Note 7: Cutting Off the Tails
| Author | Michael Feldman |
|---|---|
| OIP-Number | 1 |
| Status | WIP |
| Created | 2021-06-12 |
Issue to address with this note:
- How do we prevent payouts that would otherwise bankrupt the system? i.e. avoid becoming Taleb’s turkey
Context
Talib $Titan Turkey pic.twitter.com/6JvUjkYddn
— Flood (@ThinkingUSD) June 16, 2021
With Overlay, a death spiral can occur through the protocol’s OVL-X inverse markets, where X is ETH, DAI, etc., when all open interest on these markets is bearish OVL and a significant price drop of OVL relative to X occurs.
The payout of bearish OVL trades trends toward infinity as the price of OVL relative to X goes to zero, given the nature of the inverse market. Infinite printing by the market contract from this payout would lead to a collapse of the system.
Background
Recall, passive OVL holders are effectively the counterparty for any PnL resulting from an imbalance in open interest on a market, taking on dilution risk – the protocol prints more OVL to the circulating supply to cover imbalance profits. Funding payments from longs (shorts) to shorts (longs) draw down this risk over time.
The protocol remains liable for any PnL associated with an imbalance in open interest on a market. The amount of OVL the protocol will need to print at some time in the future \( m \) to cover this PnL is
\[ \mathrm{PnL}(m) = \mathrm{OI}_{imb}(0) \cdot (1 - 2k)^{m} \cdot \bigg[ \frac{P(m)}{P(0)} - 1 \bigg] \]
where we make the simplifying assumption here that all existing positions were built at the same time \( 0 \). An appropriate funding constant \( k \in [0, \frac{1}{2}] \) for each market can be determined through statistical methods, modeling price \( P(m) \) at time \( m \) as a stochastic process. Governance can then adjust \( k \) on-chain based on value at risk metrics all token holders are comfortable with.
However, what if imbalance risk isn’t drawn down quick enough through funding or if our models are incorrect? How do we definitively cap dilution risk over a set amount of time?
This is particularly worrisome for the OVL-X inverse markets, as PnL to be printed for a bearish OVL imbalance looks like
where we’ve normalized with respect to initial imbalance and initial price. As the price of OVL relative to X trends toward 0, the protocol is forced to print more and more OVL for profitable bearish trades. Bears can then dump this OVL on spot alongside further bearish positions on the inverse market to collapse the spot price more, print more OVL, and continue dumping. Rinse and repeat, and the system eventually prints to infinity.
Stopping the Death Spiral
With this note, we build on prior caps work from WPv1.
To mitigate the death spiral, we can include:
-
Payoff caps – limits the maximum percent change in price offered to traders in a position’s payoff function
-
Dynamic OI caps – limits new position builds when the market has printed an excessive amount of OVL over a prior period of time (cooldown on trading)
Constant payoff and OI caps make it possible to enforce a worst case amount printed per trade. Dynamic OI caps take this a step further and make it possible to enforce the worst case amount printed over a period of time, by limiting trading if excessive printing has happened in the recent past. Thus, the combination of payoff caps with dynamic OI caps offers us an avenue to enforce a worst case inflation rate.
The maximum amount the system is allowed to print over a given period of time will then degenerate to
\[ \mathrm{PnL} |_{max} = C_{\mathrm{OI}} \cdot C_{P} \]
irrespective of future price.
-
\( C_{P} \) is the payoff cap putting a maximum on \( [\frac{P(m)}{P(0)} - 1] \) offered to traders. This can be set
-
\( C_{\mathrm{OI}} \) is the dynamic OI cap putting a maximum on \( \mathrm{OI}_{imb} \) taken on by the market contract. This can be set and lowers/raises dependent on the amount of printing that has occurred in the prior \( \omega \) blocks.
