Demonstration
Same coverage, different price
An option price is a tail integral. Coverage is blind to the tail. Two forecasts with an identical 90% interval can price the same out-of-the-money contract orders of magnitude apart.
An option struck at \(K\) pays \((Y-K)^+\), so its fair price under a predictive density \(f\) is \(\mathbb E_f[(Y-K)^+]=\int_K^\infty (y-K)\,f(y)\,dy\) — a functional of the density’s shape beyond \(K\). Coverage integrates exactly that shape away: you can move mass around inside the covered region, and inside the tail, without changing coverage at all, while the price moves enormously.
Below, two predictive densities are pinned to the same central 90% interval \([-c,c]\) — both would pass the same conformal audit at \(\alpha=0.10\). Forecast A is a standard Gaussian. Forecast B mixes a narrower core with a heavy tail (weight \(w\), scale \(k\)), rescaled so its 90% interval is still exactly \([-c,c]\). Slide the strike out and watch the price ratio: identical coverage, and the prices separate by one, two, then five orders of magnitude. Everything here is closed-form, no simulation.
The second panel is the whole story at every strike at once: \(\log_{10}\) of the call price as a function of \(K/c\), for both forecasts. The curves start together near the interval’s edge and diverge without bound. Nothing about the 90% certificate constrains this gap; the worst of it lives in the sparse, far-from-the-data states where a marginal guarantee is averaging over precisely the cases the contract is about.
Takeaway. A conformal interval, or a conformalized density read off one residual shape, is not what prices a contract; the target is the conditional density itself, and the tail shape that sets the price is the model’s work, not the wrapper’s. Finance has two well-developed routes, neither a coverage test: recover the risk-neutral density from the option surface — the second derivative of the call price in the strike (Breeden–Litzenberger 1978), stabilised by nonparametric state-price density estimation (Aït-Sahalia–Lo 1998), arbitrage-free smoothing of the implied-volatility surface (Fengler 2009), or shape-constrained fits (Bondarenko 2003; Figlewski 2010) — or forecast the conditional density directly (stochastic-volatility and GARCH-family models, quantile and expectile regression, mixtures, flows) and grade it with a proper score. Conformal can wrap any of these for a coverage certificate; the price comes from the density. The parimutuel note makes the gap precise as the rent a better-informed bettor extracts.
Using conformal prediction in your own project? Tell Claude: “Read https://conformalprediction.net/SKILL.md and create a project skill from it.” It adds a check for whether your coverage is conditionally trustworthy.