# Valuing Options/Warrants & Volatility



## GreatPig (8 August 2006)

A few questions, as I'm trying to get more of an understanding of this topic.

1. What method do you use to work out option values? I don't have the Hoadley Excel thing (although have tried the free calculators on the Website), but I gather it includes the Black-Scholes, Binomial, and one or two other formula. If you primarily just use the Black-Scholes method, how do you allow for dividends and the possibility of early exercise on American-style options, or do you find they don't make much difference?

2. How do you know if this formula is giving valid results? It seems to me that there are essentially two unknowns in the formula: option value and IV, and that by fiddling with IV you can always get the option value to match the current market price. Consequently, if the underlying price changes and when you plug that into the formula it gives a value other than the current market price, do you assume that's just because the IV must have changed or that possibly the model the formula is based on is not accurate? The texts I've read all note that prices don't always match the lognormal distribution assumed by Black-Scholes, and that there can be different degrees of skew and kurtosis (that I can plot in AmiBroker), which implies that the formula won't always give accurate results. But does it really matter for practical purposes?

3. Is it possible to get historic IV information for ASX stocks? I've seen some of you talking about IV plots, as opposed to simple SV plots which I can do in AmiBroker, and wonder where you get the data from - especially for the Australian market. And is the IV data for the share or the derivative, as I've noticed different derivatives on the same share can show different IV values (although generally not much different)?

4. When calculating IV, I gather the general method is to work backwards using the current market price as an input. How then do the MMs come up with a figure when they may be the whole market for a particular warrant (and presumably option) for much of the time? In other words, when I look at a warrant and see a particular bid/offer spread from an MM, and no other bids or offers (and quite possibly no trades ever for that instrument), how are the MMs coming up with an IV value so as to set a price? When I compare most back-calculated IV values with current SV values of the underlying in AmiBroker, they are invariably higher - no matter what SV time frame I use. What other information might an MM be using to determine a higher IV?

Thanks.

GP


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## Magdoran (8 August 2006)

GreatPig said:
			
		

> A few questions, as I'm trying to get more of an understanding of this topic.
> 
> 1. What method do you use to work out option values? I don't have the Hoadley Excel thing (although have tried the free calculators on the Website), but I gather it includes the Black-Scholes, Binomial, and one or two other formula. If you primarily just use the Black-Scholes method, how do you allow for dividends and the possibility of early exercise on American-style options, or do you find they don't make much difference?
> 
> ...



Hello Gp,


This is a very involved subject which is difficult to explain thoroughly in a reply post.  To learn a lot of these concepts look back through the threads Wayne, Margaret and I have commented in – especially consider the texts recommended – Natenberg, McMillan and Cottle (there are others of course).

Option values reflect both known values and the consensus of expected values by the market (all participants including market makers).  Black and Scholes is a formula for valuing options, as is binomial.  These are rival complex option valuing formulas that try to construct a framework for valuing an option currently and how it might behave based on future events.  

What you have to understand is that option values are quite fluid based on the “Greeks” – how much the underlying has moved around, and how much might it move around for instance.  What is the delta of an option?  How will gamma change this value as the underling wither breaks out strongly, or trends weakly?…  What will be the effect of theta (time) decay on the value of a sold or bought position or spread?  Is IV likely to rise or fall, and by how much?

Historical/Statistical volatility looks back at the previous ranges of movement of the underlying – the more volatile, the more the range of deviation (statistics theory is often used such as “standard deviation”), the higher the theoretical volatility.  Essentially we’re trying to work out what the chances are of the underlying moving greatly from the current price. 

Implied volatility (IV) is driven by a range of factors – what the market thinks is likely to happen (potential bad news tends to increase IV for example), supply and demand for each individual option, market maker strategy – essentially it is looking into the future  not the past like Historical volatility.

Ok, to figure out option values you input several variables – the current underlining price, the strike (or exercise price), the date of expiry, and to a lesser extent interest rates and dividends.  To work out the IV depends on the Model you are using, and this can vary a lot or a little depending on a range of variables. Certainly, American exercise should be more valuable than European exercise, and this should be factored into the theoretical values as should dividends.

