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Smile Modeling; What Is SKEW?
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# Smile Modeling; What Is SKEW?

April 27, 2012

I am sending this out, but will not be able to respond to questions till Monday as I am on an annual “family” businessÂ trip this weekend.

This post will be a bit dry, but hopefully interesting to those who trade options.Â  This may sound educational, but hopefully it encourages some deep thought. To be blunt- all of this- is so I can say I have a vega of 3 today and a vega of 4.5 in two weeks. 99% of traders cannot do that (or they think they can but their math is wrong). Yet the guy on the other end of the trade (market maker) clearly knows about smiles, convexity, or stochastic volatility. This is why call rape (patent pending), skew rape (patent pending), and retail IV crushes (patent pending) all occur. If you believe it or not, as tricky as the SPX is concerning IV’s inverse correlation to price and skew’s positive correlation to price, I have seen more blowups in Corn futures when novice traders do not fully evaluate IV and IV smiles.

First off, what is an IV smile? Smiles are the shape of the IV for a particular contract at 1 point in time. Each option has its own IV. I use a simple polynomial regression (eg beta means smile width, skew means direction, IV means the smile intercept). Its a “quadratic” equation, but also a simple “right click, add trendline, click polynomial, order 3” in excel. OrÂ  {=linest(YY,XX^{1,2}) }if you prefer.

Why does this matter? Well first off, it is hard to value options without modeling the smile- let alone evaluate risks in the form of using greeks. If you think you have a -50 delta, and the market moves south, you may or may not have a -1 delta at a lower price. That depends on how much theta or vega you have on your book. In any case, you have to understand what is moving your P&L- otherwise you might wake up one morning being completely correct on market direction but wrong on vol.Â  Corn futures do this often.

Without EVIL ado let’s dig in to the data.

Here is what a corn IV smile looks like (recently).

Here is the SPX (a while back if I remember).

Clearly the Emini looks completely opposite of Corn. Ignore the corn prices on the bottom of this chart, I applied the SPX smile to corn prices to compare the smile “apples to apples.”

What does this mean?

Well, in general, the IV smile tends to reflect market distribution. Distribution meaning probability of up versus down (averages mean little if you have read any of my previous posts). BSM (Black Scholes Merton) option pricing is based on lognormal (gaussian, normal etal) distribution. Normal meaning the up deviation and down deviation are equal. However, we know that the S&P has a bias to grind higher and flush lower; therefore, it tends to have a negative skew. Or the deviation from average has a negativeÂ tenancyÂ (bias). To adjust for this real world problem, option pricing models “charge” a higher IV for OTM puts to compensate for real distribution. Many claim it is due to lack of liquidity- I do not really agree as options are priced at both the BID and the ASK and there no way the EMINI puts are illiquid and Corn puts are not inasmuch, relatively speaking. So it is more likely the smile reflects market distribution and not lowÂ liquidity Â IMO.

Here is where this gets complicated. If Â smile reflects historical distribution, then historical data should Jive with option smiles.

Here is SPX/EMINI historical distribution (since 1990 i believe).

You can see it clearly does have a negative skew. So I presume this makes sense with the smile – which is negative (above).

However, corn is different.

Here is CORN since 1957:

It has a negative skew, yet the smile is currently (+)???

Now here is Corn over the past few years- just to confirm that there is no positive skew.

The skew here is 0, but yet not positive like the smile suggests? What is going on? Is theory fubar or is something else going on?

Well let’s look at historical IV skews for corn (past 3 years daily).

This tells me it generally has a positive skew.. which does not jive with historical real price distribution. Clearly it stays well above 0 until about 25 days till expiration.

Ok so what then? WTF is going on? SPX makes sense according to historic data but corn does not? If one just blindly traded on theory they (1) would have sold risk reversals on corn to arbitrage the skew and (2) probably lost money.

I think there are a few explanations.

(1) perception of a bullish bias for corn outweighs historical reality. I believe this, but I would also say China sells puts on grains since 2008 as well- which will in turn changes the smile. Both are “fundamental shifts” to the price of grain that I believe should be considered. I would like to believe this simpleÂ answer, but myÂ mathematicallyÂ inclined head says something else is different.

(2) The SABR (stochastic alpha beta rho) volatility model should be considered.

