Home » Theory » Understanding non-linear nature of inverse futures

As of today, inverse futures contracts, devised back in 2011, are the most popular derivative for trading Bitcoin and other cryptocurrencies, with volumes reaching billions of dollars in a single day. As you probably know, these derivatives are not your usual direct futures contracts and are a bit more complicated. If you want to familiarise yourself with the nature of this instrument and get a better understanding of what will follow here, please read these articles first (preferably in the same order):

## Linear futures contracts

Apart from being inverse, as the name suggests, these contracts also have non-linear nature. In short, this means that your profit/loss in BTC on each contract does not correlate linearly with the movement of BTC/USD pair. You do not gain or lose the same amount of BTC on each, let’s say, \$100 move in the price of Bitcoin. Let’s take a look at an example of the direct futures contract first and understand why it is linear.

You buy ten futures contracts, each for one ton of aluminium at \$2,000 with three months delivery time (10 tons for \$20,000). One month in, the price of aluminium rises to \$2,100, your current unrealised profit is \$1,000, since you can buy something for \$20,000 according to contract, and sell it immediately at the market for \$21,000 (\$2,100*10 ton = \$21,000). A bit later the price moves another \$100 up, and your unrealised profit becomes \$2,000. And then another \$100, and the price of one ton becomes \$2,300. Do you see where this is going? For each fixed move in a price of aluminium, your profit on these contracts changes by the same amount, \$1000 for each \$100, or \$100 for each \$10, or \$10 for each \$1, et cetera. These relationships between price movement and profit/loss on contracts called linear, and if you plot a graph of it, you will see a straight, upward-sloping line.

## Non-linear futures example

Now, let’s take a look at the example with non-linear inverse futures. Suppose you have 1 Bitcoin, the current price is \$200 BTC/USD, and one contract is worth \$50. Priced in BTC one contract costs \$50/\$200 = 0.25 BTC. You buy four long contracts at \$200 BTC/USD, which costs you \$200, or one BTC exactly (neglect the fees for the sake of simplification). You went long on BTC and short on USD as you’re, essentially, short \$200 priced in BTC. Now, if the price of bitcoin goes up, the price of \$200 in BTC will go down, since it is now can buy you fewer bitcoins.

The price then goes to \$300 BTC/USD, or \$50/\$300 = ~0.166 BTC per \$50 (one contract). Remember that you entered short on USD at 0.25 BTC per \$50 contract, and it is now 0.166 BTC, so you gain 0.25-0.166 = 0.084 BTC on each contract, and 0.084*4 = 0.336 on four contracts. So, this \$100 move in BTC/USD pair gained you 0.084 BTC per contract.

Next, the price goes to \$400, and we do the same calculations. One contract is now worth \$50/\$400 = 0.125 BTC, your total gain on one contract is 0.25-0.125 = 0.125 BTC, or 0.5 o BTC on 4 contracts. Notice, that the first \$100 move in BTC price gained you 0.084 per contract, and the second \$100 move gave you a smaller sum of 0.041 per contract (0.166-0.125 = 0.041).

Let’s do the last one. Price goes to \$500, and one contract is \$50/\$500 = 0.1 BTC. Total gain per contract on the USD you shorted against BTC is 0.25 – 0.1 = 0.15 or 0.6 on 4 contracts. This move of \$100 gave you even smaller gain in BTC: 0.125 – 0.1 = 0.025 per contract.

Again, see where this is going? Each subsequent move up of the same distance in BTC/USD pair gives you smaller and smaller gains in BTC, that is non-linear nature of inverse futures contracts. Your profits in USD are still amplified compared to spot trading since bitcoins used to open contracts are rising in value as well as those you gain on your long position.

## Why do we have nonlinearity in inverse futures?

Why does this happen you may ask? Remember, that “under the hood”, we’re essentially trading inverted USD/BTC pair, as opposed to the standard BTC/USD pair (this allows for settling trades in BTC only), we’re shorting our USD priced in bitcoins when we go long on BTC/USD and vice versa. So when the standard pair of BTC/USD “travels” the same \$100 distance up, again and again, the inverted pair of USD/BTC travels less and less in the opposite direction, because each new \$100 move constitutes a smaller and smaller part of a whole bitcoin if its price is rising. For example, at the price of bitcoin at \$500, a \$100 move up represents 20% of a single coin (\$100/\$500 = 0.2), at the price of \$600, a \$100 move up represents 16.6% (\$100/\$600 = 0.16.6), at the price of BTC/USD at \$700, a \$100 move represents 14.2% (\$100/\$700 = 0.142), et cetera.

The opposite happens when the price of BTC goes down. Each \$100 move constitutes a bigger and bigger part of a single coin since it’s price getting smaller, so the gains in the inverted USD/BTC pair, grow larger and larger with every move down of the same size in BTC/USD.

The chart below shows you non-linear relationships between profit/loss in BTC and price movements in BTC/USD in the inverse futures contracts.

## Conclusion

Inverse bitcoin futures have non-linear payout curve in terms of bitcoins. Each succeeding move of the same extent in the price of BTC/USD does not produce the same amount of profit or loss denominated in BTC. However, if you convert inverse futures profit/loss into the quote currency (USD), the payout will be linear (explained in the links at the top of the article).