equals the option premium.

If a Uniswap pool is priced so LPs have a zero PnL, or perhaps some equilibrium gamma risk premium, the fees generated for each range equal their expected dollar gamma. One then just uses various ranges to create arbitrary option payouts, and the realized fee revenue will generate the requisite option premium for the realized gamma. This differs from standard options, where a buyer pays the option premium upfront.

Thus, Panoptic presents the standard out-of-the-money LP position as a covered call, which seemingly can be captured via a liquidity range within an AMM pool.

At 30k feet, the ability to create options like those above seems plausible. The problem is that the options within an AMM LP position are a set of micro-straddles the seller can exercise at any time. For any one block, the option is only active in a small range, 0.1% wide. Outside this range, the LP’s options generate no fees and incur no convexity costs.

For example, consider the following range covering 104 to 106 (the same logic would apply to a range of 0.1%). This shows the standard covered call or short put nature of LP positions, in that if the current price is 108 to the right, the LP position is short a put; if the current price is to the left at 100, the LP position is effectively in a covered call position.

What makes this weird is the gamma for these ranges is zero outside their range.

Up to the lower range bound, the seller of this option (the LP) gets no option decay premium; gamma is zero, and so is the fee revenue.

As a practical matter, this means arbitraging an LP using listed options would be a nightmare. Consider this LP range and a 1-business-day option with a similar gamma.

The smooth Gaussian distribution to the standard option, even with a 1-day maturity, is a poor fit. See the net delta and gamma below.

The spiked changes in delta and gamma at the end of this +/-5% range imply that standard options will not help hedge one's LP position, and it would be better to simply delta hedge the LP position. This highlights how the LP position looks like a standard option only from far away. When the LP position is out of the money, it is not option-like; when it is active, the shortest-dated options traded on Deribit will be informative but otherwise useless.

Out-of-the-money Option as a Synthetic Put/Call Dynamic Strategy

Standard Put Option

While it's easy to generate a graph showing how an LP position is like an out-of-the-money put or call, a better analog is a synthetic version of these options. This is because a standard option has a positive value at the start, paid by the buyer to the seller, which does not apply to LP ranges. For example, if the current price is $3500, a put can be calculated using a simple lattice for the future value of an asset and its corresponding asset prices.

Long Stock with Put

Expected Value = $3,559

Minimum Value = $3,500

Option cost $59

Without the put option, the expected price is $3500 (multiply final prices with final probabilities and add them up). The put option replaces the bottom price with the strike price of $3500. When multiplied by their various probabilities, the expected value is now $3559, implying the put value of $59. A buyer can pay $59 to raise his worst-case scenario from $3274 to $3500. These two scenarios have the same expected value, and some find the option scenario preferable.

Stop-Loss Strategy

Now, consider the stop-loss strategy. Here, one owns an asset at $3500 but instead of paying for a put option, he adopts the dynamic strategy to sell it when the price falls. This also raises the price in the worst-case scenario, but here, the expected value of the asset is the same, $3500, and there is no upfront cost.

Stop-Loss Strategy

Expected Value = $3500

Minimum Value = $3385

Another way to see this is to note that with the stop-loss, while one can raise the minimum stock sale price up to the strike, the cost comes from missing out on those times when the price rebounds. The expected value of all these scenarios is the same (once one includes the cost of the put option).

Put Replication Strategy

A better analog to the Panoptic put option is a replication strategy that sells when the price falls below a certain level and then repurchases it if it goes back above. This enables the owner to capture more of the upside. An out-of-the-money range is like a synthetic option replication strategy in that it generates no cost or revenue when not touched.

Consider the trinomial lattice shown below. With a trinomial lattice, the price moves up, down, or stays the same (roughly ¼ for up and down, ½ for staying the same). The price starts at $3500, and each period experiences a 1% volatility. He sells when the price falls below $3400, and when it rises above $3400, he repurchases it. Thus, in the first green cell with 6.3%, in the next period 1.6% chance the price will fall below $3400, and a sale will occur. In the period just to its right, in period 4, we do not assume 3.2% of these cases will sell, only those that are moving downward, so that would be about  2.4% (~0.25*0.095). Similarly, we estimate the buy-backs using only those probabilities below $3402 that rise in the next period. So, in period 7, we have about 3.1% of that period buying at $3402 (0.325*0.055).