The SABR model was a concept originally published in Wilmott finance to adjust smiles for skews, correlation of skew to price, and vol of vol. It models fixed income very well, but I do think the “theory” does help to explain option IV skews and IV “contangos.”

Ok I know that sounds complicated, but lets discuss the “basic” concept.

Information is abundant at Wilmott or if you want the code you can go here.

The “basic” concept is that option prices should price in the RISK of future price movements (eg distributions). However, options truly must price in future OPTION valuation distributions- this includes IV curves. Remember an option’s value before expiration is dependent upon more than the price of the underlying asset. Consequently, if price (SPX speaking here) is to rise one must mathematically compensate for IV to drop and the SKEW to rise. So one cannot just model something linear as up moves decrease IV and increase skew, while down moves increase IV but decrease skew. So the present value of price paths for an option is not the same for up and down due to inherent smile correlations.

I am sure that soundsÂ over complicated, bear with me…

We know from many posts here that SKEW moves with price and IV (eg VIX) is inversely correlated to SPX movements. Many claim this to be the black swan index (including myself). This is mostly likely the case- however- I do believe there is more to this story. Otherwise live cattle futures would not have a negative skew, corn a positive, and SPX negative when the underlying distributions are ratherÂ similar. Go look at JUL corn options versus DEC corn options. Notice from a fundamental standpoint DEC has MORE upside risk than JUL given the former \$105 new crop old crop spread. Furthermore, old crop is planted- soÂ theoreticallyÂ there is no weather risk like DEC (new crop, just being planted). YET JUL has a larger (+) skew than DEC….

Here is SPX versus a skew. Skew moves with price for the most part.

In practicality, what does this mean? Sure higher prices mean more risk, but Â why then does the VIX not rise with SKEW? Again there are holes in both the black swan and the VIX sentimentÂ argument. The reason is “arbitrage.” If vol does not equal IV gamma or option selling should make money.

So I am attempting to argue that SKEW reflects more than simple “downside distribution,” it affects theoretical correlation of IV to price (according to SABR theory). Just think when the VIX hit 80 or so the SKEW went to 0 because (1) there is obviously little downside risk after an epic collapse, but also consider what IV at 80% was pricing in? Risk was priced in, and SKEW was low. You can see my line of thought…. IV may reflect complacency, but the SKEW is an adjustment for option selling risk.Â FurthermoreÂ in 2008 the panic priced in high volatility, but one would ALSO expect the correlation of IV to price to weaken (say go near -40 as opposed to -99). Why? Well there was a large risk that prices could fall and IV not rise much above 80. Both the VIX:VXV ratio and actual resultsÂ confirmedÂ this in MAR of 2009. The VIX did not make new highs when the SPX made new lows- vol was 30% not 80%.

This further supports that skews and time spreads (VIX:VXV) also likely are affected by other variables such as actual volatility and correlation of IV to price.

As for corn, we know it tends to explode up and down along with volatility… but the correlation is mostly positive with IV; hence, the IV smile is slightly positive- especially when there is a break in price. So corn and the S&P may actually have similar downside tail distributions, but the smiles are different due to the different inherent correlations of IV to price.

Now another thing that many confuse is the VIX:VXV ratio (IV time spread). Yes it reflects market makers and sentiment, but it also is an adjustment for theoretical mean reversion of volatility (just like the lower low in SPX and lower high in VIX comment). Â Ever wonder why it is so seasonal? Or why it follows price? Volatility dries up around holidays, so 1M vol drops and 3M vol rises (relatively speaking).Â  Ok so that explains the holidays, but why is the IV time-spread correlated to price?

Here is my best explanation (using SABR theory). (1) the market grinds higher and many know that can’t last forever so they bid up back-dated IV (just like skew). So volatility is low now, but common sense tells us the risk of it rising is high; therefore, the IV spread widens as volatility drops. Actual volatility forces the VIX lower, but estimated mean reversion volatility stays higher in the VXV.

This also explains why both corn and spx smiles go crazy into expiration. Here how the smile changes over time for corn.

Here is what the SPX looks like. You can see that time affects the smile. Moneyness means “% in the money.” Given that prices are “lognormal” we us ln(strike/current price); this is also the first part of the BSM pricing equation.

