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I am learning about quantitative finance and currently learning about delta neutral strategies. The examples given are most of the times of the form: if you buy call options, you can hedge your position by short selling a certain amount of the underlying stock.

My question is as follows: suppose you think that the stock is going up in 3 weeks and you call options accordingly. Now we want to hedge our position by buying put options (in the case that the price does not go up). Is it possible to get a delta neutral position in this way and is it used often?

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  • $\begingroup$ You think that the stock is going up (bullish on the stock) but you also want to have a delta neutral position (neutral on the stock). These two are not consistent... If you are bullish you want a positive delta position, if you want to limit the losses on the downside you can indeed buy a put for this purpose but IMO you should keep a positive overall delta. "hedging" in the sense of limiting downside losses is different from "hedging" in the sense of eliminating upward and downward movements. Black and Scholes are using "hedging" in the latter sense, the neutralization of risk. $\endgroup$ – noob2 Oct 22 '18 at 19:34
  • $\begingroup$ @noob2 I am also talking about neutralization of risk (as in Black-Scholes) $\endgroup$ – etotheipi Oct 23 '18 at 21:07
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You could, for a particular vol, time to maturity, spot and strike have a delta neutral position by buying a certain amount of the put.

But consider what would happen if the spot goes up: The delta of the call will increase and the delta of the put will decrease. Plus, you just spent an additional premium compared to delta hedging with the underlying.

Basically, its not a good ideia. If the real world exhibited the Black Scholes assumptions, of continuous costless trading, you could in theory preserve any intrinsic value of an option position by continuously delta hedging with the underlying to have a delta neutral position.

But if you use options to delta hedge, not only will it be more costly because of premiums, you will also have the Greeks of your hedges to worry about...

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Assuming that, after buying the options, you will no longer trade in the underlying or options, it only makes sense to buy a put, in addition to the call, if you want to bet that the price of the underlying (PU) will experience a big increase or a big decrease. Note that you will be paying the premiums of both the call and of the put. Therefore, the increase or decrease in the PU needs to be high enough to generate a payoff greater than the sum of both premiums.

Of course you can choose the strikes and the volume of the call and the put to have a delta neutral position at the time you buy the two options. But this only means that your bet roughly equally to an increase or a decrease in the PU. However, as soon as the UP changes, your delta will not be zero: this is not a delta neutral strategy, i.e., a strategy of continuously trading to keep the delta equal to zero during the whole life of the options.

In a delta-neutral strategy during the whole live of the option(s) you are in theory completely hedged. Therefore, you will not make or lose any money (as long as the observed volatility is equal to the implied vol during the life of the option), which means that you are not betting on an increase of the PU.

The moral of the story: You cannot bet and hedge at the same time. Actually, hedging is a strategy that aims to avoid any bet, and therefore, any risk.

If you want to bet that the PU will increase and not decrease, you have only to buy and to pay the premium of the call.

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