Posted by: Nunchi Research
Abstract:
A fundamental challenge in on-chain finance is hedging continuously compounding, variable-rate liabilities, such as those found in protocols like Aave. A standard linear interest rate swap provides an imperfect hedge, leading to basis risk as its PnL does not grow in lockstep with the non-linear, exponential growth of the debt. This paper details the methodology behind the Nunchi Borrow Rate Perpetual, which is engineered to provide a “pathwise-perfect hedge” by tracking a compounding index rather than a raw rate.
1. The Nunchi Approach: From Rate to a Compounding Index
The innovation lies in what the perpetual contract tracks. Instead of the raw, instantaneous borrow rate (r_t), the perpetual’s price is mapped to a Cumulative Borrow Multiplier Index (J_t) .
- Index Construction: The index is constructed by integrating the annualized Aave variable borrow rate (r_t) over time to create a cumulative log-index, K_t, and then taking its exponential to create the direct multiplier J_t.
Jt=exp(∫0trsY ds) J_t = \exp\left(\int_0^t \frac{r_s}{Y} \, ds\right) Jt=exp(∫0tYrsds)
This J_t index starts at J_0=1 and grows exponentially, perfectly mirroring how a debt balance D_0 compounds on Aave to become D_t = D_0 * J_t.
- From Index to PnL: The perpetual’s mark price is an affine transformation of this index. The infinitesimal PnL is directly proportional to the change in the index: dPnL_t = N * dJ_t, where N is the notional. This direct mapping is the source of the perfect hedge property.
2. Worked Example: A Pathwise Hedge
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Scenario: A user borrows 1,000,000 USDC on Aave (D_0) and simultaneously longs the AAVEBORROW-PERP with the same notional (N).
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Assumptions: The Aave borrow rate is constant at 5% APR for 90 days.
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Aave Interest Cost: After 90 days, the Borrow Multiplier Index J_90 is ≈ 1.012422. The total interest accrued on the loan is:
ΔD_90 = $1,000,000 * (1.012422 - 1) = $12,422. -
Perpetual’s PnL: The cumulative PnL of the long perpetual is PnL_90 = N * (J_90 - 1). Since N = D_0:
PnL_90 = $1,000,000 * (1.012422 - 1) = $12,422.
3. Conclusion
The mark-to-market profit from the perpetual (+
12,422)exactlycancelstheinterestcostfromtheloan(−12,422) exactly cancels the interest cost from the loan (-12,422)exactlycancelstheinterestcostfromtheloan(−
12,422). Because the PnL is tied to the same compounding index J_t that drives the debt, this hedge holds perfectly not just at the end of the period, but at every infinitesimal moment along the way.
This design methodology transforms a complex, protocol-level liability into a clean, continuously tradable, and perfectly hedgeable risk factor, representing a powerful new technique for the financialization of on-chain mechanics. We welcome peer review and discussion on this model.