The result is to better match the put option regarding the sale price. As with the put option, the sales-price PDF dominates the base case.

Of course, I'm missing something because the put raised the expected sale price but cost money. Here, the synthetic put cost nothing but a bit of work. The cost comes from the extra trades reflected in the fact that the synthetic put's PDF is much higher than the other two distributions. This extra selling comes from extra buying. Every buy before the final period represents an extra round-trip trade compared to if we just ignored the price until the end.

In those extra transactions we sell low ($3355) and buy high ($3402), a loss. Unsurprising, the net cost of these trades exactly equals the cost of the put. This is an excellent way to intuit the earlier assertion that the expected convexity costs of a continuously delta-hedged option equals the option premium,

Out-of-the-money (OTM) puts and calls would provide users with an attractive strategy for implementing this strategy, removing the need to place all those orders as in a dynamic strategy. The problem is that I don't see anyone wanting to sell them.

Unattractive Short Out-of-the-Money Put PnL

For the foreseeable future, one will not arbitrage volatility selling an LP position against another, as that requires low spreads. It took decades for that sort of liquidity to build up in tradFi, and we are not close. Even today in tradFi, those who arbitrage relative vol between two assets are almost always professionals with the lowest fees, capital requirements, and fastest trading platforms. Retail option traders generally have a view about the future price, not implied option movements, because they do not have the low-cost access that would make trading this profitable.

The OTM option sellers on Panoptic will see an unfamiliar payoff. Consider the retail tradFi payoff

Revenue: option premium (expected convexity cost)

Retail Cost: Max(strike – price, 0)

In Panoptic, assuming the AMM is priced at cost, we have

Revenue: realized fees (realized convexity costs)

Retail Cost: Max(strike – price, 0)

For Panoptic the revenue switches from expected to realized convexity (which, if priced correctly, shows up in LP fees). This dramatically changes the payoffs for the OTM option seller. For the happy case where the price rises, the payoff for the standard OTM put seller gets his option premium as pure profit. The Panoptic put seller, however, gets nothing unless the price ends close to his strike. For the unhappy case where the price goes down, the scenario is similar. [I had to break these two cases into different graphs because the downside pnl is so much larger.]

I don't think retail put sellers will find this payoff attractive. The Panoptic put seller will be betting that the price will not go much below or above their strike, while a standard retail put seller thinks the price will increase. This is because in Panoptic the seller's revenue looks like the graph below, quite different than what one sees in a standard option.

Panoptic's OTM put is more like an out-of-the-money straddle. That's a unicorn in retail option strategy because no one wants it.

In contrast, the standard at-the-money straddle is potentially attractive, as the differences are relatively minor between the Panoptic and standard option PnL.

More Problems

Several unwelcome features remind me of another Uniswap-promoted complementary product, Squeeths. This product was sold as a way to hedge LP risk by allowing people to go long perpetual gamma, the opposite of what is generated by the LP position (short perpetual gamma). It traded on the price of the underlying squared, such as squared eth.

However, to match an LP position’s gamma, the squeeth requires at least ten times the delta, which must be hedged elsewhere. The capital required would explode. A good hedge reduces required capital.1 Another terrible thing about squeeths is they are not like buying a variance swap, which is like a futures contract on variance. Instead, they are a sequence of forward-starting one-day variance swaps, meaning, if you thought volatility was cheap, by the next day, it would be repriced, so subsequent days are all afresh. Going long a squeeth was like a position on the future daily conditional expected variance over the realized variance, something not seen off the blockchain because no one wants it.

[I just checked the squeeth team's Twitter, and they are hiring developers, highlighting no one cares whether something has or will work, just that it’s plausible. I used to think squeeths were the worst idea I have ever seen on the blockchain, but that's reserved for in-game leveraged sports betting (the creators thought the standard 5% vig would magically disappear in-game). As if a bet where the payoff is +100 or 0 needs leverage.]

