The forward price is the agreed upon price of an asset in a forward contract. Using the rational pricing assumption, for a forward contract on an underlying asset that is tradeable, we can express the forward price in terms of the spot price and any dividends. For forwards on non-tradeables, pricing the forward may be a complex task.
Forward price formula
If the underlying asset is tradable and a dividend exists, the forward price is given by: where
Proof of the forward price formula
The two questions here are what price the short position should offer to maximize his gain, and what price the long position should 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 priceor else the transaction cannot occur. An economic articulation would be: The future value of that asset's dividends is calculated using the risk-free force of interest. This is because we are in a risk-free situation so why would the owner of the asset take any chances? He would reinvest at the risk-free rate. The spot price of the asset is simply the market value at the instant in time when the forward contract is entered into. So and his net gain can only come from the opportunity cost of keeping the asset for that time period. let Solving for fair price and substituting mathematics we get: where: where ci is the ith dividend paid at time t i. Doing some reduction we end up with: Notice that implicit in the above derivation is the assumption that the underlying can be traded. This assumption does not hold for certain kinds of forwards.
Forward versus futures prices
There is a difference between forward and futures prices when interest rates are stochastic. This difference disappears when interest rates are deterministic. In the language of stochastic processes, the forward price is a martingale under the forward measure, whereas the futures price is a martingale under the risk-neutral measure. The forward measure and the risk neutral measure are the same when interest rates are deterministic. See Musiela and Rutkowski's book on Martingale Methods in Financial Markets for a continuous-time proof of this result. See van der Hoek and Elliott's book on Binomial Models in Finance for the discrete-time version of this result.
