Note 13: No-Arbitrage Imbalance
| Author | Michael Feldman |
|---|---|
| OIP-Number | 1 |
| Status | WIP |
| Created | 2022-05-19 |
This note aims to address the following issues:
- What is the open interest imbalance in the no-arbitrage limit?
- What is the exposure the protocol assumes in the no-arbitrage limit?
Summary
The open interest imbalance we should expect to approach in the no-arbitrage limit is
\[ \frac{\mathrm{OI}_{imb}}{\mathrm{OI}_{tot}} = \frac{\Delta r}{2k} \]
where \( \Delta r \) is the difference in “risk-free” rates between the quote and base currencies for the market. \( k \) is our funding governance parameter.
“Risk-free” rates for each respective currency are likely to be the native staking yields, given covered and uncovered interest rate parity arguments. See this series by Lily Francus for more detail.
The exposure the protocol is liable for in the no-arbitrage limit is near zero given the funding “burn” mechanism – the protocol is paid in full for the risk assumed. To achieve this level of stability, however, governors should ensure \( k \) satisfies
\[ \Delta r \ll 2k \]
This condition results in concavity of the profit liability function \( \mathrm{PnL}_{liability}(\tau) \) (i.e. printing risk protocol takes on) in the no-arbitrage limit.
Risk-Neutral Imbalance Level
From Equations (29)-(31) of the V1 Core whitepaper, funding is given by
\[ f(I) = 2k \cdot I \]
where
\[ I(\tau) \equiv \frac{\mathrm{OI}_{imb}(\tau)}{\mathrm{OI}_{tot}(\tau)} \]
\( f(I) > 0 \) implies longs pay shorts and the opposite for \( f(I) < 0 \).
Covered interest rate parity implies the expected funding rate paid by the longs to the shorts in the no-arbitrage limit \( f(I) = f_Q \) will be
\[ f_Q = r_{quote} - r_{base} = \Delta r \]
the difference in “risk-free” rates between the quote and base currencies of the inverse market. In practice, the “risk-free” rate for OVL should be the yield on the single-sided staking liquidity mining pool (Overlay’s version of T-notes). Similarly for ETH, the “risk-free” rate should be the yield from ETH 2 staking.
The risk-neutral open interest imbalance \( I(\tau) = I_{Q} \) would then be
\[ I_{Q} = \frac{\Delta r}{2k} \]
PnL Liability
From Equation (32) of the V1 Core WP, the protocol is liable for
\[ \mathrm{PnL}_{liability}(\tau) = -\mathrm{OI}_{b_r}(\tau) \cdot P(0) + \mathrm{OI}_{imb}(\tau) \cdot [P(\tau) - P(0)] \]
at a time \( t=\tau \) in the future. \( t = 0 \) is now. \( \mathrm{OI}_{imb} \) is the open interest imbalance over time. \( \mathrm{OI}_{b_r} \) is the open interest removed from the system given the protocol’s exposure to the market due to the imbalance.
The profit liability is the amount of OVL the protocol has to print given the initial open interest imbalance at the current time.
Note that
\[\begin{eqnarray} \mathrm{OI}_{imb}(\tau) &=& \mathrm{OI}_{imb}(0) \cdot e^{-2k\tau} \\ \mathrm{OI}_{b_r}(\tau) &=& \mathrm{OI}_{tot}(0) \cdot \bigg[ 1 - \sqrt{1 - I(0)^2 \cdot ( 1 - e^{-4k\tau})} \bigg] \end{eqnarray}\]To simplify the math for the purposes of this note, take the spot price to be driven by a Wiener process
\[ \frac{P(\tau)}{P(0)} = e^{(\mu - \frac{\sigma^2}{2}) \tau + \sigma W_{\tau}} \]
with drift \( \mu \) and volatility \( \sigma \). Then, the expected value is given by
\[ \mathbb{E}_Q\bigg[ \frac{P(\tau)}{P(0)} \bigg] = e^{\Delta r \cdot \tau} \]
under the risk-neutral measure \( Q \).
General
Graph of PnL liability not necessarily assuming no-arbitrage imbalance level at \( \tau=0 \).
No-Arbitrage
Graph of PnL liability assuming no-arbitrage imbalance level at \( \tau=0 \).
Stability
The system appears very stable in the no-arbitrage limit when \( \Delta r \ll 2k \). Stable in the sense of inflation of currency supply is nominal given the profit liability charts above.
The system appears to potentially become unstable when \( \mu \geq 2k \) and no-arbitrage breaks down. Additional circuit breaker and cap mechanisms we have limit the damage to these types of events.
To verify stability, one can look at the first and second derivative of the expected value under the risk-neutral measure of the profit liability with respect to time in the limit as \( t \to 0 \)
\[\begin{eqnarray} \lim_{\tau \to 0} \frac{d}{d\tau} \mathbb{E}_Q\bigg[\mathrm{PnL}_{liability}(\tau)\bigg] &=& P(0) \cdot \mathrm{OI}_{imb}(0) \cdot \bigg( \Delta r - f_Q \bigg) = 0 \\ \lim_{\tau \to 0} \frac{d^2}{d\tau^2} \mathbb{E}_Q\bigg[\mathrm{PnL}_{liability}(\tau)\bigg] &=& P(0) \cdot \mathrm{OI}_{imb}(0) \cdot \Delta r \cdot \bigg( \Delta r - 2k \bigg) \end{eqnarray}\]This imposes a condition on our governance chosen value for \( k \) in order to achieve a concave liability function in time i.e. \( \lim_{\tau \to 0} \frac{d^2}{d\tau^2} \mathbb{E}_Q\bigg[\mathrm{PnL}_{liability}(\tau)\bigg] < 0 \):
\[ \Delta r < 2k \]
Fortunately, we calibrate \( k \) using VaR at 95, 99% levels, which will produce \( \Delta r \ll 2k \). As long as this condition holds, the system should be stable.
Governance will need to monitor \( \Delta r \) vs \( 2k \) continuously and adjust \( k \) accordingly in the event the spread in “risk-free” rates between OVL and e.g. ETH increases significantly. Potentially in V2, \( k \) could be adjusted dynamically based on current staking rates in a manipulation-resistant way.