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Can someone give a mathematically explanation why a delta call will move toward 1 if it’s ITM and 0 if it’s OTM? Similar for puts. It will mean that if it’s ITM, the call will move like the stock?

Also, why if the call and the put have the same strike price, the sum of the deltas is equal to 1?

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You dont need mathematically sense. You just need to see whats going on as the expiration date nears..

if the strike is $100 and the stock is current worth $80, the delta will continue to move towards zero as expiration nears.. why? Because time value is going away. Let’s say theres 3 days left, it wouldn’t matter if the stock moved up by $5 bucks, the odds of it over $100 become very very slim. So even if the stock moves 10%, the option will likely not move much.. hense delta is zero.

Opposite is true when you’re in the money. As you’re approaching expiration, your time value is getting smaller.. WHen you’re deep in the money, your value will move similar to the stock.

Perfect, thanks!

And for my last question, is it because I cover all the cumulative distribution?

Regarding your last question, it’s just a mathematical property of the hedge ratio.  Note that it’s technically the sum of the absolute value of the hedge ratios - well, depending on how you define hedge ratio for a put option.  Here’s a proof below for the most common case where S+ > X > S-:

Haha, thanks iceman!