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PrinceQnt
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American Implied Volatility from quotes

October 28th, 2020, 9:53 pm

Hi all, Big question. 

1)What is a fast, reliable and market consistent methodology for calculating IV->Smile->Surface for American style, single stocks names, that have dividends, from snap-shot quote data, (1-20 min) about 4000 symbols. 

Notes: Whaley, HHL, Bjerk, Binomial, Jump etc.. give approximations that are great given the inputs; in practice the market disagrees with inputs. Dividend estimates, borrows are unobtainable, and even short rates are inconsistently used. Other models seem very "sensitive", and complicated SVI/SSVI S3, CEV... and I remain unsure of what direction to go to handle a huge data set. Butterfly/Cal arb removal, well maybe in next the questions/threads... 
 
Additional couple of questions: 
1A) The market disagrees with dividend estimates, is there an implied method used in practice 
1B) The market will also disagrees with short rate and borrows (just no obtainable), same question about implied method that is used in practice. 

I hope that this thread can become a reference point for myself and others who are working on this issue with Vanillas. Thank you.
 
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Alan
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Re: American Implied Volatility from quotes

November 1st, 2020, 6:52 pm

I don't know the full answer, so these are just some thinking-out-loud ideas on how to approach it. I am relying partly on ch9,  "Back to basics: an update on the discrete dividend problem", in my book "Option valuation under stochastic volatility II". 

First, suppose you knew the market's assumption for the dividend estimates, ex-dates, and interest rates over a particular option period. Then, the  Vellekoop-Nieuwenhuis (VN) tree algorithm (discussed in my book chapter) is a fast, convergent GBM option value, consistent with discrete dividends. I haven't tried this, but suspect you can easily invert it for the American-style implied vol parameter for say the option midquote -- just like you can invert the CRR model.

So, then my idea would be to try various methods for the cost-of-carry estimates, and convince yourself that the VN implied vols are not too sensitive to the method. I discuss a couple of cost-of-carry methods in my recent ERP article here. One is based upon the VIX white paper, and the other is based on put-call parity for general strikes. While put-call parity technically fails under early exercise, I suspect if you stick to close-to-the money options where early exercise is not too likely, you could get some sensible results. A third method would be to simply project the ex-dates and dividends from history, and use the at-the-money option data to infer interest rates. Since the interest rates should be similar across non-dividend paying, non-hard-to-borrow stocks, that should provides some checks. 

Then, armed with these various cost-of-carry sets, see how much difference it makes to the VN implied vol inversions.

For arbitrage-free smoothing the IV surface, there is a pretty big literature for this for euro-style options. Now, given a set of American-style implied vols (from the above construction), a common approach in a lot of academic literature is to simply convert all the option prices to euro-style options prices by using the same (American-style) implied vols. Whether this procedure is any good is something to investigate. Once you take this route, then you can apply the arb-free smoothing literature, and then move back to American-style again after smoothing.  
 
PrinceQnt
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Re: American Implied Volatility from quotes

November 7th, 2020, 12:22 am

Honored to have you chime in Alan. At this moment in time, for me, your book -> Option valuation under stochastic volatility II, seems like an essential resource and a great collection of practical modern derivatives theory. I will have to get a copy of part I and Part II both! 

Thanks for the reference to your ERP paper, book, and VN model.

Some initial thoughts. 

1. Yes, close to the money seems to generate some sensible results for "q"; after carefully constructing my rates the residuals(borrows) are more or less in line from the few I have checked with steady divs; will have to check some extreme cases from this year that are good HTB examples. looking at the box spreads on the other hand, is quite volatile from quotes, for equities the microstructure swings these too much on 4 legs; perhaps microprices could help. Direct intra-strategy quotes for boxes on futures options are quite accurate, do not know if there is a quoted market of that sort for US equties, preparing RFQs on 4000 names for that strat is not feasible while its fine for one-offs. The quick-and-dirty approach with near strikes gives "good" results, for 5-min snapshots and daily calcs; closer than taking entire chains or many symmetric strikes ( this method will likely average out better over wider time windows and there are a lot of different empirical est methods for that)  

2. Projecting Ex-Dates will be important for for divs cash model or proportional model. However, for near divs a projection which is updated once declaration data is received seems like a good recipe for incorporating "hard" data to the model. This of course becomes muddy with less liquid names that do not pay "steady" enough divs; and matters less for expirations much further out in time where my gut tells me that a continuous div assumption may suffice. 

