Editing Futures contract (section)

===Arbitrage arguments===
Arbitrage arguments ("[[rational pricing]]") apply when the deliverable asset exists in plentiful supply or may be freely created. Here, the forward price represents the expected future value of the underlying [[discounting|discounted]] at the [[risk-free interest rate|risk-free rate]]—as any deviation from the theoretical price will afford investors a risk-free profit opportunity and should be arbitraged away. We define the forward price to be the strike K such that the contract has 0 value at the present time. Assuming interest rates are constant the forward price of the futures is equal to the forward price of the forward contract with the same strike and maturity. It is also the same if the underlying asset is uncorrelated with interest rates. Otherwise, the difference between the forward price on the futures (futures price) and the forward price on the asset, is proportional to the covariance between the underlying asset price and interest rates. For example, a futures contract on a zero-coupon bond will have a futures price lower than the forward price. This is called the futures "convexity correction".

Thus, assuming constant rates, for a simple, non-dividend paying asset, the value of the futures/forward price, ''F(t,T)'', will be found by compounding the present value ''S(t)'' at time ''t'' to maturity ''T'' by the rate of risk-free return ''r''.

:<math>F(t,T) = S(t)\times (1+r)^{(T-t)}</math>

or, with ''[[continuous compounding]]''

:<math>F(t,T) = S(t)e^{r(T-t)} \,</math>

This relationship may be modified for storage costs ''u'', dividend or income yields ''q'', and convenience yields ''y''. Storage costs are costs involved in storing a commodity to sell at the futures price. Investors selling the asset at the spot price to arbitrage a futures price earns the storage costs they would have paid to store the asset to sell at the futures price. Convenience yields are benefits of holding an asset for sale at the futures price beyond the cash received from the sale. Such benefits could include the ability to meet unexpected demand, or the ability to use the asset as an input in production.<ref>{{Cite journal|title=Commodity Futures Prices: Some Evidence on Forecast Power, Premiums, and the Theory of Storage|publisher=The University of Chicago Press|last1=Fama|first1=Eugene F.|last2=French|first2=Kenneth R.|journal = The Journal of Business|year = 1987|volume = 60|issue = 1|pages = 55–73|doi = 10.1086/296385|jstor = 2352947}}</ref> Investors pay or give up the convenience yield when selling at the spot price because they give up these benefits. Such a relationship can be summarized as:

:<math>F(t,T) = S(t)e^{(r+u-y)(T-t)} \,</math>

The convenience yield is not easily observable or measured, so ''y'' is often calculated, when ''r'' and ''u'' are known, as the extraneous yield paid by investors selling at spot to arbitrage the futures price.<ref>{{cite book|title=Options, Futures, and Other Derivatives|pages=122–123|publisher=Pearson|last1=Hull|first1=John C.|edition=9th|year=2015}}</ref> Dividend or income yields ''q'' are more easily observed or estimated, and can be incorporated in the same way:<ref>{{cite book|title=Options, Futures, and Other Derivatives|page=112|publisher=Pearson|last1=Hull|first1=John C.|edition=9th|year=2015}}</ref>

:<math>F(t,T) = S(t)e^{(r+u-q)(T-t)} \,</math>

In a perfect market, the relationship between futures and spot prices depends only on the above variables; in practice, there are various market imperfections (transaction costs, differential borrowing, and lending rates, restrictions on short selling) that prevent complete arbitrage. Thus, the futures price in fact varies within arbitrage boundaries around the theoretical price.