FRM Fun 6 (Mon)

[Suzanne Evans]

FRM Fun 6 (optimal versus effective hedge)

In the FRM, both authors Hull and Geman define the futures basis (b) = S(t) - F(t). In words, the basis is the difference between the spot price and one of the (any of several quotable) futures prices.

Geman further defines basis risk as the variance[S(t) - F(t)]; i.e., the variance of the difference between two random variables. Then Geman defines a "classical measure of effectiveness of hedging a spot position with Futures contracts" as given by h = 1 - variance[basis]/variance[S(t)] = 1 - variance[S - F]/variance[S(t)].

While Hull defines the minimum variance hedge ratio (aka, optimal) as given by h* = correlation(S,F)*volatility(S)/volatility(F).

What is the relationship between Geman's hedge effectiveness (h) and Hull's optimal (h*), if any? for example, are they demonstrably equivalent? unrelated? is one superior?

 

[ShaktiRathore]

Now According to my solution:
the Geman defined h=1-(Var(S-F))/Var(S)
or h=1-(VarS+VarF-2*corr(S,F)*sqrt(Var(S)*Var(F)/Var(S))
or h=1-(1+(VarF-2*corr(S,F)*sqrt(Var(S)*Var(F)/Var(S))
or h=(-VarF+2*corr(S,F)*sqrt(Var(S)*Var(F))/Var(S)
or h= -(VarF/VarS)+(2*corr(S,F)*sqrt(Var(S)*Var(F)/Var(S))
or h= -(VarF/VarS)+(2*corr(S,F)*sqrt(Var(F)/Var(S))...(1)
for minimum hedge effectiveness h(opt.) dh/dS=0
thus from above
dh/dS=(VarF/VarS^2)-(1*corr(S,F)*sqrt(Var(F)/Var(S))*(1/VarS))=0
=> after some calculation corr(S,F)=sqrt(VarF/VarS)=vol(F)/vol(S)...(2)
Putting (2) in (1) implies
h(opt.)=-corr(S,F)^2+2*corr(S,F)*corr(S,F)=corr(S,F)^2 ..(3)
This is the optimal hedge effectiveness required to hedge the spot price according to geman.
According to Hull optimal hedge ratio h*= corr(S,F)*(vol(S)/vol(F))…(4)
From (3) and (4),
h(opt.)=corr(S,F)^2=(h*^2)*(vol(F)^2/vol(S)^2)…(5)
Also the relation b/w the geman hedge effectivenss and hull’s minimum hedge ratio is
h=(vol(F)^2/vol(S)^2)*(-1+2*h*)…(6)

from (6) it is clear that the gemans and hulls hedge ratio are not equivalent also that optimal hedge ration arrived at by geman is different from that of hull’s min. hedge ratio. For two optimal ratios to be equal 2,3 and 4 implies vol(S)=vol(F).h(opt) measures the correlation of spot and futures prices directly and hence gives better picture of how one moves relative to another and how effective is futures in covering up the spot price movements. An correlation of -1 implies h(opt)=1 which is a perfect hedge and lower correlation implies less than perfect hedge.
Whereas in case of Hull’s hedge ratio it also measures the relative volatility of spot price w.r.t volatility of futures which might not give better picture of relative movement of spot and futures prices directly but gives magnitude of optimal hedge ratio which gives perfect hedging.

 

[David Harper CFA FRM]

Hi ShaktiRathore,

Star for the win (ie, you were just entered in the weekly drawing for two $15 gift certificates), and THANK YOU for your generous contribution! I am trying to figure out whether we differ, as I get a little stuck on the advisability of dh/dS ... wouldn't this be finding the (S) that maximizes/minimizes (h)? If so, I am not following this step ... below is how I think about it, which may be consistent with your solution. If you observe the reconciliation, please do let me know? Thanks!

I agree with you they are not equivalent because Geman's hedge effectiveness is simply given the variance of a 1:1 hedge (1 future: 1 spot).
In this sense, Geman's hedge effectiveness could be generalized with:
var (s - h*f) = var(s) + var (h*f) - 2*cov(s,h*f) =var(s) +h^2*var(f) - 2*h*cov(s,f); i.e., h is a constant, which Geman's formula simply assumes is 1.0 in a 1:1 hedge.

The inclusion of (h) allows this measure of risk (variance of the basis) to generalize such that there is an (h) which minimizes the variance. This optimal (h) is (h*), which is different than 1.0, or as you say, only 1.0 under a restricted special case. If we use (h*) then:
if h = h* = rho*SD(s)/SD(f),
= (2*rho*SD(s)/SD(f)*rho*SD(s)*SD(f) - [rho*SD(s)/SD(f)]^2*var(f)) / var(s)
= (2*rho*SD(s)/SD(f)*rho*SD(s)*SD(f) - rho^2*SD(s)^2/SD(f)^2*var(f)) / var(s)
= (2*rho*SD(s)*rho*SD(s)- rho^2*SD(s)^2) / var(s)
= (2*rho^2*SD(s)^2- rho^2*SD(s)^2) / var(s)
= rho^2*SD(s)^2 / var(s)
= rho^2

So I conclude:

• Geman's hedge effectiveness, as it assumes 1:1, will not be optimal (but merely returns the variance of the basis in the 1:1 case)
• But it generalizes to: var (s - h*f) .... allowing it to accept a hedge ratio (h) that is different than 1.0
• Which will indeed be minimized by Hull's (h*)
• As Hull's optimal hedge (h*) will minimize Geman's generalized hedge effectiveness, they are different but totally consistent in their treatment of basis risk as the variance of the basis.