Negative Interest: Zombie Edition

[jalextowle]

Negative Interest: 🧟 Edition

When _applyCheckpoint is used to close positions at maturity, we apply the share proceeds of closing these positions to the share reserves. At this point, the reserves will reflect any negative interest that accrued during the life of these positions.

Whenever a new checkpoint is minted, the zombie interest mechanism will deduct positive interest that accrues on the matured positions. Thus, after applying the checkpoint, the base amount reserved for the zombie positions will be equal the total base proceeds required to pay out all of the zombie positions. This fact makes zombie positions sensitive to negative interest that accrues in the current checkpoint as the following example demonstrates.

Example

Suppose that Hyperdrive has no outstanding longs or shorts, $c = 1$, $z = 1,000$, and $z_{zombie} = 0$.

1. Alice pays 95 base for a long of 100 bonds.
   • $z \mathrel{+}= 95 \implies z = 1,095$
2. The term passes and some interest accrues. The share price is now $c = 1.05$.
3. checkpoint is called. Alice's proceeds are deducted from the share reserves and placed into the zombie share reserves.
   • $z \mathrel{-}= \tfrac{100}{1.05} \implies z = 999.76$
   • $z_{zombie} \mathrel{+}= \tfrac{100}{1.05} \implies z_{zombie} = 95.24$
4. A checkpoint passes and more interest accrues. The share price is now $c = 1.06$.
5. checkpoint is called and the zombie interest is deducted from the zombie share reserves and placed into the share reserves.
   • $z \mathrel{+}= \tfrac{1.06 - 1.05}{1.06} \cdot 95.24 \implies z = 1,000.66$
   • $z_{zombie} -= \tfrac{1.06 - 1.05}{1.06} \cdot 95.24 \implies z_{zombie} = 94.34$
6. A negative interest accrues. The share price is now $1.055$.
7. Alice attempts to close her position. Since the starting share price of $1$ is less than the ending share price of $1.05$, negative interest isn't assessed. Her share proceeds are calculated as $94.76$ which is greater than the zombie share reserves. This will be handled gracefully by zeroing out the zombie share reserves; however, the share reserves plus the zombie share reserves will be greater than the total number of shares tracked by the pool.

Proposed Fix #1 (Doesn't Work)

Whenever zombie interest accrues, we record the maximum share price at which we collected zombie interest, $c_{max}$. Having this variable around allows us to detect when negative interest accrued on the zombie share reserves and also has the side benefit of letting us easily collect zombie interest across checkpoint gaps.

Using $c_{max}$, we apply the full negative interest haircut to zombie longs and shorts. In the example, Alice's share proceeds would be adjusted as:

$$ 94.76 + \tfrac{1.055 - 1.06}{1.055} \cdot 94.76 = 94.31 $$

This is smaller than the zombie share reserves. The discrepancy of $0.02$ is caused by rounding, and the true result would be much closer to the zombie share reserves.

Problem

The problem with this fix is that it doesn't track which zombie positions were effected by the negative interest. Consider the following example:

1. The pool is initialized. $c = 1$.
2. Alice opens a long of 1,000 bonds.
3. The term passes and interest accrues. $c = 1.05$.
4. A checkpoint is minted. Alice's proceeds of 1,000 base are moved into the zombie share reserves and $c_{max}$ is set to $1.05$. $z_{zombie} = 952.38$.
5. Negative interest accrues and $c = 1.04$.
6. Bob opens a long of 100.
7. The term passes and no interest accrues. $c = 1.04$.
8. A checkpoint is minted. Bob's proceeds of 100 base are moved into the zombie share reserves. $z_{zombie} \mathrel{+}= 96.15 \implies z_{zombie} = 1,048.53$.

If Alice and Bob both redeemed their matured positions, the negative interest haircut would be applied to both of them. This means that Alice would receive $1000 \cdot (1 + \tfrac{1.04 - 1.05}{1.04}) = 990.47$. Bob would receive $100 \cdot (1 + \tfrac{1.04 - 1.05}{1.04}) = 99.04$. Collectively, this is equal to $\tfrac{990.47 + 99.04}{1.04} = 1047.60$, which is less than the zombie share reserves.

To summarize the problem, this solution incorrectly socializes losses across the entire zombie share reserves, which leaves some of the zombie share reserves unspent if all of zombie positions are closed after negative interest accrues.

Proposed Fix #2 (Works)

Anytime we move matured positions into (or out of) the zombie share reserves, we add (or subtract) the share proceeds to the zombie share reserves, $z_{zombie}$ AND we add (or subtract) the base proceeds to the new field called "zombie base reserves," $x_{zombie}$.

When we collect zombie interest, all we do is take the difference of $c \cdot z_{zombie}$ and $x_{zombie}$, check to see the difference is greater than zero, and move that amount of base to the share reserves.

When a trader closes a matured position, we check to see if $c \cdot z_{zombie} < x_{zombie}$. If the inequality holds, then negative interest has accrued on the zombie share reserves and we discount the trader's base proceeds by $\tfrac{c \cdot z_{zombie}}{x_{zombie}}$. Otherwise, we just pay out the base proceeds normally.

