Black-Scholes Underprices AMM Puts by About 6%

If a token only gets its price from an automated market maker, old-school Black-Scholes pricing starts to miss the shape of the market. This paper shows that such tokens follow a constant elasticity of variance (CEV) process, which means volatility rises as price falls — a leverage effect built into the pool itself. Under that model, a 20% out-of-the-money put is underpriced by about 6% in implied volatility terms, and that shortfall shows up at every pool depth. As pools get deeper, the dollar gap shrinks, but the pattern in the wings remains. The paper also tests the idea against data from 90 Bittensor subnets, finding strongly negative price elasticity in realized return variance after controlling for pool depth and flow volatility. A separate delta-hedged backtest across 82 subnets finds nearly identical hedging errors at the money, which matches the claim that the biggest pricing differences sit away from the center of the distribution.

An AMM-only token does not get its price from a crowd of bids. It gets it from a pool. That pool can make risk bend in a strange way. In this setup, price swings grow as price falls. That matters if you hold the token, sell options on it, or hedge a treasury. The classic Black-Scholes option model, the old school formula on many desks, assumes steadier swings. This paper shows a different shape. A 20%-out-of-the-money put, a bet that pays if price drops, looks about 6% too cheap in implied volatility terms. And that gap shows up at every pool depth.

When the pool sets the market

This paper derives the price path for tokens with only one price source. That source is a constant-product AMM, a pool that keeps two reserves linked by one rule. When net flow into the pool moves like a diffusion, the token price follows a constant elasticity of variance, or CEV, process. That is a model where volatility changes with price. Here, it rises when price falls. Black-Scholes sits inside this story as the limit of infinite liquidity. The new shape creates a leverage effect and a skew in implied volatility. The skew depends on the pool weight, not on pool depth. So deeper pools shrink dollars lost, but not the skew itself.

How the model meets the chain data

The math starts with a constant-product AMM. It keeps the product of the two reserves fixed. The token price comes from the reserve ratio. The model then lets net flow wander like a diffusion. That choice turns the price path into CEV. From there, it gets closed-form European option prices. That means the price can be written down exactly for options used only at expiry. It also builds liquidity-adjusted Greeks, the usual option risk gauges, but tuned for pool depth. The empirical test then checks 90 Bittensor subnets for the variance pattern and 82 subnets for hedging errors.

Why the skew matters

That matters because option desks care most about the wings, not just the center. At-the-money means the option sits near the current price. The backtest found near-identical hedging errors there across 82 subnets. That fits the model's story. The biggest differences sit far from the center. For AMM-native tokens, a deeper pool makes the dollar gap smaller. But it does not remove the risk shape. So Black-Scholes can still miss the put side, even when the pool is deep. The new pricing rule gives a cleaner way to match risk with liquidity.

What to test next

The surprise is that pool depth only shrinks the dollar gap. It does not change the skew's shape. That makes AMM-native tokens feel like a curved track, not a flat line. The next test is simple. Watch new Bittensor subnets as they open. Then check if the negative price elasticity stays after the pool mix changes. For anyone hedging an AMM-only token, this means Black-Scholes should not be the only ruler in the drawer. The pool sets the price. The pool also sets the risk.