Option Pricing on Automated Market Maker Tokens

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 Bittensor subnets exhibits a strongly negative price elasticity, decisively rejecting geometric Brownian motion and consistent with the CEV prediction. A complementary delta-hedged backtest across 82 subnets confirms near-identical hedging errors at the money, consistent with the prediction that pricing differences are concentrated in the wings.