Delta

Delta is important for two main reasons. First, delta is a widely used measure of the
exposure of an option position to the underlying. Option dealers are guided by delta
in determining option hedges. Second, delta is a widely used measure of the degree
to which an option is in- or out-of-the-money.
The option delta is expressed in percent or as a decimal. Market parlance drops
the word ‘percent’, and drops the minus sign on the put delta: a ‘25-delta put’ is a
put with a delta of ñ0.25 or minus 25%.
Delta is the part of a move in the underlying price that shows up in the price or
value of the option. When a call option is deep in-the-money, its value increases
almost one-for-one with the underlying asset price, so delta is close to unity. When
a call option is deep out-of-the-money, its value is virtually unchanged when the
underlying asset price changes, so delta is close to zero. Figure 1.14 illustrates the
relationship between the call delta and the rate of change of the option value with
respect to the underlying asset price.
1.58 1.59 1.60 1.61 1.62
Spot
0.005
0.010
0.015
0.020
Call value
delta at 1.60
delta at 1.61
Figure 1.14 Delta as the slope of the call function.
The reader may find a summary of the technical properties of delta useful for
reference. (Figure 1.15 displays the delta for a typical call option):
Ω The call option delta must be greater than 0 and less than or equal to the present
value of one currency unit (slightly less than 1). For example, if the discount or
risk-free rate is 5%, then the delta of a three-month call cannot exceed 1.0125ñ1
(eñ0.0125, to be exact).
Ω Similarly, the put option delta must be less than 0 and greater or equal to the
negative of the present value of one currency unit. For example, if the discount or
risk-free rate is 5%, then the delta of a three-month call cannot be less than
ñ1.0125ñ1.
Ω The put delta is equal to the call delta minus the present value of one currency
unit.

Ω Put-call parity implies that puts and calls with the same exercise price must have
identical implied volatilities. For example, the volatility of a 25-delta put equals
the volatility of a 75-delta call.