# Hull's method for the optimal hedge ratio: why?

To illustrate my question, let's assume that the owner of one unit of asset (unit spot price variable S) needs to sell this asset at time T in the future.

In order to hedge against a possible fall in the price of the asset, at time 0 the owner shorts h units of futures contracts (unit forward price variable F, F = F0 at time 0) over same or similar assets with delivery date at time T. He has a portfolio of value [S + h(F0 - F)].

Intuitively, the optimal hedge ratio should enable the value of the portfolio at time T to be equal to that at time 0, namely ST + hF0 - hFT = S0, namely ST - S0 = h(FT - F0). If we regress (St - S0) over (Ft - T0), h will be the slope, which is equal to Cov(St, Ft) / [Var(St)*Var(Ft)]^(1/2) = Cov(S, F) / [Var(S)*Var(F)]^(1/2).

Hull (2000) defines the "optimal hedge ratio" h as one that minimizes the variance of the value of a portfolio [S + h(F0 - F)] during the life of the futures contract (from time 0 to T), namely Var([S + h(F0 - F)]) = Var(S) + h^2 * Var(F) - 2*h*Cov(S, F), where Cov = covariance. Thus h = Cov(S, F) / Var(F).

Thus Hull's method is different from our intuition in that: Our intuition gets an h by regressing (ST - S0) over (FT - F0) to get the slope, is equivalent to that the sum of squares of the value of the portfolio from time 0 to time T (S + h(F0 - F)) is minimized; Hull's method get an h by regressing (St - St-1) over (Ft - Ft-1) to get the slope, is equivalent to minimizing the variance of the value of the portfolio from time 0 to time T.

Could someone explain why Hull regresses(St - St-1) over (Ft - Ft-1) instead of (ST - S0) over (FT - F0)?

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because T is in the future? –  user508 Feb 18 at 2:59