Free lesson · DeFi models

LP inventory after price changes

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A constant-product pool sells the rising asset and accumulates the falling asset as arbitrage aligns its reserve ratio with an external price.

For a no-fee pool with invariant k, external price p in quote per base implies y/x=p. Solve this together with xy=k to determine inventory.

A provider’s claim is a share of the resulting reserves, not a fixed number of the original tokens. Fees and deposits change invariant or ownership share; hold them fixed in this first model.

Apply it in a strategy

• Explain how liquidity provision changes token exposure as market prices move.
• Record the input timestamp, executable quantity, currency and horizon. Reconcile the result with a cash-flow or state table.
• Stress this failure condition: An oracle or arbitrage delay means the pool may not actually be aligned with the external price at the observation time.

Research deliverable

Build and explain a lp inventory after price changes worksheet. Explain how liquidity provision changes token exposure as market prices move.

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.

# Python 3.10+; standard library unless NumPy is imported below.
# Inputs and outputs use the units defined in this lesson. Synthetic teaching example.
from math import sqrt
def pool_inventory(invariant,price):
    if invariant<=0 or price<=0: raise ValueError("Positive invariant and price required")
    return sqrt(invariant/price),sqrt(invariant*price)

print(pool_inventory(1_000_000,400))