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saying you do equally well with the Call-Hedge combo as with a Put-Hedge combo regardless of how the stock moves. Let's say your Call-Hedge Combo has you all-in with 100 total shares sold at say $80/sh and your Put-Hedge combo has you all-in with 100 total shares bought at $40/sh. Due to put-call parity you paid the same price for them. Now suppose due to the extreme underlying upward "Trend" in the stock - assuming there could be such a thing - the stock triples in price straight to $180/sh. Your Call-Hedge strategy has made you something like $10/sh on 50 shares. Your Put-Hedge strategy has made you what? As the stock moves up you gradually sell your 50 shares until it hits $80/sh and you make something like $10/sh on 50 shares.


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A. long the 60 call for $1, short 50 shares at $60.
let's do this as discrete one time period to expiration model.

1. stock goes to $80. sell 50 shares. net sold 100 shares at average price of $70. exercise call to buy 100 shares for $60. all positions closed. profit = $20 on 50 shares and $0 on 50 shares, minus $1 on call contract.

2. stock goes to $40. buy 50 shares. net zero stock position (sold 50, bought 50). call expires worthles. so no position. profit = $20 on 50 shares, minus $1 on call contract.

B. long the 60 put for $1, long 50 shares at $60.

1. stock goes to $80. sell 50 shares. net zero stock position (bought 50, sold 50). put expires worthless. so no position. profit = $20 on 50 shares, minus $1 on put contract.

2. stock goes to $40. buy 50 shares. net bought 100 shares at average price of $50. exercise put option to sell 100 shares at $60. all positions closed. profit = $20 on 50 shares and $0 on 50 shares, minus $1 on put contract.

now you said stock goes to $180. we add one time period. it matters whether it went to $180 from $40 or from $80. consider each of the four scenarios above as a starting point.

A.1. stock goes from $80 to $180. you have sold 100 shares (but you haven't exercised your call yet and it hasn't expired because there's a time period left). your position is neutral. profit/loss is zero.

B.1. stock goes from $80 to $180. you sold the 50 shares you bought already. your stock position is zero. your put is way out of the money. your position is neutral. profit/loss is zero.

A.2. stock goes from $40 to $180. you had zero stock at $40 because you bought back your short 50 shares. now on this gap move to $180 you will sell 100 shares at $180. your call expires and is exercised so you cover your short 100 shares buying for $60. you make $120 on 100 shares in addition to the scalp you made in time period one.

B.2. stock $40 to $180. you were long 100 shares total at $40 against your puts. on this gap move to $180 you will sell those 100 shares at $180. you have zero stock position and your put expires worthless, out of the money. you make $120 on 100 shares in addition to the scalp you made in the first time period.


Here's how it works from the trader's perspective, analytically. you buy a call or a put with a given strike and expiration. let's say the 60 strike again, but this time stock is at $70 when you do the trade. you look at your sheets to get the delta of the option. the call delta is 65. the put delta is 35.

A. buy ITM call for $11 ($10 parity plus $1 intrinsic/time value). sell 65 shares. now you are delta neutral. you have 35 shares to sell as stock goes up and 65 to buy as stock goes down.

B. buy OTM put for $1 ($0 parity, plus same $1 intrinsic/time value). buy 35 shares. now you are delta neutral. you have 35 shares to sell as stock goes up and 65 to buy as stock goes down.

the next morning stock is up $5, to $75. you look at your sheets to see what your new delta is. the call and the put had the same gamma, let's say 2 (deltas per $1 of stock move). so in both cases your delta on your sheets is now long 10. so you say, I better sell 10 shares.

A. you now have sold a total of 75 shares. you have 25 more to sell as stock goes up and 75 to buy as it goes down. you are delta neutral again. and your gamma is now a less (stock farther from strike). your gamma is like 1.5 now.

B. you bought 35 shares before and now you sold back 10. you have 25 more to sell as stock goes up and 75 to buy as it goes down. delta neutral. gamma 1.5.

both are the same. the gamma (convexity) ratchets your delta up or down as stock moves. that signals you to trade stock and improve your average price or scalp. as stock moves away from strike you run out of gamma, so run out of stock to trade and end up fully hedged--and then you hope that stock reverses and goes back to strike so you get gamma again and can scalp again.

graphically you know how a call or a put is a curve inside a hockey stick (laid flat on it's shaft). when you trade stock you're applying a linear transformation to that graph which means that you rotate it. so whichever way you started--the call was pointing to the right, the put was pointing to the left--after the stock hedge the graphs have rotated to be exactly the same bowl. delta neutral means the level point on the bowl is where stock is. and the graph curves up in either direction. that curve is your gamma. the curve up is also your profit as stock moves. as stock moves up the bowl, that means you are getting a delta. so you trade stock to get neutral and rotate the bowl flat again, so it's symmetric. the stock trades that rotate the bowl flat lock in little bits of profit by either improving your average price as you compelete your hedge or scalping as you take off your hedge.