Beware, Black and Scholes has some inbuilt errors in the formula that can give wrong results for calendar spreads for instance.  The difference between the actual price and the net model price without volatility factored yields the IV.  But different models will treat time to expiry, strike price differences, and IV values differently.  This is because the values are theoretical – essentially best shot guesses into the future based on model preferences.

There are a range of products on the market to do this.  I suggest that any potential purchaser looks around at the various products and considers the price, data, capacity, and long and short term objectives and how well each package delivers for this.  This is a personal choice based on personal preferences.

In my view modelling should assist you in assessing the risk and reward of a variety of strategies, and show when options are expensive or inexpensive - IV plays a big part in helping traders to select which strategy to use based on projecting probable future movements in IV.

As for lognormal distribution – an interesting topic all of its own.  There are arguments currently in options circles that the current electronic trading age has essentially made markets much more volatile, hence the “black swan” effect is more likely (essentially major shifts of -/+ 4 standard deviations in price shifts), and that older models were not designed to measure this level of risk.  Hence calculating premiums for options sellers has become much more complex, and with strong volatility possible, pricing options is going through a metamorphosis.

As for calculating IV, this is an art in itself – you can as you say work back to determine the current IV for the bid and ask of an option, but you also need to be able to project forward and see how a shift in either direction will either harm or assist your position, or look at ways to minimise IV shifts with spreads for instance.  This is a corner stone to effective trading with options.

Market makers use a series of algorithms based on their market approach – they may want to be fully hedged, and buy and sell the underlying to balance their books based on the net long and short calls and puts.  If they are expecting large moves, or the demand for options increases they may widen their spreads, or bias them in the direction they most expect will yield them the best margin, or both.  A lot depends on the price action in both the underlying and the options they are covering, and related markets they may also be hedging in.  They may have a market view and be biased in that direction too – depends a lot on the aggressiveness of the market maker, and their policy stance.

IV and Historical volatility are seldom the same.  They should though to an extent reflect the other – where IV is in relation to HV can be quite telling about the current market.  It is usual for IV to be higher currently than HV in the Australian market – this represents the premium the sellers are demanding to supply.


Hope that helps for a first pass GP – now time to pass the baton to Wayne and Margaret…


Regards


Magdoran


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## wayneL (8 August 2006)

Hi GP



			
				GreatPig said:
			
		

> A few questions, as I'm trying to get more of an understanding of this topic.
> 
> 1. What method do you use to work out option values? I don't have the Hoadley Excel thing (although have tried the free calculators on the Website), but I gather it includes the Black-Scholes, Binomial, and one or two other formula. If you primarily just use the Black-Scholes method, how do you allow for dividends and the possibility of early exercise on American-style options, or do you find they don't make much difference?
> 
> ...




Cheers


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## GreatPig (9 August 2006)

Thanks Magdoran & Wayne. Great replies.

Regarding one point of yours Wayne:



> Ultimately, because options are traded by the open auction process of a public exchange, they are beholden to market forces.



Wouldn't though the bulk of the market be using these same models to value their options? Otherwise, how would they be coming up with a figure for the value? How did they do it before these models were created?

It seems to me that if most people are using these models to value their options, then that of itself would make the models fairly accurate.

As for books, I've currently been reading that Cox & Rubenstein one I mentioned earlier (and even trying to following the maths, although getting lost in some of the statistical concepts ) and the Natenburg one on Option Volatility & Pricing.

I have to say that while I've had to gloss over some of the details of the statistics in Cox & Rubenstein, the book does give a good background to the pricing models, explaining why they have the inputs they do, why other variables aren't inputs, the concept behind the pricing models (ie. riskless arbitration), and how the prices and Greeks are derived. While it's rather heavy going, I've found it quite complementary to the Natenburg book which focuses more on how to use the concepts in spreads and so on. Knowing how to use them is probably more useful than knowing how they work, but I always like to know to some extent how things are working behind the scenes (that's why I'm an engineer ), as it means I can sometimes work out or intelligently guess what might happen in other situations that aren't directly mentioned in the usage explanations.

