Toy Skew Model Thread

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.


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.


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


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


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.


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?


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?


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

9/9 end