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If a linear sum of options is constructed such that the premium payout is zero, then does it mean that resultant greeks of the cumulated options positions will be nearly zero. For simplicity, lets consider only single expiry case.

For example,

a*(Put@X1) + b*(Call@X2) + c*(Put@X2) = 0, for integer values of a,b,c, then the greeks from such a position should result in nearly zero greeks.

If they result in providing us with non-zero greeks, then there is a possibility of being able to build positions such that we only have greeks exposure, without any premium outlay.

Can this question be better worded?

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It might not work the way you think. Note first that nobody sells options for free so at least one of your integers ($a,b,c$) is negative, meaning you will have nonzero risk of losing money on your short option leg.

More specifically, let's pretend $b \geq |c|$. Then since the value of a forward contract is the same as the call minus the put plus strike $X_2$ you can also think of yourself as owning

$$ a P(X_1) + b C(X_2) + e^{-rt} (b-|c|) (F - X_2) $$

Obviously you can only make money on the first two components, but that forward contract could end up costing you a ton.

If you really don't want to pay upfront, it is quite simple to choose $a$ and $b$, look at the prices of the three options in the market, and then set

$$ c =- \frac{a P(X_1) + b C(X_2)}{P(X_2)} $$

which of course results in no upfront cost to the combination.

You may be interested in a well-known combination called the butterfly. It is (theoretically) possible to trade a butterfly for no initial premium, by choosing just the right strikes.

Ignoring bid-offer spreads, and assuming the Black-Scholes model for calls and puts as a function of strike is given by $C(K)$ and $P(K)$, the way to find some strikes for a zero premium is to

  • Choose a central strike $K_C$, perhaps the current undelying price $S_0$
  • Use a root finder to solve for a in the equation

$$ 0 = P(K_C-a) - 2 P(K_C) + P(K_C+a) $$

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  • $\begingroup$ Sorry about not making it clearer - If the option is sold, then Premium is recd and positive and if it is sold, then premium is given and negative. So that takes care of the sign issues. Else we can use real numbers instead of integers but not together. $\endgroup$
    – shoonya
    Oct 10, 2012 at 11:16
  • $\begingroup$ I agree about the Butterfly thing, Here a,b,c for 1,-2,1. What I am asking is Would it work if we made a linear combination of options for premiums and delta and other greeks and check which greeks can be maximized by solving these equations. Maybe just take positions for maximizing Speed or Vomma or any other greek. $\endgroup$
    – shoonya
    Oct 10, 2012 at 11:22
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I would not go into the feasibility of construction for this type of structure since this has already been discussed.

Assuming this is done , your linear greeks could be zero at one point in time ( like a delta hedge portfolio at a specific time ) but these could drift as the prices and correlations move.

For higher order greeks , they would not be zero.

And yes , it's possible to expose yourself to just greeks without premium , example butterfly , risk reversals etc...

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