# How does delta of an option change with time if underlying price is constant?

In his book 'Option Volatility and Pricing', Sheldon Natenberg writes that if an option has a delta of 92 with 3 months to expiry; and then 2 months pass and the underlying contract does not change in price,the delta will increase.

Why is this? Delta is rate of change of option price w.r.t underlying price.

I am not following why delta will increase if underlying remains constant.

So, this isn't always the case, but in the example provided the option is most likely in the money or near the money since the delta is nearly 1 - indicating that a \$1 move in the underlying results in a \$0.92 move in the option - this will happen when the expiration is very far out or the option is in the money.

As expiration gets closer, movements in the underlying become more pronounced in the options because the probability of the stock price moving from its current position is lower. As the probability of the stock price moving goes down, the delta of in the money options approaches 100, eventually reaching 100 at expiration.

Another way to word this is that the premium on in the money options shrinks as expiration approaches and the intrinsic value of the option increases as percentage of total value so that movements in the underlying stock price become a greater influence on the option price - hence a greater delta.

Again, if the option is out of the money, this is not the case.

As the option approaches expiry, the delta will approach zero or one, depending on whether you're in or out of the money.

This might be easiest to conceptualise if you look at the option value as a function of the stock price, and then realise that the delta is the slope of that curve. Now, as we get closer to expiry, time value fades away, and we get closer and closer to the intrinsic value, which looks like this hockey stick: __/

As you see, close to expiry, if you're out of the money, you have nothing (with delta zero), while if you're in the money, you have a forward (with delta one).

The question is always one of whether people think they can reliably predict that the option will be a good bet. The closer you get to its expiration, the easier it is to make that guess and the less risk there is. That may either increase or decrease the value of the option.