Braindumping a thread on how i think option traders could usefully think about realized and implied skew

Here's a toy model I introduce in trader and quant interviews, just to see how someone thinks, to start a discussion.

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Imagine the simplest discrete process: it just randomly goes 1,-1, -1, -1, 1 etc. The numbers of +/- are the same to keep the process flat in the long run. I can increase the variance by going to +2,-2 etc.

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Now, how about just moving the -1's to-2's, and leave the +1's alone. I'm going to need 2 times as many +1's for every -2, to compensate to keep the forward flat. -2, +1, +1, -2, +1, +1 etc. I can scale the whole thing down to keep the variance the same as the +1,-1 series

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So now I have a series that gaps down, and climbs back up just enough to stay flat, same variance as +1,-1. It's skewed. Its a nice simple little model, you can even safely set it up in Excel and check out its skew() and skew.p() functions

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Repeat the whole thing for n=3,4,5,6, what you're building is a series that shows larger and larger gaps down, and climbs steadily back up until another gap The mathematical skewness of the process is related nicely to n.

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Now imagine you own an option, discretely delta hedged. and you are simulating its value under these skewed paths.

Write down the Taylor series PnL for the 2 types of days: n lots of the +1's, 1 of the -n's.

How does the total PnL depend on n, including your delta?

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Go greek by greek, and go to third order (skew is the reason why third order greeks are ever documented, named)

Can you see that a downside option is worth more than an upside one, by an amount dependent on n?

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If so, you have a breakeven - what excess theta would you have to be compensated by to be short a downside option compared to an upside one?

Now you're talking implied skew versus realized, relative value trading, in quite an advanced way - all from a toy model

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