{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T15:29:19Z","timestamp":1787585359022,"version":"build-2736575974"},"reference-count":24,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T00:00:00Z","timestamp":1787529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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\u2019s $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.<\/jats:p>","DOI":"10.3390\/e28090950","type":"journal-article","created":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T14:49:34Z","timestamp":1787582974000},"page":"950","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Perpetual Futures for Stocks: The SpaceX Pre-IPO Market"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-9675-1434","authenticated-orcid":false,"given":"Aditya","family":"Gupta","sequence":"first","affiliation":[{"name":"Stochastic Processes, New York, NY 10013, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicholas G.","family":"Polson","sequence":"additional","affiliation":[{"name":"Booth School of Business, University of Chicago, Chicago, IL 60637, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,8,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1111\/j.1540-6261.1993.tb04024.x","article-title":"Measuring Asset Values for Cash Settlement in Derivative Markets: Hedonic Repeated Measures Indices and Perpetual Futures","volume":"48","author":"Shiller","year":"1993","journal-title":"J. Financ."},{"key":"ref_2","unstructured":"Shiller, R.J. (1993). Macro Markets: Creating Institutions for Managing Society\u2019s Largest Economic Risks, Oxford University Press."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"He, S., Manela, A., Ross, O., and von Wachter, V. (2022). Fundamentals of Perpetual Futures. arXiv.","DOI":"10.2139\/ssrn.4301150"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"SC17","DOI":"10.1137\/22M1520931","article-title":"A primer on perpetuals","volume":"14","author":"Angeris","year":"2023","journal-title":"SIAM J. Financ. Math."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Ackerer, D., Hugonnier, J., and Jermann, U. (2024). Perpetual Futures Pricing, EPFL and Wharton. 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Stochastic Processes, Wiley."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0304-4149(81)90026-0","article-title":"Martingales and stochastic integrals in the theory of continuous trading","volume":"11","author":"Harrison","year":"1981","journal-title":"Stoch. Process. Their Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"400","DOI":"10.1214\/aoms\/1177729586","article-title":"A stochastic approximation method","volume":"22","author":"Robbins","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_17","unstructured":"Binance (2026, July 29). Binance Public Data, USD-M Futures Archive: SPCXUSDT Klines, Funding, and Metrics. Available online: https:\/\/data.binance.vision."},{"key":"ref_18","unstructured":"Hyperliquid (2026, July 29). Public Info API Data for HIP-3 Asset xyz:SPCX, Including Daily Candles and Funding History. 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Econ."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/950\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T14:53:54Z","timestamp":1787583234000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/9\/950"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,8,24]]},"references-count":24,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2026,9]]}},"alternative-id":["e28090950"],"URL":"https:\/\/doi.org\/10.3390\/e28090950","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,8,24]]}}}