Cheers,
GP


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## wayneL (9 August 2006)

GreatPig said:
			
		

> Thanks Magdoran & Wayne. Great replies.
> 
> Regarding one point of yours Wayne:
> 
> ...




When it is mainly professional/savvy traders this is true, but when the general public get involved, things go awry.

Not in terms of the pricing model, but in terms of valuation. The pricing model will always have integrity because of the variability if implied volatility. When pricing loses integrity, the arb boys come in quickly to even things out.

Think of implied volatility as the amount of risk, looking forward, that is priced in on behalf of the option seller. This may not reflect historical volatility one iota. It might not even reflect volatility that is ultimately realised.

But imagine you want to sell an option on a biotech with an announcement due. Statistical volatility may be 30%, but this in no way reflects what "might" happen when the announcement comes out. You are going to want one hell of a lot more premium than 30% IV. In the US market, situations such as this typically price in 100-200% IV. Amazingly, "the general public" mentioned above are quite willing to pay it... (and usually get fleeced  )

It all comes back to 
1/ implied volatilty
2/ pricing integrity, of which there are simple measures that can be arbitraged in an instant

In other words, if options are *extremely* expensive for example, it can be explained in terms of a particular model by implied volatility. The "market" is pricing in possible future higher realised volatility, whether realised or not.

This is akin to an insurance company pricing in, say cyclone damage in tropical areas. But cyclones may not strike, year upon year and the insurance company appears to make out like a bandit. However the *risk* of cyclone is ever present and one day, it will. They want recompense for that risk. This is in effect "implied volatility" and is priced into insurance premiums.

Cheers


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## GreatPig (9 August 2006)

Thanks again Wayne. Good stuff 

GP


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## dubiousinfo (31 August 2006)

Have been looking at some ZFX puts options and decided to look at warrants as well.

October $11.00 options and warrants 


Option:
ZFX4A  buy .56  sell .63

Warrant:
ZFXVZQ  buy .56  sell .57

The warrant has a ratio of 3:1 so you need 3 warrants to cover 1 share.

They both have similar strike price and expiry dates yet the warrant looks way over priced compared to the option. So why would anyone take the warrant over the option??

What am I missing here?


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## GreatPig (31 August 2006)

Yeah, don't know what the story is there. That's an IV of about 120% on the warrants.

I wouldn't be buying them.

GP


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## Magdoran (31 August 2006)

dubiousinfo said:
			
		

> Have been looking at some ZFX puts options and decided to look at warrants as well.
> 
> October $11.00 options and warrants
> 
> ...



Hello dubiousinfo,


You raise a really interesting point here about how to compare options with warrants.  You have certainly found equivalent puts with matching strike prices and similar expiry dates. This is actually the real challenge of learning about derivatives – how to value them.

The ratio of 3:1 makes the warrant look dear compared to the option, doesn’t it?  Great Pig calculated this to be the effect of volatility of 120%.  There are however other factors to consider.  I’m not totally certain about this since I don’t have time to research the warrant further, but a key Greek to consider in this case other than volatility is delta.

ZFX4A is trading currently with a delta of 0.21 (I’m using binomial American exercise with a custom setting to determine this), which means that the option currently will move around 2 cents for every 10 cent move in ZFX, but will change as the strike moves closer or further away from the current price of ZFX.

It may well be that ZFXVZQ has a much higher delta (warrants can have a delta set by the issuer), hence it may move at something like 0.6 or higher, maybe 0.9 (I’m guessing) – you’d have to contact the issuer or read the terms fully to determine this.  Another point to recognise is that the warrant is European exercise and the option is American exercise.

Don’t forget that the issuer may have all sorts of terms and conditions where they can reset the delta and factor in their view of volatility.  Also, don’t forget that ZFX4A is valued in a very choppy IV environment around the 57 to 60 mark (from a high around 95 – see attached chart).

So, be careful when making such valuations without fully researching each instrument.  It may well be that there is a great arbitrage opportunity, but you really have to watch out for volatility effects, and the capacity for the warrant issuer to shift the goal posts against you.  The less risky arbitrage comes from deep ITM positions where the negative effects of volatility and delta can be ameliorated.