Let's try the example from "Proposed Fix #1" and see if we get a different result:

1. The pool is initialized. $c = 1$.
2. Alice opens a long of 1,000 bonds.
3. The term passes and interest accrues. $c = 1.05$.
4. A checkpoint is minted. Alice's proceeds of 1,000 base are moved into the zombie share reserves and the zombie base reserves. $z_{zombie} = 952.38$ and $x_{zombie} = 1,000$.
5. Negative interest accrues and $c = 1.04$.
6. Bob opens a long of 100.
7. The term passes and no interest accrues. $c = 1.04$.
8. A checkpoint is minted. Bob's proceeds of 100 base are moved into the zombie share reserves. $z_{zombie} \mathrel{+}= 96.15 \implies z_{zombie} = 1,048.53$ and $x_{zombie} = 1,100$.

If Alice and Bob both redeemed their matured positions, we discount both of them by the ratio $\tfrac{1.04 \cdot 1,048.53}{1,100} = \tfrac{1090.47}{1,100}$. This means that Alice would receive $1,000 \cdot \tfrac{1090.47}{1,100} = 991.33$. Bob would receive $100 \cdot \tfrac{1090.47}{1,100} = 99.1$. Collectively, this is equal to $\tfrac{991.33 + 99.1}{1.04} = 1048.49$, which is approximately equal to the zombie share reserves (the small discrepancy is caused by rounding to 2 significant digits).

This solution isn't perfect considering that Bob still takes a hit due to negative interest; however, it does ensure that all of the zombie share reserves can be paid out. Doing the accounting perfectly (by properly attributing negative interest to each checkpoint) appears to be computationally infeasible on Ethereum (I'm happy to be proven wrong here, but I haven't thought of a way to do it). IMO this solution is good enough.

[github-actions]

Hyperdrive Gas Benchmark

Benchmark suite	Current: 60742ab	Previous: c2e0f6c	Deviation	Status
addLiquidity: min	1600 gas	1600 gas	0%	🟰
addLiquidity: avg	65742 gas	64248 gas	2.3254%	🚨
addLiquidity: max	275068 gas	275046 gas	0.0080%	🚨
checkpoint: min	1150 gas	1172 gas	-1.8771%	✅
checkpoint: avg	47693 gas	47975 gas	-0.5878%	✅
checkpoint: max	191154 gas	202123 gas	-5.4269%	✅
closeLong: min	1558 gas	1580 gas	-1.3924%	✅
closeLong: avg	27677 gas	27908 gas	-0.8277%	✅
closeLong: max	150993 gas	148684 gas	1.5530%	🚨
closeShort: min	1549 gas	1549 gas	0%	🟰
closeShort: avg	29944 gas	29962 gas	-0.0601%	✅
closeShort: max	147229 gas	147189 gas	0.0272%	🚨
initialize: min	1538 gas	1538 gas	0%	🟰
initialize: avg	213117 gas	214974 gas	-0.8638%	✅
initialize: max	254775 gas	256760 gas	-0.7731%	✅
openLong: min	1487 gas	1509 gas	-1.4579%	✅
openLong: avg	50043 gas	50152 gas	-0.2173%	✅
openLong: max	184695 gas	185904 gas	-0.6503%	✅
openShort: min	1608 gas	1519 gas	5.8591%	🚨
openShort: avg	49285 gas	47423 gas	3.9264%	🚨
openShort: max	179656 gas	161536 gas	11.2173%	🚨
redeemWithdrawalShares: min	1575 gas	1575 gas	0%	🟰
redeemWithdrawalShares: avg	20026 gas	22540 gas	-11.1535%	✅
redeemWithdrawalShares: max	105972 gas	105958 gas	0.0132%	🚨
removeLiquidity: min	1639 gas	1661 gas	-1.3245%	✅
removeLiquidity: avg	147194 gas	149144 gas	-1.3075%	✅
removeLiquidity: max	323566 gas	323560 gas	0.0019%	🚨

This comment was automatically generated by workflow using github-action-benchmark.

[coveralls]

coverage: 95.069% (-0.2%) from 95.257%
when pulling 60742ab on jalextowle/feature/negative-interest-zombie
into 0953627 on main.

[jrhea]

This is fantastic! I'd be good if you pasted the spec text into the PR description. I'll wait to approve until you have time to make changes we discussed

[MrToph]

If Alice and Bob both redeemed their matured positions, we discount both of them by the ratio 1.04⋅1,048.531,100=1090.471,100. This means that Alice would receive 1,000⋅1090.471,100=991.33. Bob would receive 100⋅1090.471,100=99.1. Collectively, this is equal to 991.33+99.11.04=1048.49, which is approximately equal to the zombie share reserves (the small discrepancy is caused by rounding to 2 significant digits).

Another way to express this is that both receive their fair share of the remaining zombie reserves: shareProceeds = (dy / x_zombie) * z_zombie. I.e., Alice receives 1000/1100 * z_zombie, and Bob 100/1100 * z_zombie. Which means the losses are socialized proportionally to their bonds or original base proceeds.