Forward pricing implied volatility

Cadavre

Junior member
41 0
Been working for the last year or so on forward P/L models for combo options. This is a forward priced P/L for a single straddle:


The dividend yield and interest rates used now match the stock price and the time remaining to maturity. Since this example is modeled for the same option on the same day, the rate is the same but the dividend is adjusted for the stock price to caculate the forward spot for the option.

First thing needed was more accurate forward price modeling for a single option. Began with intra-day prices. Started with 3 chains downloaded on the same market day for a SPY 105 strike. Notice in image below the model (at the top of the image below) calculated the forward spots for the options when S=110.09 @ 2.22 and for S=110.14 @ 2.21 while the actual market prices are 2.18 and 2.14.


Not bad but still no cigar cause the combo P/Ls to be modeled will allow for multiple contracts and up to five options. The problem is the current model is using the same IV (.4476) for all the (18 day put) options.

Since chain IVs are priced to the Strike and model prices to the Stock, an approach was needed to make the strike IV on the chain being used to model the forward prices close to the IV on the actual market chains with the prices the model was being tested to. Using the closed form pricing constants where S = Stock Price and X = Strike price and whole lot of hacks and failures, finally landed on the solution (called SEX and EXCESS :devilish: ) below:


The image below show the results stepping (in pennies each of the three market chains (S=109.43, 110.09 and 110.14) the 105 put from closer to the money, S=109.43, to "furthest" from the money, S=110.14. The 110.09 prices both directions. The S=109.43 prices (only) away from the money/ The S=110.09 prices both directions. And, the S=110.14 prices (only) towards the money:


What surprised most was the the IV's, although they are heading in the right direction, are as much as 20 cents off the market IVs, BUT, the prices calculated with the forward IV's are much closer to the market prices than the "current" model using a static IV.

BTW, I do have a "kludge" that will bring the forward IV's much closer to the actual IVs. Also, since these are "free" chain options, the prices for the options might be priced to a different market price for the stock.

Someone told me I had found "sticky delta". Is anyone aware of published documents that might offer a better understanding of the processes involved in addressing this issue.

Any advice appreciated. Thanks!
 

Martinghoul

Senior member
2,690 276
Yeah, there's a lot of stuff written on "sticky delta" vs other "sticky" stuff. I think I have a few Derman and/or Dupire papers on these subjects.

Looks like you've done some nice work, but let me read in more detail.
 

Cadavre

Junior member
41 0
... but let me read in more detail.
Thanks Martin (great last name for Halloween! :wicked:). There were a few problems with the dat, but I don't know how critical it is. I have since corrected what I could but the presentation was developed before the corrections that could be made were made.

The one I could not correct for was the SPY dividend. The chain parser grabs a real time dividend amount from Yahoo when the chain is parsed. I think the dividend SPY was paying was different from the dividend SPY was paying when the market chain was grabbed.

A cubic spline using LIBOR term rates determines the "rate" to calculate the IV. At the time this presentation was built the shortest term was one month. The over night LIBOR is now used for the shortest rate term.

When the downloaded chain is parsed into the chain viewer, the IV is not rounded. I took out the rounding feature because it reduced from 6 seconds to 4 seconds the amount of time the chain parser requires to calculate 2400 IVs when all 1200 (or so) of the the SPY "composite" straddles are parsed to the viewer. The proper IV for the 105 @ 109.43 is 44.76, not 44.75.

Am working on a presentation that uses a cubic spine to develop the term IV's so that the IVs can be priced through to expiration. Am still considering wither or not to use a spline to approximate IV's for speed (as opposed to a slope intercept used to develop this presentation).

The method shown has be tested with up to 12 chains during a single market with deeper price spreads and it works out pretty much within the same tolerances the 3 (stock) price.

Note the Y1 and X1 values are the "SEX" and IV closest to the strike. So if the "SEX" for forward (stock) price for the 105 is at the 105 or 104 (or 103 etc) strike, Y1 will be less than Y2, and X1 will be greater than X2. Y1 > Y2, and X1 < X2, should XES point to a strike higher than 105.

Again, thanks.
 

Cadavre

Junior member
41 0
A 109.43 SPY 105 PUT IS ESSENTIALLY the SAME option that a 110.09 SPY 105.63 PUT (maybe anyway?).

Using a synthetic option (if a synthetic strike is, indeed, a hallmark of one of those critters) to pick apart the method offered in the initial post (why does the cursor behave as thought i am keying into an online chat box?).

Used market option closest to money SPY (109.43 / IV = 44.75% / X=105) @ market and ran some charts and second derivatives. Then repriced using 110.09, adjusted the cost to carry at the same IV as the 109.43 (44.75%) and substitutes the XES (excess) value used to forward price the 109.43 105 to 110.09.

Then ran the same vega chart and D2's. Vega is supposed to be LEG agnostic (whatever that means). The only significant (metric) change bigger than a basis point was the "Driftless Theta" second derivative (and no - I have no idea what that guy is or how he is used).

One reason for this exercise was to expose holes in the proposal that started this thread. The other was to look for an "indicator" that could be used to trigger a "kludge" (the method used to calculate the Y intercept (which is the forward IV). After some thought and a recognition that the chains used are totally retail and the likely hood that the S=110.09 option price was a stale quote (The option's "last" had no relation to the CBOE's SPY "last"). Maybe a "stale quote spotter agent"?

