Perpetual Futures for Stocks: The SpaceX Pre-IPO Market

Robert Shiller proposed perpetual futures in 1993 to create derivative markets for assets that are illiquid or whose price cannot be observed directly. Cryptocurrency markets later built the instrument under a different funding rule. We give a single no-arbitrage result that nests both designs: the perpetual price is the present value of a benchmark flow discounted at the funding rate, so the funding rule fixes both the benchmark and the discount. A random time change represents the price as the expected spot at the first event of a clock whose intensity is the funding rate. This yields the main structural result, that stochastic volatility moves the basis only through the carry, so a volatility risk premium, and not volatility itself, can break the peg. We then read price discovery as nonlinear filtering in which the funding rule is a feedback observer whose gain is the funding intensity and the peg the fixed point of a stochastic approximation, and we give a segmented market equilibrium under which the pre-listing premium is structural rather than behavioral. In the June 2026 SpaceX market, the last pre-listing closes were $172.84 on Hyperliquid and $170.82 on Binance, compared with the listed equity's $185 close on 18 June and the $135 bookbuilt offer. Simulation matches the pricing results to their closed forms. Generative Bayesian computation recovers the funding intensity sharply but not the softness of the anchor.