Funding Rate

The funding rate is a mechanism exclusive to crypto perpetual pairs that periodically transfers payments between longs and shorts—longs pay shorts when the rate is positive, and shorts pay longs when it’s negative. It’s a zero-sum fee settled directly among traders, designed to incentivize traders to neutralize their delta exposure.

It’s determined by three components:

1. Open-Interest Imbalance (a bigger imbalance leads to bigger funding rates)

2. Theoretical Funding Rate (based on open-interest imbalance)

3. Convergence Regime (how quickly the current rate moves toward the theoretical rate)

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1. Open-Interest Imbalance

We normalize long/short open interest into a single variable $x$:

$$x = \frac{\mathrm{OI}_{\mathrm{long}} - \mathrm{OI}_{\mathrm{short}}}{\mathrm{OI}_{\mathrm{caps}}} \,,\qquad x \in [-1,1]$$

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2. Theoretical Funding Rate

The Hill function maps imbalance $x$ to an equilibrium funding rate $H(x)$:

$$H(x) = \begin{cases} R_{1}\,\displaystyle\frac{(a\,x)^{n}}{(a\,x)^{n} + b}\;+\;C & x \ge 0,\\[12pt] -\,R_{2}\,\displaystyle\frac{(a\,x)^{n}}{(a\,x)^{n} + b}\;+\;C & x < 0. \end{cases}$$

• $R_{1}, R_{2}$​: maximum upward/downward rate contributions
• $a, b, n$: shape parameters controlling steepness and midpoint
• $C$: base funding offset

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3. Convergence Dynamics

Holding $x$ constant ( $\frac{dx}{dt}=0$ ), the actual funding rate $y(t)$ relaxes toward $H(x)$ at a speed set by $A(x)$:

$$\frac{dy(t)}{dt} \;=\; -\,A(x)\,\bigl[y(t) - H(x)\bigr].$$

• If $y(t)>H(x)$, the funding rate decays downward; if $y(t)<H(x)$, it rises upward.
• $A(x)$ varies by market regime:

  Regime	Condition	Speed
  Slow (Dumping)	Imbalance is decreasing	Slowest
  Default (Pulling)	Imbalance is increasing	Moderate
  Fast (Dumping + Pulling)	Imbalance switches sign	Fastest

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With this structure, the funding rate always moves smoothly toward the imbalance-driven equilibrium, adapting speed based on how participants’ positions shift.