Payoff Caps
Payoff caps limit downside exposure the protocol has to tail events in the price of the underlying feed, enforcing a predictable (non-random) worst case scenario per trade. Consider a more empirically accurate model than log-normal for the price of the TWAP market:
\[ P(t) = P(0) e^{\mu t + \sigma L_t} \]
where \( L_{t} \) is a Levy stable stochastic process with increments that follow a stable distribution \( L_{t+\tau} - L_{t} \sim S(a, b, 0, (\frac{\tau}{a})^{\frac{1}{a}}) \). \( a \) is the stable distribution stability parameter, \( b \) is the skewness parameter, and \( c = (\frac{\tau}{a})^{\frac{1}{a}} \) is the scale parameter, for time increments of length \( \tau \). \( \mu \) is a drift parameter and \( \sigma \) is a volatility parameter.
While this reduces to Geometric Brownian motion for \( a = 2 \), most price feeds exhibit fat tails, delivering a model parameter value of \( a < 2 \). PDF plots of the stable distribution give a sense for the effect fatter tails can have. Comparing \( a = 1 \) (Cauchy distribution: blue line), \( a = 1.5 \) (orange line), and \( a = 2 \) (Normal distribution: green line) with zero skewness and standard \( c = 1 \) scale:
As we increase the fatness of the tails of the stable distribution (as \( a \) moves from \( 2 \to 1 \) in the plot), more weight is assigned to values near the peak of the distribution but with less of a drop off as we approach the extreme ends of the distribution.
Value at risk (VaR) metrics like what we used in our prior risk note can be misleading. While we might be confident we won’t lose more than the VaR amount with a probability of \( 1-\alpha \), any losses that do occur in the accepted level \( \alpha \) of uncertainty can be catastrophic. In the language of the inverse market payoff, while we might expect that 99% of the time OVL price won’t decrease more than 85% relative to X, VaR won’t tell us how much of a loss to expect the 1% of the time it actually does decrease more.
Worse, improper modeling of tail behavior makes it such that the 99% confidence level we thought we had, actually turned out to be closer to 90%. It’s easy to see these issues when examining the upper range of the CDF of the stable distribution for the same values of \( a \in [1, 1.5, 2] \):
where the horizontal red dashed line represents a CDF value of 0.99. An anticipated upper bound of 3.3x on the percentage change in log price when modeling with the normal distribution (green) with a confidence level of 99%, can turn out to be an actual upper bound of 32x when modeling with the Cauchy distribution (blue) using the same degree of confidence. This is an order of magnitude difference that can result from model or calibration error.
How do we then ensure the protocol is robust with respect to tail events in the price of the underlying feed? Guided by Taleb’s work, we cut off the damage associated with the tails: simply by setting a maximum value in the contract payoff for the price delta each trade can have. For the inverse market, the prior PnL plot would reduce to something like
where any price changes that occur in the tails (orange shaded area) lead to the same finite PnL paid out by the system (top border of orange shaded area). This definitively eliminates the possibility of a single trade printing an infinite amount of OVL.
The price delta at which to place this cap can be linked to our prior VaR work.
Dynamic OI Caps
Looking at prior \( n \) update periods to see how much market has printed over a rolling window. Compare with max worst case local inflation rate willing to tolerate (market param).
If larger than max inflation rate, market dynamically lowers OI cap to zero to prevent new positions from being built until long enough time has passed for window to see rolling inflation rate below max. Then, we are setting a worst case inflation rate per market in stone instead of being solely probabilistic.
In practice, what does this look like in readjusting the inflation cap upward after cooldown period? Additionally, how should we gradually ease cap down if rate seems to be increasing over time (acceleration), so dynamic cap is somewhat smooth in readjustments?
Mitigates death spiral through two layers:
-
Cap on contract payoff limits infinite printing from one existing set of short positions on inverse market. We know what is the worst case amount we can print on one “cycle” of OI builds.
-
Dynamic OI cap prevents recycling of collateral from capped short payout immediately into a set of new short positions on the inverse market, slowing traders ability to ride price down further after dumping prior profits – a circuit breaker. Effectively can ensure worst case inflation rate by dynamically limiting max aggregate OI market is willing to take on for prolonged period of time.
Do we have problems if keep bumping up against max inflation rate cap and lowering caps or short circuiting trading constantly? Is this a reason for having smooth increases/decreases in dynamic cap by market contract? Or are sudden stops better?