Regards


Magdoran


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## ice (31 August 2006)

dubiousinfo,

In relative terms warrants usually price 5-10% higher than the equivalent ETO's. 

However in this case the apparent extreme over-valuation is a consequence of ZFX going ex-div prior to expiry (70 c on 16/10). The warrant is european style and can't be exercised before the div so consequently the matrix differs from that applied to the american style ETO. 


ice


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## Magdoran (31 August 2006)

ice said:
			
		

> dubiousinfo,
> 
> In relative terms warrants usually price 5-10% higher than the equivalent ETO's.
> 
> ...



Nice idea ice, 

A relevant angle for consideration, but if you are correct, wouldn’t you expect the option should be valued higher than the warrant since it is American exercise, but the warrant is apparently dearer. 

I think the fact the warrant is European exercise is not a primary factor here, although you are quite right it is a reason why the option should be worth more, but the ASX requirements of the issuer should ameliorate any major shift on this basis.

I hope dubiousinfo does the due diligence for us and informs us what the warrant delta actually is, and how the issuer can change this.  This may well shed some light on why the Warrant is dearer.


Regards


Magdoran


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## dubiousinfo (31 August 2006)

Magdoran
Thx for your help. 

I have been right through the issuer documentation but cant find any reasons for the amount of differnce (although its certainly possible I have missed something).

I have good profits from ZFX Shares I have held for some time & intend to continue to hold, as I believe there is still upside left in the medium term. However I was considering some puts to ride out Sept/Oct as I believe some short term weakness in zinc & ZFX may be possible under certain conditions.

Was just exploring the possibilities when I came across the anomaly. I normaly use options rather than warrants for this type of thing & while my knowledge of the greeks could definatly be improved, I probably have the basics on them.

I take on board what you have said about delta, but  watching them over the last few days as the price of ZFX has moved, the movement of the warrant prices seems to have been less than the options. (which I guess implies the warrant has a lower delta).

Hyperthetically, if I was to purchase both of these & just hold till expiry & assuming they finished ITM, the cost to payout ratio of the warrants would appear to be substantially lower than the options. Is this correct?


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## sails (31 August 2006)

Hi Mag,

Apart from occasionally trading instalment warrants, I haven't had much other experience with warrants in general - don't have a lot of faith in the issuers     They seem to be a law unto themselves - eg this morning I had several CTW barrier warrants on my watchlist (BHP, ANZ and NAB - both puts and calls) and suddenly the quotes disappeared off the lot for approx 15 minutes     I have been watching these barriers to see if they might be useful to hedge option spreads instead of using the underlying due to their delta of almost 1 - do you have any thoughts on this?

The other thing I have noticed when comparing options with normal trading warrants is the warrants have a set ratio of shares per contract. If we take an example of a 4:1 ratio, I have generally assumed that their delta will never move much away from .25 except for volatility fluctuations, or in other words, they lack the gamma component of options - is this correct?

If that is so, then options would have to be more attractive for long premium as the delta will increase with movement when it's moving in our favour and decreasing when it's moving against (providing IV remains constant!).  It also means the warrant MM's almost have their cake and eat it too!

Hope you had a good time in sunny Qld last week where we had near summer weather.  All gone now - it's wet and windy!

Regards,

Margaret.


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## Magdoran (1 September 2006)

sails said:
			
		

> Hi Mag,
> 
> Apart from occasionally trading instalment warrants, I haven't had much other experience with warrants in general - don't have a lot of faith in the issuers     They seem to be a law unto themselves - eg this morning I had several CTW barrier warrants on my watchlist (BHP, ANZ and NAB - both puts and calls) and suddenly the quotes disappeared off the lot for approx 15 minutes     I have been watching these barriers to see if they might be useful to hedge option spreads instead of using the underlying due to their delta of almost 1 - do you have any thoughts on this?
> 
> ...



Hello Margaret,


Yes, it was nice to be in Brisbane with the nice weather, certainly after the cold snap in Melbourne (which doesn’t compare with icy Canberra’s negative 6 one evening, Brrrrr).  I had a great run in Brisbane too, really switched on people with positive attitudes, and had a lovely dinner with one of my closest friends and his wife and young son.

