# How to Find the Forward Price

A forward contract obligates the two parties to the contract to transact on a certain day at a certain price. (it is not an option)

In a forward contract, the long is obligated to purchase at the contract price and the short is obligated to sell at the contract price.

Therefore if the spot price at time T1 is greater than the contractually obligated price the long profits. For example, the contract price is 100 and the spot price (market value of underlying) is 120. The long gets the asset for 100 and sells for 120.

Contracts can be closed out via offsetting, for example, if you are long a September contract you can short a September contract, the two offsetting each others price moves and absolving you of the obligation to take delivery.

### At what price do we contract to transact in the future? Forward Price

This depends on two things"

1) The current price of the underlying

2) The risk-free rate

3) Any other costs or benefits

Forward price at t0 = (spot price at t0 - PVbenefits+PVcosts)(1+r)^t

Essentially what this implies is that the forward price should be equal to the future value of the current spot and any costs or benefits using the risk-free rate.

The reason this relationship holds true because in any other circumstance an arbitrage opportunity exists

Consider:

1) The risk-free rate of borrowing is 10%

2) the underlying is 100\$

3)the forward price is 112\$

We can buy the underlying and short the contract guaranteeing a 12\$ profit. To fund this transaction we can borrow at a cost of only 10\$, the risk-free rate. Meaning we make 2\$ with no investment and no risk.

Conversely

1)The risk-free rate is 10%

2) The underlying is 100

3) The forward price is 109

We can go short the underlying and long a forward, guaranteeing a 9\$ cost of borrowing. We then invest at the risk-free rate. All without any risk or money out of our pocket. This is called arbitrage.

### Value at Some Time Other Than Inception

suppose the contract was originally 1 year and half the year has passed.

1) Find the future price as shown above

2) underlying value at the current time - forward price discounted at the risk-free rate for remaining time = value (also need to find the PV of any costs and benefits)

3) Example forward price is 110 given 10% risk-free rate. After half a year the underlying is 106

Value = 106 - 110*(1.1^-.5)

Value = \$1.12

Or

This equation implies that as the risk-free rate increases we are subtracting smaller and smaller values. This implies that if interest rates rise the value of the forward will also rise.