Paper: arXiv 2603.29763

Authors: Philip Z. Maymin

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.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a strong theoretical bridge between automated market maker (AMM) mechanics and continuous-time option pricing by rigorously deriving a CEV price process from pool flows. The empirical validation across dozens of Bittensor subnets and the delta-hedged backtest significantly elevate its practical relevance and empirical rigor.

Research Flowchart

  flowchart TD
    A[Research Goal: Price options on AMM tokens with limited liquidity] --> B[Theoretical Derivation: Net flow diffusion to CEV process]
    B --> C[Model Outputs: Closed-form European option prices & liquidity-adjusted Greeks]
    D[(Empirical Inputs: Data from 90 Bittensor subnets)] --> E[Empirical Validation: Variance elasticity & delta-hedged backtest]
    C --> E
    E --> F[Key Outcome: BS underprices 20% OTM puts by ~6% IV due to leverage effect]
    E --> G[Key Outcome: Rejection of GBM; volatility skew depends on pool weights, not depth]

Cited References (35)

  1. Option Contracts in the DeFi Ecosystem: Opportunities, Solutions, and Technical Challenges (2025) — link
  2. A Derivative Pricing Perspective on Liquidity Tokens in Constant Product Market Makers — → 2409.11339
  3. Unified approach for hedging impermanent loss of liquidity provision — → 2407.05146
  4. Decentralised Finance and Automated Market Making: Predictable Loss and Optimal Liquidity Provision — → 2309.08431
  5. The Conceptual Flaws of Decentralized Automated Market Making (2023) — link
  6. Weighted variance swaps hedge against impermanent loss (2023) — link
  7. Automated Market Making and Loss-Versus-Rebalancing (2022) — link
  8. Optimal Routing for Constant Function Market Makers (2022) — link
and 27 more
  1. Impermanent Loss in Uniswap v3 (2021) — link
  2. Replicating market makers (2021) — link
  3. An analysis of Uniswap markets (2019) — link
  4. Pricing Options on Scalar Diffusions: An Eigenfunction Expansion Approach (2003) — link
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  21. Option pricing when underlying stock returns are discontinuous (1976) — link
  22. Delta-hedge daily for 14 days using each model’s delta, updating k and recomputing deltas from observed reserves at each rebalance
  23. Using Black–Scholes to estimate the size of divergence loss for AMMs
  24. 14-day ATM European call ( K = P t ) at the model price under both the CEV model ( β = 1 / 2 , δ t = 2ˆ σ F / √ k t ) and Black–Scholes with matched ATM volatility ( σ eff = δ t P − 1 / 2 t )
  25. At expiry
  26. Liquidity-dependent volatility and the leverage effect
  27. Universal implied volatility

3 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.