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Strange


Total Posts: 1450
Joined: Jun 2004
 
Posted: 2018-05-16 14:28
Let's say I have an options database that uses incorrect forward for calculating implied vols (and does not force p/c parity). How bad of an approximation would be to take their calculated implied vols for each call/put with the same strike/expiration and take "corrected" IV as weighted average by calculated delta?

My logic is that there is wrong forward (W) and right forward (F), so the error in the implied vol would be roughtly delta*(W-F)/vega. Is that right or am I smoking crack?

I don't interest myself in 'why?'. I think more often in terms of 'when?'...sometimes 'where?'. And always how much?'

ronin


Total Posts: 361
Joined: May 2006
 
Posted: 2018-05-16 15:04
I would say


What you are showing would be the error in option price, but you are still missing on the other side. It's vol delta, not option delta.

"There is a SIX am?" -- Arthur

Strange


Total Posts: 1450
Joined: Jun 2004
 
Posted: 2018-05-16 16:26
Hmm, maybe I have not have coffee yet, here is my logic.

We are implying volatility from two options with a wrong forward, so we get the following:
C(f_r, sigma_r) = C(f_w, sigma_w_c)
P(f_r, sigma_r) = P(f_w, sigma_w_p)
delta_c = dC/dF
delta_p = dP/dF
vega = dP/d_sigma = dC/d_sigma

Making all sorts of wacky assumptions, we can say

delta_c * (f_r - f_w) = vega * (sigma_w_c - sigma_r)
delta_p * (f_r - f_w) = vega * (sigma_w_p - sigma_r)

and solving for sigma_r, we should get the delta-weighted average of the sigma_w, no?

I don't interest myself in 'why?'. I think more often in terms of 'when?'...sometimes 'where?'. And always how much?'

ronin


Total Posts: 361
Joined: May 2006
 
Posted: 2018-05-16 16:52

I've had too many coffees though... Aren't you mixing up BS delta and option delta?

option delta = BS delta + vega * d vol/dF


"There is a SIX am?" -- Arthur

Strange


Total Posts: 1450
Joined: Jun 2004
 
Posted: 2018-05-16 17:06
LOL, that would be too advanced for the type of approximation I am trying to make.

I am literally acting as if the only two options in my world are a call and a put, with the same maturity and the same strike. So my delta would be simple BS delta and my vega would be simple BS vega. That's what the database gives me and both calculated with incorrect implied vols and incorrect forward prices (so I have 2 wrong deltas and 2 wrong vegas). Which means that whatever noise comes from the difference between BS delta and some smart delta (i.e. the second term in your equation) should be fairly minor compare to the other "assumptions".

PS. It's pretty annoying that for the money we are paying them, implied vols are total shit

I don't interest myself in 'why?'. I think more often in terms of 'when?'...sometimes 'where?'. And always how much?'

ronin


Total Posts: 361
Joined: May 2006
 
Posted: 2018-05-16 17:17

TBH, it sounds like one of those where it's actually easier to do the full thing than the shortcut...



"There is a SIX am?" -- Arthur

Strange


Total Posts: 1450
Joined: Jun 2004
 
Posted: 2018-05-16 17:31
That is probably (and almost certainly) true, but calculating implied vols even for a several hundred names is a real project.

I don't interest myself in 'why?'. I think more often in terms of 'when?'...sometimes 'where?'. And always how much?'

ronin


Total Posts: 361
Joined: May 2006
 
Posted: 2018-05-17 11:07
I once looked at something vaguely similar - you had to recalc implied vols for lots of symbols for all strikes and maturities relatively quickly after a trading halt.

The solution was to just recalc some strikes and maturities, and interpolate the rest. E.g., if you just take every other strike and every other maturity, you have eliminated 75% of the calculation. And you really don't even need every other strike. So you can easily end up calculating just 1% or so of the original contracts.

The prob with your approach is that the skew term isn't some high order correction, it's order 1. You'd replace one error with another.

Having said that, some simple algebraic correction followed by arb removal will probably work. Just make sure it's the right algebraic correction...

"There is a SIX am?" -- Arthur

TakeItAndRun


Total Posts: 96
Joined: Apr 2010
 
Posted: 2018-05-17 11:35
Strange, I think your first-order approximation is right:


Strange


Total Posts: 1450
Joined: Jun 2004
 
Posted: 2018-05-18 05:31
So far I did exactly what I described above and the volatilities look OKish for the first pass. I'd love to quantify the actual error and maybe add some second order corrections.

I don't interest myself in 'why?'. I think more often in terms of 'when?'...sometimes 'where?'. And always how much?'

TakeItAndRun


Total Posts: 96
Joined: Apr 2010
 
Posted: 2018-05-21 09:50
From what I have written, one can can compute the second-order approximation. For the first-order term we have:

So we assumed:
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