# Forward price

The

**forward price**or**forward rate**is the agreed upon price of anasset in aforward contract . Using therational pricing assumption, we can express the forward price in terms of thespot price and any dividends etc., so that there is no possibility forarbitrage .**Forward Price Formula**The forward price is given by:

:$F\; =\; S\_0\; e^\{(r+q)T\}\; -\; sum\_\{i=1\}^N\; D\_i\; e^\{r(T-t\_i)\}\; ,$

where

:"F" is the forward price to be paid at time "T":"e

^{x}" is theexponential function (used for calculating compounding interests):"r" is therisk-free interest rate :"q" is thecost-of-carry :$S\_0$ is thespot price of the asset (i.e. what it would sell for at time 0):$D\_i$ is adividend which is guaranteed to be paid at time $t\_i$ where $0<\; t\_i\; <\; T.$**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:

:$K\; =\; C\; +\; S\; -\; F\; ,$

where:

:$C\; =\; S(e^\{rT\}\; -\; 1)\; ,$(since $e^\{rT\}\; =\; 1\; +\; j\; ,$ where "j" is the effective rate of interest per time period of "T" )

:$F\; =\; c\_1\; e^\{r(T\; -\; t\_1)\}\; +\; cdots\; +\; c\_n\; e^\{r(T\; -\; t\_n)\}$where "c

_{i}" is the "i^{th}" dividend paid at time "t^{i}".Doing some reduction we end up with:

:$K\; =\; (S\; -\; I)e^\{rT\}.\; ,$

**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 theforward measure , whereas the futures price is a martingale under therisk 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.

**ee also***

Forward measure

*Convenience yield

*Cost of carry

*Backwardation

*Contango

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

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