Here is another cool view.

Why does the smile change? The reasoning is the underlying properties of Â the smile change.

Volatility of Volatility is the primary reason for this change. Or volatility of the VIX, “volvol.” If volvol is low the spread remains high- or in English if SPX is grinding higher volvol drops and the VIX goes discount to VXV.

Here is some empirical evidence supportive of SABR theory of volvol.

Here you can see the correlation. VOLVOL affects time spreads for the most part. So this explains much IMO. IV smiles must account for correlations of IV to price- which also happens to account for SKEWS at the same time.

Now for the final question I often get? ” Why in general does the SPX VIX trade discount to VXV and yet Corn IV structures are the opposite?” E.g. why is DEC corn IV less than JUL corn IV? Most commods are this way just FYI.

This is complicated but the SABR does have some theoretical explanations. First off spot commodity months are more volatile than deferred commodity contracts. That is price discovery at its best, more information comes available for JUL relative to DEC; thus, more volatility for nearby months.. Very simple explanation and reasoning. As for the SPX DEC is fixed to SEP based on an arbitrage-able dividend yield. Corn spreads are a function of supply and demand differentials.

This chart is average IV (of say the smile), not ATM IV for DEC corn. ATM IV is actually much more range-bound (as previous charts above show). You can also see the crop reports here as well- notice how IV is bid up before the report, before volatility rises- no free lunch here!

BOTTOM LINE: My point is that different smiles occur on different commodities and different time horizons based upon different underlying correlations in conjunction with historical price distributions. So when you look at the MMR stuff, you now know what is really going on. This means for you rats attempting to buy OTM puts on the SPX betting on a IV spike and a price drop will not get the return you desire. You Â maybe better off buying vix futures or selling the S&P due to “smile risk.”

So there may be simple answers to the smile questions, but I do think there is quantifiable evidence for other influences.

For those of you wondering if I use SABR to model my stuff- no- I am an old fashioned modified (creative) cubic spline model guy. I just think there is evidence of the SABR theory when one steps back and compares corn to the S&P.

Also, on a side note, check out the CME data-suite. It is FREE (once you sign up) and they give live quotes for options and some futures. Here is what the calender corn spread options look like. Cool stuff if you like being creative.

Best of luck unbiased (and now creative) trading,

-Volar

Mole
Mole created Evil Speculator amidst the chaos of the financial crisis in early August of 2008. His vision for Evil Speculator is a refuge of reason, hands-on trading knowledge, and inspiration for traders of all ages and stripes. You can follow him and his nefarious schemes at various social media waterholes below.

• http://pulse.yahoo.com/_IF7HINHIP5ITYT6TDHACMWJUXQ Huisok

ouch! I got a Quant ice cream headache…. Â

volar you should change your aviatar to thisÂ

• http://practicalt.blogspot.com/ Darth_Gerb

these firehose posts are tough.
I’ll keep trying.

• Anonymous

Very nice analysis Volar. Although I do not trade or even think this deep its very interesting and I love to learn something new. I will research more into this just to pick my brain. Thanks buddy for the insight.

• Anonymous

http://content.screencast.com/users/AMCabrera/folders/Jing/media/439cd0d9-d038-4d65-b2a6-795b9319f5ce/2012-04-27_1957.png
http://content.screencast.com/users/AMCabrera/folders/Jing/media/0021e488-5804-4471-8598-4fbecb272452/2012-04-27_2001.png
http://content.screencast.com/users/AMCabrera/folders/Jing/media/b6501654-e415-4d77-9a4b-27eaae07cd89/2012-04-27_2009.png
Well I was waiting for this. Volar post=AMC massively profits. Obviously Im sort of kidding around but the correlation is pretty strange. ðŸ˜‰ Thank you very much Volar for the very good post.
Once my AUD/USD gets popped (1.0480) I will stick the new sell order at 1.0400. I’m beginning to eye the EUR/JPY which it turns out would have been the BEST option to have kept this week (damnit) went short at 107.20
“Yea but I really dont feel comfortable getting near my 50 pip loss area. SO what Ill do is take off the EUR/JPY. Then I can tolerate 50 more pips for the AUD/USD.” Dumbest week I’ve had and extremely embarrassing.Â
Anyhow, I still have that highly anticipating sell order at 1.6300 for GBP/USD but I will not tolerate no more than 20 pip loss . Â If it does not work out then I’ll wait at 1.6400 and I have a half lot at 1.6200 not so anticipating. Â Looking at the chart for the EUR/JPY I will stick a sell order at 107(full lot) and one at 106(half lot) for a sucker play. To sum up it seems I had the beating I needed to keep discipline. LOng strong EVIL!

