Free lesson · DeFi models

Constant-product swaps with an input fee

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A pool exchanges against its reserves. Buying a large fraction of the available asset moves the average price against the buyer.

Let x be base reserves and y quote reserves. In a no-fee constant-product pool, the product is unchanged by a swap. With an input fee, only the fee-adjusted input is used in the output formula, while the full input enters reserves under the assumed retained-fee convention.

Use this as a V2-style teaching model, not a universal protocol quote. Token transfer fees, hooks, concentrated ranges and integer rounding require additional rules. An external price comparison is needed to judge whether a swap is economically favorable.

Apply it in a strategy

• Compare AMM execution prices and fees with an order-book route at the same trade size.
• Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
• Stress this failure condition: Reserve snapshots can change before execution; minimum-output protection limits price but cannot ensure inclusion.

Research deliverable

Build and explain a constant-product swaps with an input fee worksheet. Compare AMM execution prices and fees with an order-book route at the same trade size.

Evidence boundary: Synthetic arithmetic and scenarios illustrate mechanics. They are not historical returns, a paper replication, or evidence of an executable edge. Research sources and their access limitations are recorded at the end of this module.

Research sources, review dates and limitations

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
def cp_swap(base_reserve,quote_reserve,base_in,fee):
    if min(base_reserve,quote_reserve)<=0 or base_in<0 or not 0<=fee<1: raise ValueError("Invalid pool inputs")
    effective=base_in*(1-fee)
    output=quote_reserve*effective/(base_reserve+effective)
    return output,base_reserve+base_in,quote_reserve-output

print(cp_swap(100,10000,10,.003))