3. Alan your comments about arb-free smoothing by converting to european and back, intrigues me. I am not familiar with this methodology or what it is even called; but I imagine there are limitations and getting back to American IVs will likely not be so simple. Hope I am wrong, but really want to know more; mind to please point me in the right direction on this. 
 
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Alan
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Re: American Implied Volatility from quotes

November 7th, 2020, 6:27 pm

Thanks for the kind remarks and interest in the books.

Re 3, I don't know what it's called, but to explain it better, consider this setup and a question for you.

Suppose an exchange trades euro-style options on a stock with no dividends, and it's the old days where the riskless rate was, let's say, 5%. Suppose you have a nice smooth arbitrage-free curve of IV's for the put options, say from the mid-quotes. (With exactly same IV-curve for the calls).

Now the exchange announces it will also introduce for trading American-style options on the same underlying. Your task is to predict, before they start trading, the new mid-quote American option prices, esp. the put prices. How do you do it? 
 
PrinceQnt
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Re: American Implied Volatility from quotes

November 29th, 2020, 11:41 am

Firstly to quickly answer Alan's question, which is intended to guide me to understanding the "americanization/europeanization method" which I still do not understand but want to: In that scenario, one would have to calculate the early exercise premium to be added to the existing european options. I feel as though I am stating the obvious and perhaps not taking your guidance in the right direction. Please chime in when you can Alan to point me towards north. 

Goal:
Some methods considered so far, in generating an American IV Surface in the presence of dividends. The goal is to be able to “quickly” and accurately accomplish this, in a way to the surface may be compared over time historically, and that the resulting Greeks are accurate.

Failed avenues:
1.   BAW, RGW, Bjerk & Stens 2002. These approximations seem quite crude, continuous or rough handling of discrete cash divs is also not satisfactory.
2.   Accelerated LR/Tian/CRR trees. Compute quickly with continuous div assumptions. Using actual rates, and sorting out implied borrows/divs provides a better outcome but I still feel as though it will not accomplish goal above with respect to accuracy; speed is also an issue.
Pricing is done in 3.0E10-3 time; implied volatility is done in roughly 5 iterations so 15E10-3 time and the operation can be parallelized to increase speed for computing the universe of all options; however with a 1 minute granularity; a precomputed NDim-array lookup may be the only way to calculate the entire universe in under a minute. This is possible only in the presence of continuous dividends. Discrete cash divs slow the procedure and when using a table method would result in an unmanageablely large NDim-array.


Avenues of Consideration:
There are several vendors out there that claim to be able to handle American IV sufaces with cash divs in under 1 second on the back of proper methodology; and some which even calculate in real time combining great methodology with HUGE computing power; leading me to believe that I am simply not looking at the right way to do this. 

Working with Vellekoop-Nieuwenhuis Tree at the moment. As suggested by Alan, this methodology handles cash divs well, though the interpolation mechanism that re-combs the tree. These trees can be accelerated, the methodology can be extended to use LR which can be further accelerated as well for speed. However my feeling is that the “end” of the V-N methodology will again be to create a precomputed NDim array to lookup IV values and will be too large to manage realistically.


Observations and Question:
All of the aforementioned methods are quite brute-force, while the trees can be accelerated and precomputed NDim array for IV lookups can be truncated at a tradeoff of accuracy; there must exist a better means of extracting smiles/surfaces from existing data for American Options with cash divs in one shot. I am eagerly awaiting for someone to chime in about what modern method may have potential. 

One more thought on Continuous Maturities compares over time:
Part of the goal is to be able to compare surfaces across time. To that end, continuous maturities must be calculated and will be an interpolation between existing maturities. For example when looking at 20-delta put option over the last 10 years, the iv must be ascertained for a 30-day continuous maturity so that comparison makes sense day-to-day and it will likely consist of an interpolation between a weekly and monthly maturity closest to 30 days. The issue here for comparison is that As dividend ex-dates will cycle in an out of those interpolating maturities which may cause the constant maturity to not accurately possess the correct dividend information. Remain very curious to learn about methods for correctly comparing surfaces/smiles/and options across time as constant maturities, in the presence of dividend cycles.


