  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.