Similar to squeeths, Panoptic positions have some unattractive practical aspects (though, not that bad). For example, there does not seem to be a reason to use actual LP positions. These are created when the option seller mints the option, but if a buyer arrives, they are instantly exited. One could simply read the GlobalFee and liquidity for the relevant ticks off the Uniswap contract, get the requisite information, and avoid wasteful expense.

Then, there is the cost of a seller canceling. As the essence of an LP position is that the option seller can costlessly exit each block, this is a significant change to the base short gamma position. TradFi options require a premium because people must be paid for locking them in over time, as only the buyer can exercise early. Daily options are popular but they too require a payment, so it's not like demanding the sellers don't revoke their straddles for only a few days is irrelevant.

Lastly, the option seller pays a commission. Considering LPs usually lose money, I don't think anyone will find selling options using LP positions as a base would make it an attractive stand-alone investment. The best targets are current LPs. One could give them the option to take their LP positions, and buyers would promise to give them the same payout, with an extra 5% or 10% on the fees. This would be easy to do, and the option sellers would be better off without cost or risk. Buyers could get long volatility without the vega premium one sees in tradFi, which could work.

Strangely, Guillaume occasionally mentions the low implied volatilities for Uniswap pools but has never documented this thoroughly. I would think showing how one can pay LPs 10% more and get access to long vega without paying the tradFi vega risk premium would be an excellent selling pitch. He could present various back-tests showing exactly how to make money or hedge volatility. Few people in defi—VCs, developers, or protocol CEOs—know much about the profitability of the most popular defi contracts, let alone how to take advantage of that.

I'm unsure what motivates this willful blindness to the actual profitability of the most active Defi contract. One cannot fix a problem that is considered taboo. I speculate that the lottery returns to prior projects that ultimately became zombies taught investors in this space to focus on the short term. Create something that looks like it implements something popular off the blockchain, note how it will be decentralized, etc., and ignore the details (eg, Augur). When everyone figures out the project is not viable, the insiders are all rich, and they move on to fund similar projects. Many faux-decentralized L2s and protocols exemplify this strategy.


Note that for an active LP position, we have

LP delta = liquidity / sqrt(p)

LP gamma = liq / (2*p^1.5)

the delta/gamma is 200, regardless of range width

In contrast, for squeeths

Squeeth delta: 2*p

Squeeth gamma: 2

Given ETH price of $3000, that's a delta/gamma of 3000

Friday, May 10, 2024

More Proof Perp Funding Rates are an Insider Scam

 On Tuesday, I argued that the main reason why perps do not dominate spot markets, as in TradFi, is that big players rightfully do not trust them for the following reasons:

  • LOB perps have to be centralized to lower costs and latency, creating a huge incentive for insider preferential access with no accountability

  • The prevalence of pump-and-dump tactics like staking, which, in the context of a perp governance token, has the sole purpose of reducing supply

  • No equilibrium theory justifies the 11% average perp funding rates

  • Perp funding rates uniquely rise after a rally and fall after bearish moves, unlike any TradFi futures market, taking advantage of bettor psychology

  • The academic work supposedly justifying the perp funding rate mechanism is profoundly different

Here, I want to add some data points provided by BitMEX perps. It highlights the market sentiment has nothing to do with equilibrium perp premiums, which makes sense because expected returns show up in the spot price, not the basis. We need grown-ups to take over, like when Walt took over Jesse’s lab on Breaking Bad.

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Basic Perp Premium Fairy Tale

First, it’s helpful to see the absurdity of the mechanism that presumably ties the perp price to the spot price. A perp is a derivative, and all derivative markets are built on an arbitrage, whether a futures or an option.

Define the Perp premium as the perp price over the spot price, in percent.

At the end of 8 hours, the average perp premium is translated into a funding rate debited by the longs and credited to the shorts.

Long FundingPayment(t+1) = - FundingRate(t+1) * Notional

Short FundingPayment(t+1) = + FundingRate(t+1) * Notional

FundingRate(t+1) = avg perpPremium (t to t+1) / 3

The story is that if the perp price is too high, it generates higher funding, encouraging shorts and discouraging longs, bringing the perp price back to the spot.

