Suppose the spot price is S, forward price is F. S and F are the prices of two correlated asset. John Hull?s book directly gives a method to calculate Minimum Variance Hedge Ratio http://financetrain.com/minimum-variance-hedge-ratio/as
h=rho*sigma_S/sigma_Fwhere rho is correlation between dS and dF during the hedging period.The number of optimal Contract N is N= h*QA/QFWhere QA is Size of position being hedged (units) and QF is Size of one futures contract (units).John Hull?s book doesn?t give details on the derivation process of both equations. And I find some problem on them.Suppose one long the spot S, and short forward F for hedging by holding N contracts, then the portfolio return Rh should be:Rh=(QA*dS-QF*dF*N)/(QA*S)=dS/S - (QF*N*F)/(QA*S)*(dF/F)dS/S and dF/F are the spot return Rs and forward return RF respectively.And leth=(QF*N*F)/(QA*S) then Rh= Rs - h* RFSo VAR(Rh)= VAR(Rs) + h^2*VAR(RF)-2h*COV(Rs,RF)Let dVAR(Rh)/dh=0, so that 2h*VAR(RF)-2COV(Rs,RF)=0h=COV(Rs,RF)/VAR(RF)= rho*sqrt(VAR(RF)* VAR(Rs))/ VAR(RF)=rho*sigma_S/sigma_FThe hedge ratio here is the same as John Hull, but optimal contracts N becomesN = h*QA*S/(QF*F)Which is different. The calculation of optimal contracts here requires S and F.So what?s wrong here?