The forward price is the agreed upon price of an asset in a forward contract. The forward price is calculated by assuming that the long position / short position will not have have an arbitrage opportunity via this transaction. There are equivalent ways of expressing this (simply put, neither the short position nor the long position should be able to make any money off the contract alone). It has been suggested that Definiton of asset be merged into this article or section. ... A forward contract is an agreement between two parties to buy or sell an asset (which can be of any kind) at a pre-agreed future point in time. ... This article needs to be cleaned up to conform to a higher standard of quality. ... The short position refers to the selling entity in a forward contract. ... In economics, arbitrage is the practice of taking advantage of a state of imbalance between two (or possibly more) markets: a combination of matching deals are struck that exploit the imbalance, the profit being the difference between the market prices. ...

Notation

F is the forward price to be paid at time T

C is any rent/dividend/coupon that the asset procures (with 100% certainty) during (0, T )

e^x is the exponential function

S is the spot price of the asset (i.e. what it would sell for at time 0)

The forward price is given by: The word probability derives from the Latin probare (to prove, or to test). ... The exponential function is one of the most important functions in mathematics. ... The spot price of a commodity or a security or a currency is the price that is quoted for settlement (payment and delivery) of the transaction immediately. ...

Proof of the forward price formula

The main dilemma here is what price should the short position (the seller of the asset) offer to maximize his gain; what price should the long position (the buyer of the asset) accept to maximize his gain?

At the very least we know that both do not want to lose any money in the deal.

The short position knows as much as the long position knows: the short/long positions are both aware of any schemes that they could partake on to gain a profit given some forward price.

So of course they will have to settle on a fair price or else the transaction cannot occur.

An economic articulation would be:

(fair price + future value of asset's dividends) - spot price of asset = cost of capital

The future value of that asset's dividends (this could also be coupons from bonds, monthly rent from a house, fruit from a crop, etc.) is calculated using the risk-free force of interest. This is because we are in a risk-free situation (the whole point of the forward contract is to get rid of risk or to at least reduce it) so why would the owner of the asset take any chances? He would reinvest at the risk-free rate (i.e. U.S. T-bills which are considered risk-free). The spot price of the asset is simply the market value at the instant in time when the forward contract is entered into. So OUT - IN = NET GAIN and his net gain can only come from the opportunity cost of keeping the asset for that time period (he could have sold it and invested the money at the risk-free rate).

let:

K = fair price

C = cost of capital

S = spot price of asset

F = future value of asset's dividend

I = present value of F (discounted using r )

r = risk-free interest rate compounded continuously

T = length of time from when the contract was entered into

Solving for fair price and substituting mathematics we get:

where:

(since where j is the effective rate of interest per time period of T )

where c_i is the i ^th dividend paid at time t ⁱ.

Doing some reduction we end up with: