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Option Pricing on Automated Market Maker Tokens

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.

Option Pricing on Automated Market Maker Tokens

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.

Paper Structure

This paper contains 41 sections, 10 theorems, 44 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. AMM theory.
  4. AMMs and options.
  5. CEV models.
  6. Concurrent work.
  7. Our contribution.
  8. Institutional Background
  9. Constant-Product Automated Market Makers
  10. Generalized Constant-Function AMMs
  11. Bittensor and Dynamic TAO
  12. The Model
  13. Setup and Notation
  14. Reserve Dynamics
  15. Derivation of the Price Process
  16. ...and 26 more sections

Key Result

Theorem 1

Under the constant-weighted-product AMM eq:cwp with net flow process eq:flow, the marginal token price $P(t)$ satisfies the CEV stochastic differential equation where the CEV exponent is $\beta = w$ (the numeraire weight), the volatility parameter is and the drift is $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Left: simulated price paths under CEV (solid) and GBM (dashed) driven by identical Brownian increments, with volatilities matched at $P_0$. The paths visibly diverge as price moves away from $P_0$: CEV produces wider swings at low prices (leverage effect) and narrower swings at high prices. Right: terminal price distribution from 50,000 Monte Carlo paths at $T = 90$ days. The CEV distribution exhibits a heavier left tail and positive skew relative to GBM, consistent with the structural leverage effect. Parameters: $P_0 = 0.025$, $k = 5 \times 10^5$, $\sigma_F = 48.7$.
  • Figure 2: Monte Carlo validation of the CEV pricing formula for a shallow pool ($k = 10^6$, left) and a deep pool ($k = 10^9$, right). Top: closed-form CEV call prices (line) vs. Monte Carlo estimates with 95% confidence intervals (points). Bottom: pricing error (MC minus CEV) as a percentage of spot. The shallow pool shows a systematic positive bias of 1--3% of spot, reflecting the Euler--Maruyama discretization error that is amplified when flow volatility is large relative to the reserve. The deep pool shows near-perfect agreement (errors $< 0.5$%). Illustrative parameters: $P_0 = 0.025$, $\sigma_F = 48.7$, $T = 30$ days, 100,000 paths.
  • Figure 3: Left: ATM call price (as % of spot) vs. pool depth $k$. The CEV (solid blue) and Black--Scholes (dashed red) curves overlap, confirming that the models agree at the money. Right: absolute pricing discrepancy $|C_{\mathrm{CEV}} - C_{\mathrm{BS}}|$ (as % of spot) on a log-log scale, for five moneyness levels. OTM options (puts in green/cyan, calls in orange/red) show discrepancies orders of magnitude larger than ATM (blue), reflecting the leverage-induced skew. All curves decline approximately as $O(k^{-1})$ (dotted reference line). Illustrative parameters: $P_0 = 0.025$, $\sigma_F = 48.7$, $T = 90$ days, $r = 5\%$.
  • Figure 4: Left: absolute Black--Scholes implied volatility extracted from CEV prices. Shallower pools produce higher absolute volatility due to larger $\delta$. Right: implied volatility normalized by the at-the-money level, isolating the skew shape. The three normalized curves overlap, confirming that the skew depends only on $\beta = 1/2$, not on pool depth. Illustrative parameters: $P_0 = 0.025$, $\sigma_F = 48.7$, $T = 90$ days.
  • Figure 5: Comparison of CEV ($\beta = 1/2$) and Black--Scholes Greeks for an ATM European call ($K = 0.025$) on a shallow pool. Left: delta. Right: gamma. Both gammas peak below the strike, but the CEV gamma is sharper and peaks further below, reflecting its concentration in the high-volatility (low-price) region. Illustrative parameters: $k = 5 \times 10^5$, $\sigma_F = 48.7$, $T = 90$ days.
  • ...and 4 more figures

Theorems & Definitions (23)

  • Example 1: A simple constant-product AMM
  • Example 2: Emission mechanics
  • Definition 1: Stochastic Flow Process
  • Theorem 1: AMM Token Price Process
  • proof
  • Corollary 2: Constant-Product AMM
  • Corollary 3: Black--Scholes Limit
  • Remark 1: Elasticity spectrum
  • Proposition 4: Volatility structure
  • Proposition 5: Implied volatility skew
  • ...and 13 more