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 rQā , the base currency at the yearly rate rBā and given a spot price S then the theoretical price Pthā to buy or sell 1 forward expiring in a time T is given by:
Pthā=Sā(1+rBā1+rQāā)T Let's consider a contract on ETHDAI expiring in 3 months (T=0.25) where the spot price is S=100DAI. Given the yearly fixed interest rate on the quote currency DAI is rQā=10% and on the base currency ETH is rQā=3% then the theoretical price to buy or sell 1 forwards would be:
Pthā=100ā(1+0.031+0.1ā)0.25=101.66DAI
the quote currency is borrowed at the yearly fixed rate rQ,bā, e.g. borrow DAI
the quote currency is lent at the yearly fixed rate rQ,lā, e.g. lend DAI
the base currency is borrowed at the yearly fixed rate rB,bā , e.g. borrow ETH
the base currency is lent at the yearly fixed rate rB,lā , e.g. lend ETH
the base currency is bought at the spot price SLā , e.g. buy ETH by selling DAI
the base currency is sold at the spot price SSā, e.g. sell ETH to buy DAI.
Theoretical short price | Theoretical long price |
---|
| |
Let's consider a contract on ETHDAI expiring in 3 months (T=0.25) :
Given one could borrow DAI at a yearly fixed rate of rQ,bā=10.10%ā, lend ETH at a yearly fixed rate rB,lā=2.90%and buy ETH on the spot market at SLā=100.10, the price to go long on 1 forward would be:
Pth,Lā=100.10ā(1+0.02901+0.1010ā)0.25=101.81DAI
Given one could borrow ETH at a yearly fixed rate of rB,bā=3.10%ā, lend DAI at a yearly fixed rate of rQ,lā=9.90% and sell ETH on the spot market at SSā=99.90, the price to go short would be:
Pth,Sā=99.90ā(1+0.03101+0.0990ā)0.25=101.51DAI
Forward price above
Pth,Sā
Let's say the price to sell 1 forward is 110.00DAI instead of Pth,Sā=101.51DAI. An arbitrageur could:
Borrow now 10000.00DAI. In 3 months 10000ā(1.11)0.25=10243.46DAI need to be given back.
Convert now those 10000DAI to 10000/100.10=99.90ETH.
Invest now the 99.90ETH to receive 99.90ā(1.029)0.25=100.62ETHā in 3 months.
Sell now a forward contract for a quantity of 100.62ETH for a price of100.62ā110=11067.83DAI.ā
Deliver the 100.62ETH at expiry to get 11067.83DAI for a cost of 10243.46DAI, locking-in a risk-free profit of 824.37DAI.
Forward price under
Pth,Lā
Let's say the price to buy 1 forward is 90.00DAI instead of Pth,Lā=101.81DAI. An arbitrageur could:
Borrow now 100ETH. In 3 months 100ā(1.029)0.25=100.77ETHā need to be given back.
Convert now those 100ETH to 100ā99.9=9990DAI.
Invest now those DAIs to receive 100000ā(1.099)0.25=10228.57DAI in 3 months.ā
Buy now a forward contract for a quantity of 100.77ETH for a price of 100.77ā90=9975.85DAI.
At expiry, the trader would receive 10228.57DAI from lending and use 9975.85DAIto buy the 100.77ETHneeded to reimburse the debt, locking-in a risk-free profit of 252.72DAI.
Pth,Sā=SSāā(1+rB,bā1+rQ,lāā)T
Pth,Lā=SLāā(1+rB,lā1+rQ,bāā)Tā