Perpetual futures are contracts without expiration date in which the anchoring of the futures price to the spot price … 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 …

conditions under which the regret-averse firm's optimal futures position is an under-hedge (over-hedge). We further show that the … firm optimally increases (decreases) its futures position when the price risk possesses more positive (negative) skewness. …