Hedging Strategy and Electricity Contract Engineering - IFOR

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5.2 Traditional hedging 105
by the position in the spot, is given by h.
The change in value of the hedged position will then be given by
Å S hÅ Fž (5.1)
The variance Æ 2 , of the change in value of the hedged position is
Æ
2
Æ
2
S h 2 Æ
2
F 2hÇKÆ SÆ F (5.2)
and the derivative with respect to the hedge ration is
2hÆ
2
h F SÆ F ž (5.3)
2ÇKÆ
Since 2
Æ 2
¡ h 2 2Æ
2
p is positive, the first order condition is sufficient to find
the h that minimizes the variance, namely
È
Æ
2
Æ
2
Æ
0
S
ž h (5.4)
Ç
h
Æ F
It is certain that the hedge ratio h, will minimize the variance, but it is debatable
if it is optimal, since we implicitly state that variance is the risk measure
of concern. If we assume that the spot price follows a geometric Brownian motion
and that the good is storable, then the cash-and-carry strategy implies that
also the future price will follow the same price process. The returns of both the
spot and the future will therefore be normally distributed, why variance or standard
deviation will be the natural risk measure, and a variance minimization is
appropriate. 3
Delta hedge For non-linear positions, where the price is non-linear in the
underlying’s price, the two simple approaches described above have to be expanded.
A static hedging strategy, like the optimal hedge ratio, cannot be used
anymore, since the dependency between a non-linear position and the underlying
position will differ with, for example, time and price of underlying. The
3 As argued in Chapter 3.4.1.