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Delta Neutral Portfolio

When would you use puts to delta hedge a portfolio?

If you are long Equity.

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You can also hedge the delta from being Long Equity with Short Call. Doesn’t really matter, with both you have to rebalance continuously. Since the Delta of both the Call and the Put will change due to market conditions (spot change, vola, interest, dividend, whatever).
There isn’t just one solution to delta hedging.

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It is typically the case the long equity will buy protection using a long put, and not necessarily use it for delta hedging. While you certainly can do it, it is more common to use a put to delta hedge a short position in equity with a short position in the put.

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So basiscally you can do it either way, hedge a short in a stock by shorting a call or shorting a put?

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A short in a stock can’t be hedged with a short call. That has to be done with a long call or a short put.

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