Cowen's Bitcoin model shows upside fading at 0.33 curvature

Bitcoin's price distribution is flattening faster at its ceiling than at its floor, a new model from quantitative analyst Benjamin Cowen shows.

"These models baked in Bitcoin's early reflexivity, when tiny capital inflows produced huge price moves," Cowen said in the paper, titled Asymmetric Tail Curvature in Bitcoin Price Quantiles. "They projected that behavior forward into a market now dominated by institutions and trillion-dollar valuations."

Cowen benchmarks three of the best-known Bitcoin price models against 16 years of daily data through May 2026. The original power-law fit, calibrated through 2018, overshot actual prices on 77.2% of trading days, with an average error of 32.1% above. PlanB's stock-to-flow model overshot on 94.9% of days, with an average error of 294.5%. Its cross-asset cousin, the S2FX model, projected a 1,699% overshoot, implying Bitcoin prices above $5 million.

The paper's headline finding is that the upper bands of Bitcoin's price distribution — those tracing speculative peaks — bend inward and flatten over time, while the lower bands stay roughly straight. The upper-tail curvature measures roughly minus 0.33 and is statistically distinguishable from zero. The lower-tail curvature sits near minus 0.02 and cannot be distinguished from a flat line. The difference is significant at the 1% level under the paper's bootstrap test. Across 27 expanding windows of historical data, the upper-tail curvature stays in a tight band near minus 0.3, and every single window rejects the symmetry test.

Cowen calls the mechanism "diminishing reflexivity." Early Bitcoin was small — a few hundred million dollars of capital could push the price up 10,000% in a year. As the market cap climbed above $1.4 trillion, the same percentage move required vastly more capital. Each cycle's blow-off top lands closer to the long-run trend than the one before. The reflexivity amplitude in his model decays as Bitcoin grows, while structural demand as a monetary asset continues compounding along a steady power-law path.

The paper devotes seven sections to its own caveats. Bitcoin has lived through only four halving cycles, and on short sub-windows the curvature parameter swings to absurd values, which Cowen flags as weak identification. Shift the model's starting point from January 2009 to January 2010, and the upper-tail curvature shrinks to roughly zero — the finding depends on how much weight the thin liquidity data from 2010 to 2011 carries. A negative curvature in log-log space implies that, as it approaches infinity, the fan eventually predicts that Bitcoin prices will decline. Cowen states explicitly that this has no literal interpretation and that the model describes a finite horizon, not a price target.

For traders tracking the four-year cycle, the practical implication is to set expectations rather than market-time. Bitcoin reached an all-time high near $126,080 in October 2025 and has since fallen about 44%. The asset is down 33% over the past year and currently trades just below $70,000, with a market cap above $1.4 trillion. It fell 4% in the past 24 hours as of press time. Cowen argues the four-year cycle still holds, but the range of plausible peak heights is now narrower than during the 2017 or 2021 cycles, consistent with lengthening Bitcoin cycles and shrinking upside.

The moonshot models did not fail because Bitcoin disappointed, Cowen writes. They failed because they extrapolated a regime that no longer exists. The paper builds on the same power-law tradition pioneered by Trolololo, Giovanni Santostasi, and the quantile-band approach of Plan C — but it offers a more honest shape for the long-run distribution.

This article is for informational purposes only and does not constitute investment advice.