• Anonymous

Great work. Will take a few readings to grasp. Before reading, I was looking at the weekly squeeze possibility supporting a spike in volatility.

http://stockcharts.com/h-sc/ui?s=\$CORN&p=W&b=5&g=0&id=p51449336855Â

• http://evilspeculator.com molecool

Volar – you just blew up my mobile phone!

• Anonymous

Great post mate, I BOW DOWN to the master ðŸ™‚

• Anonymous

Volar any chance you can make your excel file available for us?

• Anonymous

slow start to a Sunday night. Â

• Anonymous

Â i thought markets were closed!

• Anonymous

They are open. Es went to 1402ish and currently at 1400.75

• http://practicalt.blogspot.com/ Darth_Gerb

if you haven’t shorted in 4 days, I would think it’s safer now.. than then. Â he he.
http://postimage.org/image/nzswp61gf/Â
-DG

• Anonymous

What’s wrong with these people? Retail is even more short HG than last week when the were short the most in 5 years.

• http://evilspeculator.com molecool

Did everyone die? Well, the eagle has landed in Madrid – all is well.

• Anonymous

We alive. Just kicking back. Glad you had a safe flight buddy.Â

• Joe_Jones

Glad to hear it. No baby crying this time?

• http://practicalt.blogspot.com/ Darth_Gerb
• http://practicalt.blogspot.com/ Darth_Gerb

AUD/JPY – 3rd touch is a charm right?
Play Ball!
http://postimage.org/image/stgffsgxv/Â

• Schwerepunkt

Seems wrong . . .Â

• Anonymous

LOL

• Anonymous

http://content.screencast.com/users/AMCabrera/folders/Jing/media/7d30f4d2-36de-4116-9f4d-5f40f2812a6c/2012-04-30_0928.png
Penn & Teller show worked out as usual and I was please to see I was not popped at 1.0480. Glad to hear our Kaiser landed safely.

• volar

Â depends on what you need, but yes

• volar

Â lol thanks

• Anonymous

Bearfood for thought

• Anonymous
• Anonymous

http://content.screencast.com/users/AMCabrera/folders/Jing/media/190c1309-b133-44d2-9f9c-76cb7b5388d6/2012-04-30_1051.png
I find this intriguing. Maybe the buying pressure starts and selling pressure is lifted off for some time.

• Anonymous

Waiting for this div on the 5 min to evolve @ 1390:)

• Anonymous

http://content.screencast.com/users/AMCabrera/folders/Jing/media/1094533e-432f-49d8-84b2-facb068cfcdd/2012-04-30_1052.png
be aware of the potential sucker play to the downside, however, I will buy for sure at .9770

• http://iberianviews.blogspot.com/ catracho

AUDJPY Â no longerÂ riskÂ on risk off? Â Or diverging…

• Anonymous

http://content.screencast.com/users/AMCabrera/folders/Jing/media/ceb8c058-dc86-4bdf-a37c-cddf7befb01f/2012-04-30_1056.png
Sometimes technical indicators or whatever else distorts the obvious play for the longer term trade. Looking at the long to short ratio is what has had me always suspicious of AUD/JPY ever since it was above 85.00.Â

• Anonymous

I’m rubbing my hands watching this…notice the potential trendline of support on the ZL too

• Anonymous

apple losing some wax…vix up nicely

• Anonymous

Oops missed the boat…

• Anonymous

dropping stop-loss on EUR/JPY to b/e and on AUD/USD i refuse to give this market anymore of my money.

• Anonymous

LOL! Me too. Will wait to short at the gap 1400 or so.

• http://iberianviews.blogspot.com/ catracho

http://www.cftc.gov/dea/futures/deacmelf.htm

if reading correct it looks like commercials have reversed from long to short GBP in last week

• http://iberianviews.blogspot.com/ catracho

RBA rate decision 1st May…

• Anonymous

Yea, it could be nice. It kind of fits all together when you look at the move Friday. Especially if a 50bps rate cut is sprung loss. That could cause a nice drop back to 1.0200 for the AUD/USD.

