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I am new to hedging and would like to work on delta-gamma hedging. However, I still have a lot of basic questions that are unclear to me.

  1. Suppose we hold a long call option with strike $K$, with underlying asset price at $S_T$ at maturity $T$. Delta shows how the price of the option changes when the asset's price changes, but the option we hold is already bought at some initial price $S_0$, why are we even interested in delta then ? I think that after buying the option, we are not concerned about how the option's price changes due to delta directly affecting the option premium.

  2. I saw while reading stuff about delta-hedging that we try to have a portfolio with 0 delta. I also read that is close to the moneyness of an option, i.e. $\Delta \approx$ the implied probability that the option will expire in-the-money. Hence, if delta is close to $0$ then our options will likely expire out-of-the-money which is not desired. Why would we want to have a 0 delta ?

I have other questions but making these two clear can maybe help me understand many things.

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  • $\begingroup$ For 1), because you might want to sell your option again. For 2) a 0 delta w.r.t a certain underlyer means that swings in the prices of the underlier wont lead to swings in your portfolio's value. $\endgroup$
    – Arkady
    Commented Mar 14 at 1:11
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    $\begingroup$ A simple use case for hedging is the following: an Investment Bank has purchased an option from Customer A, but they have no use for it, they think they will be able to sell it to another Customer B in a little while, but in the meantime they are stuck with it. They hedge to avoid any losses (or profits) during this time, neutralizing or freezing the option so they are not hurt by its price movements. $\endgroup$
    – nbbo2
    Commented Mar 14 at 9:30

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With delta hedging, you are attempting to minimize your options risk to the directional movement in the underlying. One might do this as the option position is to express a view on the volatility of the underlying rather than the direction of the underlying. Options market makers will hedge the delta of their options positions as they are not expressing a view on the direction of the underlying and want to insulate their exposure to the underlying while making a market in options premium.

One example of a strategy to exploit the volatility of the underlying is "gamma scalping." A description of the strategy can be found here: What really is Gamma scalping?

Also, the delta hedge might not come in the form of an actual position in the underlying but on offsetting delta in different options positions. For example, one might express a view on the volatility of the underlying by taking a straddle position in the underlying. A long straddle is a long call and a long put on the same underlying with the same strike, and same maturity. On initiation, the position is essentially delta neutral as the long delta on the call is offset by the short delta of the put. The straddle doesn't care which way the stock moves, but that it moves sufficiently, up or down, to more than cover the premium paid for the options.

Of course if you are intending to express a view on the directionality of the underlying, you would not delta hedge. So in your first example of a long call, if you bought the call because you thought the stock would appreciate, you would not delta hedge as it would be neutralizing the view you were intending to express.

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