Okay, so I bought the call option for XYZ with the strike price at 50 for $2 (or $200). The at-the-money option had a delta of 0.5 and 45 days to expiry. The above diagram shows what my potential P/L would be once it crosses above the breakeven at 52. So far, so good.
Now suppose in less than 2 hours, XYZ rises to 55 and I exit the trade at $3 ($100 profit). If going strictly by the textbook example, I shouldn't be making any money before XYZ rises above the breakeven at 52. But this isn't true at all. When the price rises from 50 to 52, the option price also rises accordingly and I can almost always get out with a profit.
But my real question is regarding the delta. With a delta of 0.5, I should make $250 when XYZ rises from $50 to $55. But when does the option become "in-the-money", at which point the delta is more close to 1.0? Frankly, I don't. Hence, I have no idea what the option price would be when XYZ is trading at 55. And without this information, how would I know how many contracts to trade?