Consider a $60 stock with a call option that has a 0.50 delta and is trading
for 3.00. Considering only the delta, if the stock price increases by $1, the
theoretical value of the call will rise by 0.50. That’s 50 percent of the stock
price change. The new call value will be 3.50. If the stock price decreases
by $1, the 0.50 delta will cause the call to decrease in value by 0.50, from
3.00 to 2.50.
Puts have a negative correlation to the underlying. That is, put values
decrease when the stock price rises and vice versa. Puts, therefore, have
negative deltas. Here is a simplified example of the delta effect on a −0.40-
delta put:
As the stock rises from $60 to $61, the delta of −0.40 causes the put value
to go from $2.25 to $1.85. The put decreases by 40 percent of the stock
price increase. If the stock price instead declined by $1, the put value would
increase by $0.40, to $2.65.
Unfortunately, real life is a bit more complicated than the simplified
examples of delta used here. In reality, the value of both the call and the put
will likely be higher with the stock at $61 than was shown in these
examples. We’ll expand on this concept later when we tackle the topic of
gamma.
Definition 2 : Delta can also be described another way. Exhibit 2.1 shows
the value of a call option with three months to expiration at a variable stock
price. As the stock price rises, the call is worth more; as the stock price
declines, the call value moves toward zero. Mathematically, for any given
point on the graph, the derivative will show the rate of change of the option
price. The delta is the first derivative of the graph of the option price
relative to the stock price .