Delta measures the price (premium) change of an option relatively to a price change of $1 in the underline, given that all the other factors which affect the option price remain constant. Premiums of calls and puts are not changing in the same rate as the price of the underline security. The deeper ITM is a call or put the more its price change resembles the price change of the stock.

Suppose that we have a call at stock C with strike price $30, expiration after 6 months, premium $1.24 and the current price of C is $26.30. If the stock rise $1 the above call will rise $0.32. So, its delta is 0.32. If we had a call deep ITM, i.e. with strike price $15, then an $1 stock advance would have result to a premium rise of $0.95. In this case its delta is 0.95. Deep ITM calls or puts tend to behave like stock. If in the above call with strike price $30 the delta was 0.45 then what would have happened to the premium after an $1 advance in the stock? The premium would have advanced by $0.45, from $1.24 to $1.69. If the stock had declined $1? The call premium would have declined from $1.24 to $0.79.

In case that we had a put with the same characteristics as the call above, with premium $4.60 and a delta of $0.70 what would have happened to the value of the put if the stock price had advanced $1? Its value would have declined $0.70, from $4.60 to $3.90. If the stock had declined 1$? Then the put would have gained $0.70, from $4.60 to $5.30.

Mathematically delta is expressed by the first derivative of the option value relatively to a $1 change in the underline security and its formula is:

Call delta is positive whereas put delta is negative. This means that when the stock price rises the call premium will also rise but the put premium will decline. Delta cannot exceed absolute 1, no matter how deep ITM an option is.

Another interpretation of delta which is more empirical and less mathematical is that it expresses the possibility that an option would be ITM at expiration. If a call has a delta of 0.60 then the market perceives it to have a probability of 60% to be ITM at expiration. In that sense, if we add up the delta of a call and a put with the same characteristics on the same underline security, this must be equal to 1.

Notice also that the estimated delta of an option is an approximation of reality and not a substitute. In real markets, prices might behave differently than theoretical estimations suggest.