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cquand


Total Posts: 93
Joined: Sep 2007
 
Posted: 2010-11-17 08:41
Good morning,

My purpose is calibrate a volatility surface for single stocks.

What is the market standard option pricing model generally used to back out implied volatlity from the American option prices on single stocks?
- Dividend treatment: Discrete or continuous or mix div (Discrete for short-term and continuous for longer term: in this case, what is the breakdown of the term structure generally used?)
- Model: whaley, Barone-Adesi, Bjerksund (continous) or non recombing tree (discrete) or smtg else?

Thanks for your help

Graeme


Total Posts: 1629
Joined: Jun 2004
 
Posted: 2010-11-17 10:54
Discrete cash divs as far as there are broker forecasts, then discrete percentage divs using some or other model to forecast these percentages.

Then finite difference for finding implied vol. You can do a few tricks to make this faster than what you might think. For example, you can use one of the approximations that you mention to get the solver started and then refine with actual finite difference; or start with large steps (in time and stock) and then refine as you get closer to the solution. A first estimate might be found by assuming the option is European, for example.

Beware: smoothing out discrete dividends into a continuous dividend yield is a potential source of great toxicity.


Graeme West

cquand


Total Posts: 93
Joined: Sep 2007
 
Posted: 2010-11-17 12:05

Thanks for your reply.

Actually, regarding discrete div, I face sometimes a problem re the "call-put parity" (as in the implied vols of the call and put for a same strike are different:
- no dividend are due before expiry, the call put parity is phuked (I obtain joker vol smile!) - Two Divs and Two Expiry: I cannot manage to find a combinaison of div, which would make the smile fine for both of them

Any guess what would be the problem?

Also I heard that the standard appraoch would be to use a discrete div model for the short-term options but one should use a continuous div model for the longer term, as vol is underestimated otherwise. Is that correct?


pj


Total Posts: 3451
Joined: Jun 2004
 
Posted: 2010-11-17 12:25
>Also I heard that the
> ...
>Is that correct?
Yes.

вакансия "Программист Психологической службы" -але! у нас ошибко! не работает бля-бля-бля -вы хотите об этом поговорить?

Baltazar


Total Posts: 1769
Joined: Jul 2004
 
Posted: 2010-11-17 12:29
cp parity only works for european options
plus beware you are not using stale "last traded" prices.

that might explain your problem

Qui fait le malin tombe dans le ravin

pj


Total Posts: 3451
Joined: Jun 2004
 
Posted: 2010-11-17 12:35
What Baltazar said.

вакансия "Программист Психологической службы" -але! у нас ошибко! не работает бля-бля-бля -вы хотите об этом поговорить?

cquand


Total Posts: 93
Joined: Sep 2007
 
Posted: 2010-11-17 13:16
I did not mean the call put parity as the equality but more the fact if my forward is wrong, I will extract a vol smile which is not continous when I start to use call instaed of put. In this case, it does mean than my forward i.e. my div estimates are wrong, right?

TakeItAndRun


Total Posts: 98
Joined: Apr 2010
 
Posted: 2010-11-17 13:59
Since the cp parity is true phor European options, you get 2 volatility surphaces. Then, you use the semi-surphaces corresponding to OTM options.

Don’t phorget that a volatility surphace relies on a dividend model. As long as one uses the same consistent div model, everything is phine.

One last thing, use market data phor the phorward (div swap, synthetics, or why not assuming pc parity phor the ATM strike).

cquand


Total Posts: 93
Joined: Sep 2007
 
Posted: 2010-11-17 15:39

We have two vol surfaces? one for american and for one for European?

I went to conference and the following methodology was suggested: (I assume here one ex-date before the expiry):

- guess a div amount d1

- back out implied vol for each option using an american pricer disc. div.

- price the european call and put for the same (I guess for the nearest ATM will be sufficient)

- compute the div amount d2 from the put call parity relationship

- iterate until d1 = d2

Does that make sense?

 

 


TakeItAndRun


Total Posts: 98
Joined: Apr 2010
 
Posted: 2010-11-17 19:15
Two questions, two answers:
The two surfaces: American calls and American puts. Then, you recover the best part of them.

What I would do (I guess one iteration is sufficient): given a volatility and an initial dividend, the phirst derivative of call (and put) with respect to dividend, what is the shift of dividend so that the cp parity is satisfied? That's the idea.

Edit: phphphfff


Graeme


Total Posts: 1629
Joined: Jun 2004
 
Posted: 2010-11-17 20:55
TIAR: it was phunny exactly once. Please stop, if you want people to read what you write.

