Most likely you already know, that futures contracts trading is possible without physical delivery of the traded asset and profit/loss is settled directly in cash between parties (explained here: What is a futures contract?).

In case of Bitcoin and other cryptocurrencies, the ability to trade them versus fiat currencies without the need to interact with fiat itself can be beneficial for a lot of market participants. Exchanges could avoid excessive regulation by doing away with fiat, and traders still can get exposure to crypto vs fiat trading pairs without making fiat deposits.

The standard Bitcoin futures market would treat Bitcoin as a commodity itself, and the profit/loss would be settled in conventional currencies. For example, one contract could represent 1 BTC and would be settled in USD, so the traded pair would be BTC/USD. Now imagine if we swap BTC and USD places in such contracts. This gives us the ability to treat USD as a regular commodity just like any other asset in conventional futures markets and settle profit/loss in BTC. In this example, one contract could represent a standard size of, let’s say, \$5,000 and the price would be quoted in BTC, which at the time of writing would be 0.66 BTC per \$5,000 (\$7,500 per BTC on 04.06.2018).

The approach mentioned above is still your standard direct futures contract, just with trading pair swapped around. It also proved to be highly confusing for most of the traders, which are used to see price quotes as USD per BTC and not the other way around. In such market, to go long on Bitcoin one would have to go short on USD/BTC pair and vice versa.

To understand how we get from the example above to inverse futures contract, let’s first take a look how to calculate a profit in a standard direct futures contract with settlement in USD.

Suppose one contract is worth 1 BTC and you enter a long position at \$10,000 with five contracts (\$50,000 in total). The price moves in your direction, and you close the trade at \$11,000. Your profit is calculated like this:

Profit = (Pclose – Popen)*Ncontracts

where Popen is the price at which you opened, Pclose is the price at which you closed (both prices quoted as BTC/USD), and Ncontracts is the number of contracts you have. Simply put, it’s just difference between close and open price times number of contracts. So in this example, you stand to make \$5,000 (fees are neglected for the sake of experiment).

Now here comes the magic. We take that standard formula for the regular futures and apply it to the inverted USD/BTC pair, where the USD is an asset traded, and contracts are settled in BTC. As you remember, in this case, to go long on BTC you would need to go short on USD/BTC. To do away with this confusion, we apply an inverse sign to the calculation of profit:

Profit = -(Pclose – Popen)*Ncontracts

Now the trader doesn’t need to go short on USD/BTC pair if he wants to long BTC vs USD and the price can be quoted as BTC/USD.

But this isn’t all. Remember that the price in the original formula is quoted as BTC/USD but in reality we now trade USD/BTC. We need to quote prices as BTC per USD inside the calculations. To do that, we take the reciprocal (invert upside-down) of the price quotations, which will tell us how much is one dollar worth in terms of BTC.

The final formula will look like this:

Profit = -(1/Pclose – 1/Popen)*Ncontracts

Such approach now lets us trade USD as a standard commodity in regular futures markets without ever delivering it and settle the contracts in BTC, but the quotation of the price and opening long and short positions can be made just as in the spot BTC/USD pair.

Such profit/loss calculations introduce a non-linear payout curve to inverse futures contracts. Explained here: Understanding non-linear nature of inverse futures

Inverse futures contracts were devised by Aleksey V. Bragin of Moscow State Technical University in 2011.