TradFi Futures Arbitrage vs. Perp Arbitrage

The main problem here is this is not arbitrage. With arbitrage, profits are certain. For example, in a futures market for oil with a standard fixed maturity (unlike a perp), the theoretical relationship between futures and spot is:

The futures premium, called the basis, is the difference in the interest rates of the traded asset to its numeraire interest rates. If one trades commodities, this is just the fiat interest rate minus the ‘own interest rate’ on the commodity (often negative, as it includes storage costs or decay).

To simplify the example, assume these interest rates are zero. In that case, if an arbitrageur sees the futures price above the spot price, he can lock in a profit by selling the futures and buying the spot. His payoff at maturity will be:

If Pfut0 > Pspot0

Payoff: longSpot + shortFutures

Substituting for starting and ending prices we have

Payoff: Pspot1 - Pspot0 + Pfut0 – Pfut1

Since at maturity, Pspot1 = Pfut1, the payoff is trivial, the known (i.e., constant) initial futures and spot prices.

Payoff: Pfut0 – Pspot0

This is a certain, riskless profit. Many people will be willing to lend to someone with access to a strategy that can make riskless profits, so this will be repeated until the futures and spot are equal.

  1. Do while Pfut0 > Pspot0

  2. Buy spot, sell fut

  3. Endo

Arbitrage acts as a money pump on anyone defending a non-equilibrium basis, making markets efficient because there’s a limitless supply of capital for those with genuine arbitrage opportunities.

In contrast, in the perp market, we have a net payout of

+Pspott – Pperpt  + Pperp0 - Pspot0 + fundrate

Here, the future price difference, Pspott – Pperpt, is not zero, as there is no delivery dictating equivalence. The perp premium could widen further, necessitating more margin or an early closure at a loss. Arbitrage will be tentative without the resolution we have in standard futures markets. There is too much risk to motivate unlimited capital into this opportunity. Further, the funding rate is not necessarily equal to  Pperp0 - Pspot0. For example, in an 8-hour funding period where the prior 7-hour average perp premium was negative, the funding payment will probably be negative even if the current perp premium is positive. No one trading on funding rates would touch the perp until the next funding rate window, leaving the perp market unethered. Alternatively, the perp premium could widen further, necessitating more margin or an early closure at a loss.

A high funding rate does discourage longs and encourages shorts, but this applies to long-term investors. Such investors look over weeks, if not months, so they are not concerned by the minuscule edge on which market makers and short-term arbs are focused. For day traders, the uncertainties around potential 0.1% funding returns make the funding rate irrelevant, which is why they tolerate 50%+ funding rates (assuming a perp-spot arb, that’s 4 transactions, which would cost at least 20 basis point total, implying an unactionable range that implies funding rates with a 80% annualized range).

BitMEX Quanto Perp Premium Mainly a Covariance Estimate

In 2018, BitMEX offered a way to trade ETH with long and short leverage. People loved it then, and it’s still actively traded there today. As users only deposited Bitcoin, they had to be creative, so they created a bastard version of an FX quanto, which allows traders to, say, take a position on yen while staying in USD.

The BitMEX ETH-USD perp payoff is a notional amount times the change in the USD ETH price.

This is the platypus of perps with notional in Bitcoin and returns in USD. To translate into a pure USD return, we capture the change in value of the Bitcoin notional in USD from t to t+1. We can ignore the funding rate for the gist here, as the funding rate is determined elsewhere.

Translate from Bitcoin notional into USD at time t.

And then, once completed, back into USD

This way, the round trip starts and ends in USD

Rearranging we get

Using net returns:

The expected return of this ETH-USD perp is

Given that the ETH and BTC returns are about 80% correlated, this final term is significant and generates a convexity in the USD return.

The expected value of E(x*y) is the expected covariance of x and y.

I have intuition about volatilities and correlations, not so much on covariances. I prefer to replace this with correlations and volatilities to get the annualized covariance.

