The option delta is something like a delta of 0.7 means that for every $1 the underlying stock increases, the call option will increase by $0.70. However from the daily change in stock price compared to the option change do not seems to adhere to the definition. Like for ibm, the from on 24 may is 3 cents (close to close) but the option (strike price 115, delta 0.995) change is +50 cents from 11.05 to 11.55. if ibm price increase 3 cents than wouldn't the this option with delta 0.995 increase 2.985 cents? (0.995*3 cents)

There are two possible explanations, likely its a combination of both. First, Delta is not constant, it also has a rate of change (gamma). Therefore, the delta is only applicable for a small range around the current price. More importantly, there are 3 other factors in the price of an option. Changes in: Implied Volatility (Vega) Time (Theta), also known as Time Decay Interest Rates (Rho) For example, for both calls and puts, an increase in implied vol will increase price, and knowing the Vega can help you calculate how the price will be impacted by this input.

You're quoting the last sale of an ITM option that traded 35 contracts for the day. And the +.50 from the previous day's last sale is not an accurate measure of the change in the option's price. The last B/A was 10.55x10.75 - a more acurate measure of potential pricing. The 11.55 trade likely happened when IBM was trading at higher prices and given the low liquidity - this price may not be representative of "fair value" at the time of the sale. Overall IV changes in IBM options may account for a shift from higher to lower option pricing during the day. IBM fell .97 for the day which would, at least, partly account for the lower prices in the closing B/A of the June 115C. Delta predicts the change in the options price - for a given price change of the underlying - with all other pricing factors remaining unchanged. So, delta is, at best, an estimate of what should happen since there are other factors involved.

Reasons for discrepancy: 1) Last trade occurred earlier in the day at a different underlying price 2) Wide B/A spread makes daily change look larger or smaller 3) Change in implied volatility 4) And once in awhile, bad quotes ala Yahoo

Anyone knows how delta neutral strategy makes money? There are comments that one has to constantly hedge and readjust to delta neutral. But ultimately how does this strategy make money from large price movement from the stock? When is appropriate to close out the entire position?

Delta-neutral strategy makes money from volatility. E.g. if you are long volatility and it rises then you make money. And, yes, you do need to readjust to remain delta-neutral. The strategy is also known as gamma scalping.

How it makes money depends on whether you're long or short delta but in general, it makes money from reversals in the position as well as IV change in your favor. It's appropriate to close out the position when IV is moving against you. Do a web search on "delta neutral" and you'll find numerous articles with examples.

that's true but the major source of money is often just from keeping the position delta neutral even when you are flat vega. if you're long gamma (basically long optionality) you need to sell high and buy low to accomplish keeping the position delta neutral. the profit from this strategy should be able to offset the price of the option paid. to OP: pick up some book - there are gazillions of them available on the market, e.g. Hull, Wilmott etc..