Delta hedging and gamma risk

Individuals and firms buy or write options in order to hedge or manage a risk to
which they are already exposed. The option offsets the risk. The dealers who provide
these long and short option positions to end-users take the opposite side of the
option contracts and must manage the risks thus generated.
The standard procedure for hedging option risks is called delta or dynamic
hedging. It ordains
Ω Buying or selling forward an amount of the underlying equal to the option delta
when the option is entered into, and
Ω Adjusting that amount incrementally as the underlying price and other market
prices change and the option nears maturity.
A dealer hedging, say, a short call, would run a long forward foreign exchange
position consisting of delta units of the underlying currency. As the exchange rate
and implied volatility changes and the option nears expiration, the delta changes, so
the dealer would adjust the delta hedge incrementally by buying or selling currency.
The motivation for this hedging procedure is the fact that delta, as the first
derivative of the option value, is the basis for a linear approximation to the option
value in the vicinity of a specific point. For small moves in the exchange rate, the
value of the hedge changes in an equal, but opposite, way to changes in the value of
the option position.
The delta of the option or of a portfolio of options, multiplied by the underlying
amounts of the options, is called the delta exposure of the options. A dealer may
immediately delta hedge each option bought or sold, or hedge the net exposure of
her entire portfolio at the end of the trading session.
In trades between currency dealers, the counterparties may exchange forward
foreign exchange in the amount of the delta along with the option and option
premium. The option and forward transactions then leave both dealers with no
additional delta exposure. This practice is known as crossing the delta.
Delta hedging is a linear approximation of changes in the option’s value. However,
the option’s value changes non-linearly with changes in the value of the underlying
asset: the option’s value is convex. This mismatch between the non-linearity of the
option’s value and the linearity of the hedge is called gamma risk.
If you are long a call or put, and you delta hedge the option, then a perturbation

of the exchange rate in either direction will result in a gain on the hedged position.
This is referred to as good gamma.
Good gamma is illustrated in Figure 1.18. The graph displays a long call position
and its hedge. The option owner initially delta hedges the sterling call against the
dollar at USD1.60 by buying an amount of US dollars equal to the delta. The payoff
on the hedge is the heavy negatively sloped line. If the pound falls one cent, the value
of the call falls by an amount equal to line segment bc, but the value of the hedge
rises by ab[bc. If the pound rises one cent, the value of the hedge falls ef, but the
call rises by de[ef. Thus the hedged position gains regardless of whether sterling
rises or falls.
1.58 1.59 1.60 1.61 1.62
0.005
0
0.005
0.010
0.015
0.020
hedge
long call
a
b
c
d
e
f
Figure 1.18 Good gamma.
If you are short a call or put, and you delta hedge the option, then a perturbation
of the exchange rate in either direction will result in a loss on the hedged position.
This is referred to as bad gamma.
Bad gamma is illustrated in Figure 1.19. The graph displays a short call position
and its hedge. The option owner initially delta hedges the sterling call against the
dollar at USD1.60 by selling an amount of US dollars equal to the delta. The payoff
on the hedge is the heavy positively sloped line. If the pound falls one cent, the value
of the short call position rises by an amount equal to line segment ab, but the value
of the hedge falls by bc[ad. If the pound rises one cent, the value of the hedge
rises de, but the short call rises by ef[de. Thus the hedged position loses regardless
of whether sterling rises or falls.
Why not hedge with both delta and gamma? The problem is, if you hedge only with
the underlying asset, you cannot get a non-linear payoff on the hedge. To get a
‘curvature’ payoff, you must use a derivative. In effect, the only way to hedge gamma
risk is to lay the option position off.