Graeme West

TakeItAndRun


Total Posts: 98
Joined: Apr 2010
 
Posted: 2010-11-17 21:10
Well, I like it that way. Besides, I don't have 'ph' on my Phrench keyboard.

pj


Total Posts: 3451
Joined: Jun 2004
 
Posted: 2010-11-17 21:22
> Besides, I don't have 'ph'
har-de-har-har

вакансия "Программист Психологической службы" -але! у нас ошибко! не работает бля-бля-бля -вы хотите об этом поговорить?

Baltazar


Total Posts: 1769
Joined: Jul 2004
 
Posted: 2010-11-19 14:55
re two vol surface: that would not make sense, ideally you want one vol surface that gives both european and american prices when fed to the appropriate pricer.

I don't know of products that have liquid options both american and european at the same time so this will be less relevent in reality.

regarding fitting the dividend I would use something similar to
http://www.math.uni-frankfurt.de/~stoch/hafner_wallmeier.pdf
they do it for european options but you could do the same for american options (using an american pricer and american option prices)

to sum up: I would say if you have an american pricer, you should have two surfaces only if the div is wrong and the right div is the one that makes the surface from call and put match.
Btw I would work expiration by expiration rather than with a surface, that's probably easier.

And I would make sure I don't work with stale last traded option prices but with bid-ask quotes. Also don't rule out that your american pricer might be inaccurate (american prices are dependant on the volatility dynamics and rate/vol correlation, as Granchio said elsewere, it is not a trivial problem).

Qui fait le malin tombe dans le ravin

TakeItAndRun


Total Posts: 98
Joined: Apr 2010
 
Posted: 2010-11-19 23:33
How do I price an option?
First, I get the right forward.
Then, I apply my volatility to the appropriate pricer.

So, if you’ve got the wrong dividend, stop pricing until you get the right dividend unless you want to be arbitraged. Or, price with low/high estimates of the dividend.

Why the cp parity does not hold for American options?
Because American options can be exercised at any time up to the expiration.
So, what do you do if you want to extract an implied volatility surface when you have only American option prices?
IV(K,T) = Am Call IV(K,T) for K > ATM
IV(K,T) = Am Put IV(K,T) for ATM > K where ATM is the at-the-money strike for the given expiration.
In other terms, one looks for OTM American calls and OTM Americans puts.

Here is how I would get the implied dividend:
Assume that the cp parity holds for American options which are ATM.
Define C as (call bid + call ask)/2 (strike K, maturity T),
P as (put bid + put ask)/2,
S as (underlying bid + underlying ask)/2,
D as the initial dividend (payment date Td).

Using Taylor formula for a small shift of dividend we get:



And D + Δ D is the implied dividend but it is based on a false assumption.

So, I think, cquand, your algo is better in terms of accuracy:

- Initial dividend d1
- Get the Am Call/Am Put Implied Volatilities
- Price Eur Call(Am Call IV,d1) and Eur Put(Am Put IV,d1)
- Compute d2 using the cp parity
- Iterate

If it converges then Am Call IV = Am Put IV for the given strike. But, I am not sure it is fast…Using D + Δ D as the initial dividend will, I think, speed up the process.

Edit: typo



Baltazar


Total Posts: 1769
Joined: Jul 2004
 
Posted: 2010-11-22 08:58
I am not sure the assumption you make is valid: OTM american option=OTM european option.

Even if a call is slightly OTM, there is still a possibility that the stocks goes up fast and starts paying a large dividend before expiration. Agreed this would be small probability but non-zero.

working with an american pricer removes this problem.

Qui fait le malin tombe dans le ravin

TakeItAndRun


Total Posts: 98
Joined: Apr 2010
 
Posted: 2010-11-22 09:57
I am not sure I made such an assumption. To be clear when I price options, I usually use an American model for American options and European model for European options.

The implied dividend I computed is not the right value due to the fact that American options do not verify the cp parity.

However, one should bear in mind that the ‘expected expiration’ for American options is usually not the expiration date (due to the possibility of early exercise) contrary to European options.

Baltazar


Total Posts: 1769
Joined: Jul 2004
 
Posted: 2010-11-22 13:00
well "Assume that the cp parity holds for American options which are ATM" is similar to "assume they are european" to me (or that their "american premium" compensate, that's true).

but anyway. The thing is that there will be some handweaving in optimizing your dividend as it is fairly possible that you will find an acceptable dividend using ATM options and when you plug that dividend to your american pricer and use OTM vols (say OTM calls) to price ITM options (say corresponding puts), you won't get perfect prices.

