Pricing

All the pricing formulas are well-known in financial engineering and are extracted from the book "Options, futures and other derivatives", 6th edition, John C. Hull, Chapter 5 on futures pricing.

Theory

Formula

The price of a future $F_0$, expiring in a period of time $T$ on an asset with a spot price $S_0$ , is directly linked to the risk free rate $r$ via the following formula:

$F_0 = S_0*exp(rT)$

Note the exponential function assumes a continuous compounding. This simple theoretical formula is valid for an asset which does not provide any income and is not realistic for a currency pair on which we have different borrowing and lending rates, e.g. on ETHDAI there are different rates for DAI and ETH. However we will do some analysis and provide basic examples before generalising for futures on currencies with different prices to buy and sell futures in a later section.

Cash-and-carry trades

Two kind of arbitrages keep derivative prices close to their theoretical formula:

1. If $F_0 > S_0*exp(rt)$ the market is in contango. Arbitrageurs can buy the asset and short futures contracts on the asset to make a risk free profit. This is called a cash-and-carry trade.

2. if $F_0 > S_0*exp(rt)$the market is in backwardation. Arbitrageurs can short the asset and enter into a long futures contract to make a risk free profit. This is called a reverse cash-and-carry trade.

Example

On the 25th of March, the price of ETH on the spot market is 3500 DAI and the risk free rate is r=5% per annum. Consider a futures ETHDAI expiring on the 25th of June. We will consider below two cases to lock a risk free profit, one for a cash-and-carry arbitrage considering a futures price of 3700 DAI and one for a reversed cash-and-carry for a futures price of 3300 DAI.

	Cash-and-carry Futures price 3700 DAI	Reversed cash-and-carry Futures price 3300 DAI
Now	Borrow 3500 DAI at 5%/y for 3 months Buy 1 ETH Get into a short position in the futures contract to sell 1 ETH at 3700 DAI in 3 months	Borrow 1 ETH Short 1 ETH to make 3500 DAI Invest 3500 DAI at 5%/y for 3 months Get into a long position in the futures contract to buy 1 ETH at 3300 DAI in 3 months
At expiry	Deliver the 1 ETH and get 3700 DAI Pay back the loan with 3500*exp(0.05*3/12)=3544 DAI A profit of 3700-3544=256 DAI is realized	From the investment get 3500*exp(0.05*3/12)=3544 DAI Get 1 ETH by paying 3300 DAI Give back 1 ETH A profit of 3544-3300=244 DAI is realized

Advanced pricing

Now we will consider the more realistic case of a futures on currencies, e.g. ETHDAI, on which one could borrow the quote currency at the fixed rate $r_Q$, e.g. borrow DAI at the fixed rate $r_Q$ , and could borrow the base currency at the fixed rate $r_B$ , e.g. borrow ETH at the fixed rate $r_B$ . Given the current market structure in Defi, it is assumed the base and quote currencies cannot be lent at a fixed rate. By expanding the main pricing formula to remove any arbitrage and give a fair pricing value, the table below provides the protocol buy price, or the price at which a trader could sell, and the protocol sell price, or the price at which a trader could buy.

Protocol sell	Ask price
$F_S=S_0*exp(-r_BT)$	$F_B=S_0*exp(r_QT)$

The above formula needs to be slightly modified to take into account the execution fees collected by the protocol.

Example:

Given the price of ETHDAI is 100 on the spot market, the protocol could derive the prices at which traders could buy and sell a futures expiring in 3 months:

• If the yearly fixed borrowing rate on ETH is 4% then the price on the bid is Fbid=100exp(-0.04/4)=99.00 DAI

• If the yearly fixed borrowing rate on DAI is 5% then the price on the ask is Fask=100exp(0.05/4)=101.26 DAI

As defi matures, one could expect the development of fixed lending rates resulting in tighter bid ask spreads.