というNBER論文が上がっている（ungated版）。原題は「Perpetual Futures Pricing」で、著者はDamien Ackerer（スイス連邦工科大学ローザンヌ校）、Julien Hugonnier（同）、Urban Jermann（ペンシルベニア大）。
以下はその要旨。
Perpetual futures are contracts without expiration date in which the anchoring of the futures price to the spot price is ensured by periodic funding payments from long to short. We derive explicit expressions for the no-arbitrage price of various perpetual contracts, including linear, inverse, and quantos futures in both discrete and continuous-time. In particular, we show that the futures price is given by the risk-neutral expectation of the spot sampled at a random time that reflects the intensity of the price anchoring. Furthermore, we identify funding specifications that guarantee the coincidence of futures and spot prices, and show that for such specifications perpetual futures contracts can be replicated by dynamic trading in primitive securities.
（拙訳）
永久先物*1は、満期の無い契約であり、先物価格の現物価格への括り付けは、買い建て側から売り建て側への定期的な資金調達の支払いによって確保される。我々は、線形、インバース、およびカントス先物など様々な永久契約*2の無裁定価格の明示的な表式を、離散時間と連続時間の両方について導出した。具体的には、先物価格は、価格の括り付けの強度を反映する、ランダム時間でサンプリングされた現物のリスク中立予想で与えられることを我々は示す。また我々は、先物価格と現物価格の一致を保証する資金調達の仕様を識別し、そうした仕様では永久先物契約が原始証券の動的取引で再現できることを示す。
論文では永久先物のほかエバーラスティング・オプションの表式も導出しているが、その考案者の一人はサム・バンクマン＝フリードとの由*3。
*1:cf. 暗号通貨の永久先物契約とは何ですか?。
*2:cf. What Are Linear Contract and Inverse Contract – CoinEx Help Center、インバース型無期限契約入門、 Quanto - Wikipedia。本文では「The terms of a standard, or linear, perpetual futures contract include the underlying asset (e.g., BTC/USD) representing the value of one unit of the base currency (BTC) in units of the quote currency (USD), a contract size expressed in units of the base asset (1BTC), and a margin and settlement currency (USD) in which profits and losses are realized. Cryptocurrency trading platforms have introduced multiple variations of the linear contracts. The most important such variation is the inverse contract where the base currency itself (BTC) is used as the margin and settlement asset and the contract size is expressed in units of the quote currency (10’000USD). This innovative product allows to speculate on the exchange rate between a crypto and a fiat currency without the need to actually hold units of that fiat currency and was thus widely adapted by trading platforms that typically cannot accept deposits in fiat currencies since they do not qualify as banks under the existing regulation. Another important variation is the perpetual quanto futures that uses a third currency (ETH), different from the quote and base currencies, for margining and settlement.」と説明している。
*3:本文では「Perpetual futures contracts where the target spot price is a function of the spot price (rather than the spot price itself) have been proposed by Bankman-Fried and White (2021) under the somehow misleading name of everlasting options but, outside of power contracts (White et al., 2021; Perennial Labs, 2024) and their applications (Prospere, 2022; Clark, 2023; Clark, Leone, and Robinson, 2024) they have found limited traction so far.」と説明している。