• http://practicalt.blogspot.com/ Darth_Gerb

ps. the boat ride ends badly.
finally rented the movie over the weekend.

http://media.photobucket.com/image/recent/ppcccaps/wotw9.jpgÂ

• Schwerepunkt

USD.JPY just keeps dropping. That Yen won’t weaken no matter what GOJ does. [anyone have a low target for this pair?]

• Anonymous

Ok, whats the state of play (been in meetings).
Weak move to downside so far today, or is it a slippery slope?

• Schwerepunkt

Just watch out for the fresh dog shit. On a slope that could be deadly.

Seriously, though, \$TICK shows a down trend day to me, BWDIK? Â I’m a leech.Â

• http://practicalt.blogspot.com/ Darth_Gerb

got the turn, but it’s Month End. Â trust No-thing.
http://s11.postimage.org/7i4jc5m8z/temp1.png

• Anonymous

Â ðŸ™‚
I have taken a small long posn, but it could go either way, i suppose
\$TICK (barchart.com), seems to vascilate about null-hypothesis?

• Schwerepunkt

\$TICK was mostly down from during 9:30-10:15 period. That “usually” means a trend day. But, it did spike a bit right at 10:15 muddying the waters.

• Anonymous

EOM window dressing today, should br ripe for short tomorrow:)

• Anonymous

Â Seems like a clear T, but, as CS says, let price lead the way!

• Anonymous

NG shook off the thurs report blues…bought OXGN at .98 and awaiting repeat of last year…caught in between 50sma and 180sma…DSCO 2 FDA approved drugs..cleared 50 today, leaving 180&360 in the dust..tight stops on all…its bio!

• Anonymous

Moles trend alert suggested otherwise. Time in minutes is what most of us get for TICK but if you dissect to ticks or seconds, it makes a big difference which is why I don’t trust reading TICK on my own cause I don’t have tick data or second data in my feed.

• Anonymous

Triangle formation on the ZL…i’m guessing since its end of mth, this is probably gonna resolve upwards but atm the downward slope is steeper

• Kudos

Market opens down and stays down most of the day while yields on peripheral europe (spanish and italian 10yr notes) fall 10 basis points and close at low of day? Suspect to me…. Copper now positive on the day. Now even more suspect. These together with slight divergence = (fill in the blank)

• Anonymous

Noticed that too.

• Anonymous

and there she goes…..bam….

• Anonymous

The 5minZL triangle just never quits…and will probably flatline any minute now….hate that stuff…especially if it flatlines into the close

• Anonymous

Well…you know what happens, trend days or at least easier to trade days usually come after a few of these paint dry sessions. We had 2 trend days last week to potentially bank major coin…i guess we just will have to wait for this one to stagnate a few days.

• http://evilspeculator.com molecool

Have no fear – evil Mole is here. Let’s get that comment stream going while I finish my post though!!

• http://practicalt.blogspot.com/ Darth_Gerb

impossible captain! – Raised by Wolves is offline.
ðŸ˜‰

http://s15.postimage.org/pyo72xkaz/temp2.png

Silver, Hot or Not?

• Anonymous

Ah, took a long here @ 1390. Time for the Marines!

• http://practicalt.blogspot.com/ Darth_Gerb
• http://practicalt.blogspot.com/ Darth_Gerb

Incoming!!
(2m candle /ES)

Boom!

• Schwerepunkt

chuka-luka.

• Joe_Jones

My trading account absolutely loves it when you are away from the lair…
LOL!
;o)

• Anonymous
• Anonymous

PWEW……PWEW…….PWEW! (Skynard ducks)

• Anonymous

The EUR/JPY has already been taken. I took the 50 pips. Resetting the sell order in EUR/JPY back at 106.00. Taking the very little profit from the AUD/USD. I still have the sell order at 1.0400. Nice little profits considering what was shaping up last friday.

• Anonymous

Paging Drew. Hey check out the long/short ratio for the USD/CAD that is a pretty good chunk lopped off.

• http://evilspeculator.com molecool

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