Qui fait le malin tombe dans le ravin

nikol


Total Posts: 712
Joined: Jun 2005
 
Posted: 2011-01-18 00:41

i stuck my head into the same problem.

cquand:

to me it does not make sense because vol implied by American is sort of averaged local vol between now and maturity in such way that

  • American Put or American Call(local vol(K,T)) ==
  • American Put or Call(Vol*(K)) ==
  • Quoted Amer.Put or Call(K,T)

where in the first place I mean use of entire local vol surface and in the second place I mean some effective vol which makes theoretical Put/Call equal to observations. Assuming that Local vol here is the same as for Europeans and by using Hagan approximation one can transform Local-vol into Implied-Vol usable for Europeans. You want to use vol-implied-byAmericans to price Europeans, but I would say that 

European Put or Call (ImpliedVol(LocVol)) =/= European Put or Call(Vol Implied-by-Americans)

where the left-side is made possible by the mentioned Hagan (and I think is correct way) and the right-side is what you propose.

The question is - by how much they are different?

 


cquand


Total Posts: 93
Joined: Sep 2007
 
Posted: 2011-01-18 10:20

Nikol,

It is already a step ahead of what I'm trying to achieve, the local vol is a model for your volatlity surface and I'm not there yet.

I was just trying to have a raw market vol surface which will be achieved by investigating what is the "bridge" (.i.e. option pricer) between price and vol.

Once I got this I can use a vol point v(k,t) and I should be able to price the american option and the european, and recover the initial market price. The next step is probably to fit this surface with a vol model, but that is not my issue right now.

So far, I find that the following pricer seems quite good:
1. Non-recombining tree using discrete divs including also the borrow rate

2. Use of business tenor .i.e include holiday + theta throughout the day (non-linear)

3. Discrete Divs: Tenor < 2y => estimated using Call/Put prices following the next ex-date (Objective function: Call IV = Put IV). Note that I still get some problems with this estimate, it often gives me differrent results using the next expiry after ex-date and the one after! And it is not a problem of borrow as the difference is too big. So not sure if I'm doing something wrong.

4. Tenor >2y - I estimate longer dated divs using proportional div, i.e. I take my div yield from the discrete div <2y, and compute the div amound like that for each expiry.

Please let me know your thoughts


Baltazar


Total Posts: 1769
Joined: Jul 2004
 
Posted: 2011-01-18 11:40
I would add that certain french stocks have both european style and american style options listed. I remember one US index had too but not sure which one.

This could be usefull to test your method if the spreads are tight.

Qui fait le malin tombe dans le ravin

cquand


Total Posts: 93
Joined: Sep 2007
 
Posted: 2011-01-18 12:20

In European single stocks, you have indeed European options for the big expi (Mar/Jun/Sep/Dec) but the liquidity is basically zero, so I actually remove them from my option chains. But yes it would be an interesting test, to back out vols around ex-date, although the European will be probably questionnable.

For info: in Europe, French and Spanish names have both type of options but for even most liquid names like BNP or SAN have zero liquidity for the European. In the US in the SPX top 100 (I'm not really into it though), I know you have a few names like: Wells fargo, Disney, Pfizer and Warner. I have no idea why these names in particular...

 


nikol


Total Posts: 712
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Posted: 2011-01-18 17:05

if there are even dozens of trades/day it is way better than polling brokers for quotes.


nikol


Total Posts: 712
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Posted: 2011-01-18 18:00

cquand:

for discussion: put-call parity is a degree to which you can hedge put-call combo. for europeans exact hedge is possible (C-P = S0-PVdiv-K.exp(-r.t)). for americans this turn to be imprecise because you do not have control over exercise of your short position (counterpart does). this can be expressed as (link) : C+K.exp(-r.t)<P+S0<C+K+PVdiv  or canbe rewritten as P + S0 ~ C + K.exp(-r.t) + h.{K.(1-exp(-r.t))+PVdiv}, where 0<h<1. Notice, that under t->0 {}->0. One should try to calibrate h, because that will reflect the measure of mismatch between your Call IV and Put IV.

From this perspective I would ask myself the following questions:

  • why counterpart wants to excersize rather than just liuidate on the market? especially if everything is priced-in?
  • look from perspective of bid/ask side of the book. who sets quotes on those? people offer amer.options because they want to sell them, which means they either their liquidate long position or want to take short position (writing). reverse is true for bid side. Can I calibrate h from this process?

Baltazar: can you give those phrench names? please :)

 

EDIT: formula typo


Baltazar


Total Posts: 1769
Joined: Jul 2004
 
Posted: 2011-01-19 08:48
I remember SocGen, Cquand mentionned BNP.
There was other but I can't remember which.

regarding your point: paying extra for american over european doesn't mean I want to exercise early now. It just means that I take into account that there might be some situations where I might want to exercise instead of liquidating in the market.

for exemple an ATM call is not a candidate for early exercise now but if the stock goes up enough it might become one, hence the higher price.

Qui fait le malin tombe dans le ravin
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