Hope to hear from you, thanks.
 
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nikol
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Re: American Implied Volatility from quotes

November 29th, 2020, 6:11 pm

Why do you care about implied, but not about local vol directly? 
IVOL are usable for contracts without path dependency, otherwise, you have to convert into local vols or develop a "bridge" between IVOL and path-dependent pricing. 
 
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fomisha
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Re: American Implied Volatility from quotes

November 30th, 2020, 6:17 pm

If you need fully automated vol surface calibration from option prices which is fast, reliable, bias-free and arb-free, works with cash dividends and has any other industry standard bells and whistles (vol time, microprices, temporal filtering etc) you should check out Vola Dynamics.

PS. Disclaimer: I am affiliated.
 
PrinceQnt
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Re: American Implied Volatility from quotes

Today, 12:57 pm

Why do you care about implied, but not about local vol directly? 
IVOL are usable for contracts without path dependency, otherwise, you have to convert into local vols or develop a "bridge" between IVOL and path-dependent pricing. 
@nikol The goal is to have a surface that I can observe and measure over time consistently. Not sure that the methodology you are speaking about is conducive to that end goal, please correct me if I am seeing this wrong. 

The "bridge" would make sure correct path dep pricing come out the other end, but the local vol surfaces themselves (Continuous maturity at a couple of fixed periods) that correspond may not have meaning, when comparing across time periods in the way I am after. 

Again, please let me know your thoughts.  
 
PrinceQnt
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Re: American Implied Volatility from quotes

Today, 1:21 pm

Thanks for the kind remarks and interest in the books.

Re 3, I don't know what it's called, but to explain it better, consider this setup and a question for you.

Suppose an exchange trades euro-style options on a stock with no dividends, and it's the old days where the riskless rate was, let's say, 5%. Suppose you have a nice smooth arbitrage-free curve of IV's for the put options, say from the mid-quotes. (With exactly same IV-curve for the calls).

Now the exchange announces it will also introduce for trading American-style options on the same underlying. Your task is to predict, before they start trading, the new mid-quote American option prices, esp. the put prices. How do you do it? 
Answered guiding question in prior post, please see when you have a sec. 

Some quick thoughts after a glance at the literature about VN and Discrete divs. 

VN-> Shortcoming is divs close to "birth" of the tree process. Zoom method (trinomial) to increase resolution as well as cubic rather than linear spline interpolation to glue together the piecewise lognormal does however help to implement this tree-scheme with additional accuracy. Will have to go through this to see if it can be done efficiently to price the entire universe of US optionable securities within my time constraints. Ditto for "Stair tree" method.
References: 
 1. Option Valuation under Stochastic Volatility II: With Mathematica Code first Edition by Alan L Lewis  (Great book!)
 2. Efficient Pricing of Derivatives on Assets with Discrete Dividends February 2006Applied Mathematical Finance 13(3):265-284 (Original Paper)
 3. Efficient option pricing on stocks paying discrete or path-dependent dividends with the stair tree, Tian-Shyr Dai 

I am not sure at this stage if under these tree schemes I will run into issues when trying to retrieve implied volatilities from market quotes. The shifts, recombine, but could there be multiple solutions for implied volatility, unsure of how to set guarantee convergence (seed value or other) for the bisection (or other solver)? 
 
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Alan
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Re: American Implied Volatility from quotes

Today, 3:19 pm

Thanks for the book compliment.

Re "divs close", you can always start the tree, say, a day earlier than "today". 

Also, for calls anyway, say the ex-date is in a day or two and there are no further dividends to a particular expiration. Then, in that case, you could speed things up by 

(i) only building the tree from, let's say, yesterday T(-1) to one day after the ex-date, call that date T(2). 
(ii) make the time increment as small as needed for good accuracy, hourly or whatever.
(iii) start the backwards iteration at T(2), using the exact (no dividend) Black-Scholes call formula for node values at T(2). 

Ultimately, a good production code library might have lots of little speed-ups like this.
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