Option Pricing on Automated Market Maker Tokens
Philip Z. Maymin

TL;DR
This paper models token prices in AMMs as a CEV process, derives closed-form option prices, and empirically validates the model with real data, revealing a leverage effect and pricing discrepancies.
Contribution
It introduces a CEV-based stochastic model for AMM tokens, providing closed-form option prices and empirical validation of the model's predictions.
Findings
CEV process describes token price dynamics in AMMs.
Black-Scholes underprices 20%-out-of-the-money puts by ~6%.
Empirical data supports the CEV model over geometric Brownian motion.
Abstract
We derive the stochastic price process for tokens whose sole price discovery mechanism is a constant-product automated market maker (AMM). When the net flow into the pool follows a diffusion, the token price follows a constant elasticity of variance (CEV) process, nesting Black-Scholes as the limiting case of infinite liquidity. We obtain closed-form European option prices and introduce liquidity-adjusted Greeks. The CEV structure generates a leverage effect -- volatility rises as price falls -- whose normalized implied volatility skew depends only on the pool's weighting parameter, not on pool depth: Black-Scholes underprices 20%-out-of-the-money puts by roughly 6% in implied volatility terms at every pool depth, while the absolute pricing discrepancy vanishes as pools deepen. Empirically, after controlling for pool depth and flow volatility, realized return variance across 90…
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