Hedging Strategy and Electricity Contract Engineering - IFOR

É
È
É
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5.2 Traditional hedging 107
constant. There are however two complications. First, the delta of a non-linear
position is changing with underlying price, volatility, time to maturity and
interest rate, why the hedge will have to be revised as soon as any of these
parameters change. Since, for example, the underlying price typically will
change at least every other minute, the delta hedge has to be dynamically
rebalanced. The second complications comes from the fact that the delta
hedged position is only locally immune to changes in the underlying price and
will for any movement actually be exposed to some risk.
The value of the hedged position can be Taylor expanded, like any function
, with respect to the underlying price S
Ê”Ë
dÉ
S
È dS 1 2
É 2
È
2 S 2 dS›
ž?ž@ž
1
n
È n
É n
È
S n ž@ž?ž (5.7)
dS›
The error caused by approximating the value É , with only its first component
in the Taylor expansion grows with the changes in the underlying dS. One way
to improve the approximation is to add the second component in the expansion.
Delta/gamma hedge By requiring that not only the delta, but also the second
derivative with respect to the underlying price, the gamma
È 2
É
È
S 2 (5.8)
shall equal zero, one conducts a so-called delta/gamma hedge. The approximation
is of course better than in the delta case, but one still has the problem
of dynamic rebalancing, which because of transaction costs can be extremely
costly if done on a continuous basis.
Duration hedge The more sophisticated hedging schemes using the sensitivities
and , are typically mainly used for simple underlying contracts
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without a time component, like stocks. For interest contracts, which always
have such a time component, adding a dimension of complexity, a simpler
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approach is often called for. One such method is the duration hedge. The
duration of a bond is a measure of how long, on average, the holder of the bond