My understanding of Delta is the on the CALL side, if the delta is 0.40, for every $1.00 move up in underlying price the option price will go up by 0.40. What happens if the underlying moves down by $1.00, will the option price also move down accorindingly by 0.40? Also, for the PUT side, assuming delta is -0.30, what happens when the underlying moves down by $1.00? I had assume that this will cause the option price to be reduced by 0.30. Correct me if I'm wrong. Thanks!

For a call with a 40d, as the stock moves up or down there is the assumption that the option will move up by $0.40 and or down by $0.40 respectively. For a Put with a 40d, as the stock moves down or up there is the assumption that the option will move up by $0.40 and or down by $0.40 respectively. This assumption is only valid for short price movements because as the stock moves, the delta changes so incremental movements in the price might not be the same delta for the entire 1 point or 2 point move. If this is a high priced stock like TSLA, the delta will not change quickly over even a 10 point move but when applying the same expectation for a stock like GE, as the stock moves in and out of the money, the delta can change quickly. Also, with regard to short duration options, as they move the new delta changes faster than say an option with an expiration out 6 months. You then have to consider more than the price of the stock in relation to the strike, as it is not the only input into the pricing model. Any change in supply/demand will affect the Implied vol and will increase or decrease the value of the option too.

Just to elaborate on what Robert said. The greeks are derived at one point of time, commonly called a snapshot of the option. Greeks can change over time and may not be very accurate in some cases. OP, in your example, the put would gain .30. This assumes that all other option pricing factors remain static including zero time decay. Extreme examples are sometimes helpful. During crude oil's past few months of volatility, there were times when IV fluctuated wildly day to day. Delta was useless and Theta was fairly useless during these times because the IV fluctuations trumped all of the other greeks. This is a common occurrence with earnings announcements (EA). During the days before EA, the IV of the front month/weekly options is jacked up because the market knows that the EA event may produce extraordinary volatility. And everyone knows that there will likely be an IV crush at some point after the EA. Delta cannot be relied upon in this situation to predict the option's behavior. Typically, you need a much larger move to profit than Delta would suggest. And yet you still hear the complaint after EA, "the stock went up, how come my calls went down"