Paradox: IV of butterfly strategy

[mikhail]

I came across a kind of paradox while trying to calculate the IV of a butterfly strategy.

1) Let’s suppose that we create a 90 / 100 / 110 1-year butterfly using call options and that the IVs of call are 30% / 40% / 50%:

Link to strategy:

http://optioncreator.com/st7wkz0

2) When I tried to calculate the IV of the whole strategy it turned out to be 28% (approximately):

Link to strategy (with IVs set to 28%):

http://optioncreator.com/sti6fv4

You can see than in both cases the value of the butterfly strategy is 1,4 dollars.

The paradox is that the IV of the whole strategy (28%) is smaller than all the individual IVs of options used to build this strategy (30%, 40% and 50%).

Well, I understand that in the butterfly strategy some options were bought and others were sold, but still this may seem somewhat counterintuitive.

The big question is: does the IV of an option strategy make any sense at all? Should we calculate and use this figure?

 

[minmike]

I'm not sure why it matters. Please explain.

Also what markets are you looking at to get that IV curve? In general, puts have higher IV than calls, ATM have lower IV than wings. That curve never occurs in real life. Try using a real curve and check results.

 

[mikhail]

That curve never occurs in real life. Try using a real curve and check results.

The IV curve is somewhat theoretical, of course.

The figures were selected to make the example more striking.

The main question is: does it make sense to calculate the IV of the whole strategy? Or is IV a figure that has sense when applied to individual options only?

 

[OTM-Options]

The IV curve is somewhat theoretical, of course.

The figures were selected to make the example more striking.

The main question is: does it make sense to calculate the IV of the whole strategy? Or is IV a figure that has sense when applied to individual options only?

The main question should be: Do you think the underlying will be about $100.00 in 1 year?

Obviously you think that it will since you built the butterfly around that strike. So IV is a moot point.

 

[minmike]

The IV curve is somewhat theoretical, of course.

The figures were selected to make the example more striking.

The main question is: does it make sense to calculate the IV of the whole strategy? Or is IV a figure that has sense when applied to individual options only?

No it doesn't. You should know what a general increase/decrease in IV does and you should know how skew could affect your position. But actual IV calculations don't make sense.

 

[mikhail]

The main question should be: Do you think the underlying will be about $100.00 in 1 year?

As I already said, this is a theoretical example, with the values picked at random, just for the sake of demonstration.

The question to which I am trying to get an answer is this:

Does it make sense to calculate the IV of the whole strategy? Or is IV a figure that has sense when applied to individual options only?

 

[longthewings]

28% is absolutely a meaningful number. It is your breakeven vol. Assuming you are long the fly, if realized vol over the next year comes in below 28%, you should expect to make money....above 28%, you should expect a loss....At 28%, you expect a scratch.

This tells you an expectation (an average) only. It says nothing about P&L variance. That's where hedging comes in.

But to emphasize, yes....28% is worth knowing. It's a data point to compare to your vol forecast to see if the trade even remotely makes sense.

One last thing. Per your original post, there is no paradox here. Realized vol is realized vol. If you knew for sure that realized vol was going to be 15% over the next year, you'd price each option of the fly at 15% vol in your model in order to calculate the fair value of the fly. The skew exists because it is addressing market participants' views on the higher moments of their P&Ls, produced by stochastic vol, discrete hedging, jumps, etc. First moment/expectation is one thing (and does not depend on the skew). Skew is a higher moment phenomenon (the 3rd moment to be precise). The market is pricing at 1.40 because that's the market's view of the AVERAGE fair value of the fly. Which is why the two numbers coincide. No paradox.

Hope this helps.