Stableswap Math

Curve-style Stableswap 2-asset pools use an amplified invariant that interpolates between constant-product ( $`xy = k`$ ) and constant-sum ( $`x + y = k`$ ) behavior:

A \cdot n^n \cdot (x + y) + D = A \cdot D \cdot n^n + \frac{D^{n+1}}{n^n \cdot x \cdot y}

where $`A`$ is the amplification coefficient, $`n = 2`$ for a 2-asset pool, and $`D`$ is the pool invariant (total liquidity at equal prices). High $`A`$ concentrates liquidity around the 1:1 peg; low $`A`$ behaves more like a standard constant-product pool.

Amplification coefficient

$`A`$ controls how tightly the pool prices around peg. At $`A = 0`$ , the pool degrades to constant-product. As $`A \to \infty`$ , the pool approaches constant-sum (zero slippage at peg, catastrophic slippage off-peg). Real-world pools typically use $`A`$ values between 10 and 2000.

Impermanent loss under depeg

Stableswap IL behaves differently from constant-product IL. Near peg, IL is suppressed by the amplification factor — the pool holds close to 50/50 reserves even as prices drift slightly. Off peg, IL accelerates sharply because the amplified invariant concentrates liquidity in a narrow price band. The transition between “near peg” and “off peg” depends on $`A`$ : higher $`A`$ means a sharper cliff.

The ε ↔ δ relationship

For a depeg of magnitude $`\epsilon`$ (fractional price deviation from peg), the reserve imbalance $`\delta`$ is a function of $`A`$ and $`\epsilon`$ . This relationship is computed numerically via fixed-point iteration rather than closed-form — the invariant equation is transcendental in the general case. DeFiPy’s AssessDepegRisk primitive uses this numerical approach.

DeFiPy implementation

DeFiPy’s Stableswap math lives in StableswapPy. The amplified invariant, swap math, and D computation are implemented in stableswappy.cpt.exchg.StableswapExchange. The impermanent loss formula is implemented in stableswappy.analytics.risk.StableswapImpLoss.

The agentic primitives that consume Stableswap math:

AnalyzeStableswapPosition — full position decomposition with alpha/unreachable-alpha regime handling
SimulateStableswapPriceMove — project position value at a hypothetical price change
AssessDepegRisk — quantify exposure to a stablecoin depeg using the ε ↔ δ relationship

See agentic_primitives_by_category for executable examples.