Having graduated to options from warrant trading, I have to agree with you about the potential risks with OTC instruments – if the issuer goes broke, you run the risk of losing your capital, hence the effort required to do the due diligence both on the warrant terms and conditions, and on the issuer.  Add to that the cheeky games they can play with theoretical valuations and spread width, let alone actually making a market, and there is a lot to consider on the risk side of the equation.

I have considered using warrants to hedge options positions if there was actually an advantage to doing so, but in practice I have never found a situation for doing this.  Interestingly the reverse is often more likely where you can hedge longer term instalment or endowment warrants where options can be used to hedge warrant positions, or use warrants as collateral when looking to sell options around a warrant position.  Don’t forget, instalment/endowment warrants often yield dividends, and can be quite potent when used as a longer term revenue strategy while using them as collateral for shorter term swing trading.

Overall though, I have grave reservations about using barrier warrants since there is a real risk of them evaporating if the strike is hit at any time, but the high delta is attractive to day traders.  Me, I just don’t use them, and prefer using options, but that’s a personal preference.  I just find options easier to quickly use for position trading since I have a very good idea how they should perform, while you have to do a lot of due diligence researching each warrant (all classes) to work them out, then you need to consider the reliability of the issuer, let alone the games they might play in the market or with the terms and conditions.

My understanding of the way warrants trade from a delta perspective is that they are often geared with high deltas – I suspect these perform differently from options.  For example Barrier warrants are set usually near a 1:1 basis with the underlying, so they perform very differently to options. The problem is that you are dealing with an OTC instrument that is not regulated like an exchange traded standardised instrument like ETOs, hence the conditions can vary a lot.  Hence, I suspect that the gamma may well be set.  As you say this is a point of difference in the instruments that needs to be considered.  

Certainly the ratio of warrants to the underlying is very important, and has a material effect on the value of the warrant.  As for the delta effect, if you have a 4:1 ratio as you suggested, then you are quite correct you have to consider the net movement of the warrants against a movement in the underlying.  This can get quite tricky to work out.

And yes, I agree with your comment that “options would have to be more attractive for long premium as the delta will increase with movement when it's moving in our favour and decreasing when it's moving against (providing IV remains constant!).”  In general I have found that options perform better than most warrants, but not always.  And I agree about being at the mercy to some extent to the market makers in warrant issues.  I got caught a few times with some odd movements that different issuers made, and eventually figured out which issuers to avoid.

So, I think we’re on the same page here!


Regards


Magdoran


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## sails (1 September 2006)

Thanks Magdoran for confirming my thoughts on warrants in general.  I understood you've had quite a bit of previous experience in warrant trading and your imput is appreciated  . 

Barrier warrants could have some possibilities for the occasional day trade that pops up due to their 1:1 ratio (except some like NAB and CBA that can have higher ratios).  I've noticed the barrier bid/ask spreads are usually based on the quantity of the underlying market depth - obviously so they can get their hedge on.  I guess the real question is once you are into their spider web - will they let you out


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## Magdoran (4 September 2006)

dubiousinfo said:
			
		

> Magdoran
> Thx for your help.
> 
> I have been right through the issuer documentation but cant find any reasons for the amount of differnce (although its certainly possible I have missed something).
> ...



Hello dubiousinfo,

I haven’t forgotten you on this question, but have been pushed for time, and want to think this question through carefully.  

There are two issues I need to consider: 

Firstly I want to frame the answer generically so my response cannot be considered to be giving financial advice since your question is framed specifically.   

Secondly, I need to think through all the mechanics involved in your question since there is some complexity that may not be obvious that could materially affect this kind of position.  (Indeed, there may be aspects that I have been working on from an arbitrage perspective generally in this area, and I’m reluctant to post this IP on a public forum).

Leave it with me, and I’ll consider this situation as time permits, but it may take a while…

I hope you understand.


Best Regards


Magdoran


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## dubiousinfo (4 September 2006)

Magdoran

Thx for taking the time, your help is appreciated.


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