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This is a really basic question,so I apologise. If I have a delta neutral portfolio consisting of an option and its underlying, such that any move in the option is cancelled by an opposite move in the underlying and vice versa, then how can this portfolio ever make a profit? If the move in one is always cancelled out by the other, then it should make neither a profit or a loss.

So how a delta neutral strategy makes money?

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    (1) as soon as the stock moves you are not delta neutral any more due to the delta convexity and (2) you never know if you really are delta neutral because you have to make assumptions on the implied volatility which may or may not materialise in the future. – assylias Nov 24 '14 at 11:04
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A delta neutral position can make money from change in implied volatility, change in underlying price, and/or time decay (if short options).

Implied volatility and time decay are self explanatory so let's look at a simple price example. You buy 500 shares of XYZ at $49.75 and ten $50 puts, each with a delta of -50 so you're delta neutral. You own a synthetic long straddle.

XYZ immediately drops to $48.75 with no change in IV. The put delta is now -590 (each put has a delta of -59). You now buy 90 shares at $48.75 to restore delta neutral.

Now, XYZ immediately recovers the $1 back to $49.75 so you sell the 90 shares, restoring delta neutral and booking a $90 gain. The more times that XYZ percolates up and down or down and up, the more opportunities that you have to do this. In order to make a profit, before expiration you must book gains faster than the rate of time decay.

Since this is a straddle, money is also made if there is some terrific or terrible news that drives share price dramatically up or down, outside and beyond the break even points (approximately share price + option position cost AND share price less option position cost).

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With the netural position delta strategy under high IV returns short vega,there is a possibility to profit from a decline in IV. Of course, if volatility rises higher, the position will lose money. It is therefore best to establish short vega delta-neutral positions when implied volatility is at levels that are in the 90th percentile ranking.

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Depending on the structure of you're portfolio, it could be that your portfolio is delta neutral to take advantage of diminishing time value on options, short straddles/strangles would be an example.

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