Anyway here is the graphic. Should anyone see anything to help with a better understanding of whats going on here based on the relatively similar metrics the synthetic reported with the market metrics - please chime in:

Is it Aretha or is it Memorex? Most of the metrics in the chart legend are used by some kind of "hedge ratio" pricing tool. The reason the legend's "ASK" is "struck" is because chart was called by a calculator instead of a market chain.

Still shopping for feed back and crossing all the eyes. :cry:

Have the forward (through expiration) IV spline working - if you guys ain't too bored with this stuff I will show as much as I can when I find time to "formalize" it.

Thanks

Edit: Just noticed the second derivative, "DVegaDTime" (dunno nuthin `bout him/her) - read or heard close to expiration FX trades "might" use that guy) report the most obvious variance (are you thinking "so what" like I am?).
 
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Cadavre

Junior member
41 0
DVegaDTime refers to the change in vega in respect to changes in time. you can read more about it in this article: 26254888 Know Your Weapon 2
Bookmarked thanks.

DVegaDTime = Delta Bleed
Driftless Theta = Pure Bleed

Snag of DVDT and DT for the 109.43 / 105 PUT 18 Day:

They look like different "sexes" of the same species.

One thing for sure Reading this stuff is making my eyes bleed!

Both those guys are functions of the clock - interesting cause one issue in the closet is spline-ing IV forward to expiration.

When the option modeled forward is the oldest in the chain (next in line to expire) with more than two weeks of life left, the theta spline has some inertia, but not enough to lift the IV to the level of the same strike with only 3 days left. For example, the 18 day 109.43 105 SPY Put IV is 44.75%(Vol=43,119 A, Open Int=15,919). The weekly (3 days left) for the same strike is 47.71% (Vol=172,265 A, Open Int=306,579).

Using Cubic Spline when the 3 day is not included tails up the IV to ~49% at the 3 day in price point.

One paper I read used a spline other than Cubic and it relied on several market days of data to produce an IV surface.

Looking for a spline with enhanced acceleration at the tail.

Goal is to build the surface from the data in the chain being parsed.


This baits the question: How precise need the model be in a market where counterfeiting, and not "value" are the only thing pumping the indexes? So how accurate does it need to be? Trade direction now follows FOREX. Value through Innovation has been off-shored (now i got to rant)

BEGIN RANT - please forgive :( :
The swings in the market are all dollar carry inversions - it's not like BOA lucked into a great investment or Apple invented an iPod customers could have sex with :devilish:. We're in an oil glut but the market is pricing the stuff like we're only drops away from darkness.

What is really happening is the fed (and othe CBs) are diluting the cagatha out of their respective sovereign tokens. My royalty checks are the same as they were before - but they only buy half what they used to! They have even figured out hout to use the consequence of USD printing to fudge the inflation number - if you won a house where the principal balance is less than market value - the loss of principal is computed as a discount (WHY DIDN"T I THINK OF THAT).
END RANT:

Is Emanuel Derman a T2W contributor? :LOL:
 

Cadavre

Junior member
41 0
Interesting that both Vega and Theta are said to be leg agnostic, setting aside PUT vs CALL IV and that their second cousins (DvegaDtime {vega bleed) and Driftless Theta {pure bleed}) showed the most significant variance regarding strike stickyness (SEX/XES)!
 

Cadavre

Junior member
41 0
DVegaDTime refers to the change in vega in respect to changes in time. you can read more about it in this article: 26254888 Know Your Weapon 2
This may sound retentive. Read (or tried to read) a bunch of forward IV approaches. Some used delta, some used money-ness. Don't those metrics require an IV to calculate?

Pretty hard to do when the IV is the "quest" :)

The pricing model is a matrix with columns being a range of possible stock prices and rows can be any day from the moment the model was created through the the expiration date of the option being modeled.

The model currently relies on the chain's strike IV to produce forward spots. Ex: Assume on Dec 2, 2011 SPY has a trade at 106.66. Without a forward IV, the model price will be close, but not on the market price. Close is ok for single contracts - but spreads/combos and multi contracts can't be expected to suffer the "horseshoe" spot modeled with a "flat" IV.

So - (Question 1) - how can a metric that (should) require a reliable (forward) IV be used to thrash (model) forward IV spots?

Ex: You car's gas tank is empty. Your solution is to drive it to a gas station to gas up so you can start the engine? (i guess Buckaroo Bonsai could make sense of that) - I can't :)

Should the forward IV spots tests out (this is the question), then the matrices presenting greeks, that also suffer the limitations of the chain's IV (delta theta vega skew smile) would need to subscribe to the forward IV spots. ie - the forward iv matrix would supply an IV for each cell in the greek matrices.

IOW the forward greeks would calculated with forward iv's, instead of the "singular" flat strike/expiration iv the current model uses.

Those forward greek spots, calculated with forward IV spots, would not look like as they do when hashed with the (singular) static IV. Right?
 

Cadavre

Junior member
41 0
Interesting that both Vega and Theta are said to be leg agnostic ...
:oops:
We are allowed our Perry moments, but not every 5 seconds - apologies, i think its the vanilla Gama sharing Vega's "leg agnostic" status (leg iv's?). There is a (separate) theta for each side of the straddle.