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I'm confused on how to construct a long short portfolio using strictly options.

If I wanted to hedge stock, I would long stock A and short stock B by making sure that the quantity * price of A = quantity * price of B (excluding beta for simplicity). So total exposure long and short would be equal. Therefore if the price of both A and B increased by 1%, my portfolio wouldn't change in value.

How would I replicate this sort of hedge using a call option on A and a put option on B? I've looked into delta hedging, but that looks like it only works if one side of the hedge is using stock. Otherwise the option with the higher underlying stock price would move the portfolio value more given an equal percent price change in A and B.

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  • I'd buy a high delta put on one stock and a high delta call on the other in a ratio based on your performance analysis of the stocks. If simply based on price, use the price ratio of the two. It could get a lot more complicated if based on volatility or some other stat. FWIW, I've traded a lot of pairs and it was cleaner just using the stocks (better liquidity, narrower spreads, no time decay issue, etc.). Dec 10, 2021 at 21:44
  • @BobBaerker I was doing pairs trading using stock, but I wanted to get some more leverage. I'll probably look into the higher delta and longer expiry options in order to reduce the effects of the other greeks. Dec 10, 2021 at 22:33

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The relationship becomes: quantity * delta * price of A + quantity * delta * price of B = 0.

This reduces to the stock formula when we note that the delta of a long stock is +1 and the delta of a short stock is -1.

Just as with regular delta hedging, gamma (change of delta with price) means that an initially balanced hedge will gradually become unbalanced as the prices move.

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  • Maybe explain how many calls to buy and how many calls to sell, because "call option on A and a put option on B" doesn't work.
    – base64
    Dec 10, 2021 at 1:48
  • @base64 Why doesn't "call option on A and a put option on B" work? I mean, if the quantity of calls and the quantity of puts obey the formula I gave.
    – nanoman
    Dec 10, 2021 at 1:50
  • C + PV(K) = P + S. S = C – P + PV(K). Ignore PV(K). Goal: Sa - Sb = Ca - Pa - (Cb - Pb) = Ca - Pa - Cb + Pb.
    – base64
    Dec 10, 2021 at 1:57
  • Consider this. If the Current and Strike Price of both A and B stock were $100. Suddenly Price of A increased by $10. Did he make linear payoff of $10 profit by Ca - Pb, given that Pb option price didn't change?
    – base64
    Dec 10, 2021 at 2:14
  • @base64 The goal stated by OP is "if the price of both A and B increased by 1%, my portfolio wouldn't change in value." OP notes an analogy to delta hedging, which is only linear and neglects second-order (gamma) effects. In your example, Ca and Pb have equal and opposite deltas (0.5 and -0.5), so the change in Ca + Pb is proportional to the change in Sa - Sb, for small movements.
    – nanoman
    Dec 10, 2021 at 3:06

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