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Theoretical pricing
The formulas presented below are theoretical and used for forward pricing. Contango protocol uses similar but different formulas by taking advantage of the trader's margin to make expirables in DeFi capital-efficient (see the protocol pricing section).
In TradFi, forwards on currencies are priced using the well-known interest rate parity relationship. Given the quote currency can be borrowed or lent at the yearly fixed rate
, the base currency at the yearly rate
and given a spot price
then the theoretical price
to buy or sell 1 forward expiring in a time
is given by:
The pricing formula above is adapted from from "Options, futures and other derivatives", 6th edition, John C. Hull, Chapter 5 on futures pricing.
Let's consider a contract on ETHDAI expiring in 3 months (
) where the spot price is
. Given the yearly fixed interest rate on the quote currency
is
and on the base currency
is
then the theoretical price to buy or sell 1 forwards would be:
In the section, a more realistic formula is derived. Taking the example of the currency pair ETHDAI, it is supposed that:
- the quote currency is borrowed at the yearly fixed rate, e.g. borrow
- the quote currency is lent at the yearly fixed rate, e.g. lend
- the base currency is borrowed at the yearly fixed rate, e.g. borrow
- the base currency is lent at the yearly fixed rate, e.g. lend
- the base currency is bought at the spot price, e.g. buyby selling
- the base currency is sold at the spot price, e.g. sellto buy.
The table below provides the theoretical prices at which a trader can go long or short a forward:
Theoretical short price | Theoretical long price |
---|---|
| |
It could be noticed that, the tighter the spread between the borrowing and lending rates and the tighter the spread on the spot price then the tighter the spread between the forward prices to go long and short.
Let's consider a contract on ETHDAI expiring in 3 months (
) :
- Given one could borrow DAI at a yearly fixed rate of, lend ETH at a yearly fixed rateand buy ETH on the spot market at, the price to go long on 1 forward would be:
- Given one could borrow ETH at a yearly fixed rate of, lend DAI at a yearly fixed rate ofand sell ETH on the spot market at, the price to go short would be:
If the price of a forward is above or under the theoretical formula then an arbitrage condition arises. Since market participants can take advantage of this "free lunch", by using significant amounts of money, prices are brought back to their theoretical formulas. In the examples below, where the same numerical assumptions as in the example above are kept, the two arbitrages to bring the price at equilibrium are presented.
Let's say the price to sell 1 forward is
instead of
. An arbitrageur could:
- Borrow now. In 3 monthsneed to be given back.
- Convert now thoseto.
- Invest now theto receive in 3 months.
- Sell now a forward contract for a quantity offor a price of.
- Deliver theat expiry to getfor a cost of, locking-in a risk-free profit of.
This arbitrage could be done with much bigger amounts and would disappear when the forward price is brought back to its theoretical pricing.
Let's say the price to buy 1 forward is
instead of
. An arbitrageur could:
- Borrow now. In 3 months need to be given back.
- Convert now thoseto.
- Invest now those DAIs to receivein 3 months.
- Buy now a forward contract for a quantity offor a price of.
- At expiry, the trader would receivefrom lending and useto buy theneeded to reimburse the debt, locking-in a risk-free profit of.
This arbitrage could be done with much bigger amounts and would disappear when the forward price is brought back to its theoretical pricing.
Last modified 11mo ago