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Funding Rates

Funding rates act as a final controller to ensure our final objective

Residual risk

While the AMM addresses the marginal risk from the change in inventory using premium and rebate, there remain some residual risks, one of which is caused by the change in the price process of the underlying asset. To cover for such risk, we charge funding periodically.
Given
Xt X_t
is a function of an
nn
dimensional vector
q\mathbf{q}
, we need a way to define funding rate for each market such that the total funding accumulated across
n n
markets add up to
ρ(Xt+τ)(P+L)\rho(X_{t+\tau})-(P+L)
. To achieve this, we use a risk attribution mechanism called Euler allocation.
Let
Xt,iX_{t,i}
be the liability arising from
ii
th market. Using Euler allocation, we calculate the amount of funding needed from
ii
th market as
ρEuler(Xt+τ,iXt+τ)\rho_{Euler}(X_{t+\tau,i}|X_{t+\tau})
. Now, for each trader
jIij \in I_i
with position size
qijRq_{i_j} \in \mathbb{R}
where
jIiqij=qi\sum_{j\in I_i}q_{i_j}=q_{i}
, they are responsible for covering
ρEuler(Xt+τ,iXt+τ)qijqi\rho_{Euler}(X_{t+\tau,i}|X_{t+\tau})\frac{q_{i_j}}{q_{i}}
For example, if the market is long-heavy (
qi>0q_{i} > 0
), traders with long position
(qij>0)(q_{i_j}>0)
pays funding, while those with short position
(qij<0)(q_{i_j}< 0)
receives funding. This way, the AMM accumulates
F=i=1njIiρEuler(Xt+τ,iXt+τ)qijqi=ρ(Xt+τ)(P+L)F=\sum_{i=1}^n\sum_{j\in I_i}\rho_{Euler}(X_{t+\tau,i}|X_{t+\tau})\frac{q_{i_j}}{q_{i}}=\rho(X_{t+\tau})-(P+L)
, which is precisely the amount of funding needed to satisfy the invariant
Xt+τP+L+FX_{t+\tau}\leq P+L+F
with high probability.