Since Jan 2019, the volatility for BTC and ETH has been 67% and 83%, respectively, with a correlation of 82%, generating a covariance adjustment that annualizes to be about 46%. The annualized return premium of the BitMex ETH swap compared to the ETH USD return over that period: 94% vs. 140% (all data reflecting arithmetically annualized returns applied to 8-hour intervals).

Annualized Arithmetic Returns

Jan 2019-May 2024

Interestingly, the average funding rate over this period was around 55%. This implies an average ETH-USD perp funding rate of 10% for the shorts, net of the boost provided by the covariance term.

Annualized ETH-USD BitMex Quanto Funding Rate and Covariance

Jan 2019-May 2024

This highlights that those setting the BitMEX perp premium are good at their unpublicized objective. We see further evidence of the funding rate capturing the covariance by comparing the varying covariance with the funding rate over time. For example, when the covariance was very high in 2021, 74%, the funding rate averaged 94%, while when the covariance was low in 2023, 18%, the funding rate was 32%. The net funding rate, subtracting the covariance return, aligned with the ‘standard’ perp ETH and BTC funding rates over this period.

Annualized ETH-USD BitMex Quanto Funding Rates and Covariance

by Year

Another prime tell for manipulation is how the past, not the present or future, best explains the current funding rate. Here, I correlated the funding rate with three different returns, where P is the price of ETH in USD.

  • Funding rate: rate paid by longs at t+1 based on the average perp premium for an 8-hour window from t to t+1

  • Past return: Pt/Pt-3 days -1

  • Contemporaneous return: Pt+1/Pt -1

  • Future return: Pt+1+3days/Pt+1 -1

BitMEX ETH-USD quanto Perp Funding Rate Correlation

Jan 2019-May 2024

As funding rates are positively serially correlated, we should expect these to be the same sign. No equilibrium theory implies the past return to dominate the future or contemporaneous returns in explaining the current funding rate.

The Tether Yield Adjustment

A final data point is provided by the difference between the BTC-USD perp denominated in tether, USDT, and the perp denominated in BTC. Those going long BTC-USDT post tether, while those going long BTC-USD post bitcoin.

As tether can be lent out at a 6% higher rate than BTC, this shows up in their average funding rates. This implies that the LPs are affiliated with the exchange, because somehow they have to getting the advantage of this higher yielding coin.

BitMEX BTC Perp Annualized Funding Rate by Numeraire

Jan 2019-May 2024

How Do Users Know This?

If the perp premium merely reflected sentiment, it would not capture the BTC-ETH covariance, as with the ETH-USD quant perp, nor would it accurately capture the opportunity cost on UST. No one on the web provides tools to incorporate these adjustments, though clearly the LPs are doing the correct adjustments. When I search ‘BitMEX perp funding rate covariance,’ half of the items presented are blog posts by myself over the past five years, implying I am a significant percentage of everyone in the world aware of this issue who can talk about it, a set that effectively rounds to zero. While BitMEX provides examples of how covariance affects the total returns given various hypothetical BTC-ETH returns, they do not offer simple tools to calculate this using current data.

In traditional markets, convexity adjustments are explicit once known. The table above shows the detailed convexity adjustment needed when comparing Eurodollar futures with a Eurodollar swap. Before this relationship was understood, several large investment banks made actual arbitrage profits going long the futures and short the swaps. Around 1995, however, many prominent academic articles highlighted the arbitrage and noted the precise convexity adjustment needed. Eurodollar futures and swap traders all account for this subtle convexity adjustment, which is why such tools are ubiquitous.

BitMEX perp traders are oblivious to the current covariance or UST lending rate and just trade it like they would any other asset based on recent price movements. If the perp premium reflected bullishness, it would be quite a coincidence that bullishness correlated well with the recent ETH-BTC covariance. Those making markets, however, know what they are doing and are setting the perp premium to get their targeted funding rate.

BitMEX created perps out of necessity when there were no stablecoins. This anachronism persists because their LPs can profit from gaming this funding rate, as its typical users are generally trading noobs (who else would want 50+ leverage on an asset with a 60 vol?). Perp protocols will never dominate spot until they transition away from these sorts of cheap tricks, as institutions are rightfully wary of having significant positions on platforms that play silly games.  

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