# General Discussion on Option Modelling

#### AndrewS

##### Member
IV is the volatility implied by the price of the option at this moment. B&S or any other formula will fill in the blank variable. If you have the price then you can calculate the volatility; if you have the volatility then you can calculate the price. The problem is when you try to take current prices or volatilizes into the future and predict where they will be.

#### status1

##### Well-known member
if you have the volatility then you can calculate the price
That's my question
How do I get this volatility so I can calculate the price

#### garyw

##### Active member
Gold Member
@status1 : You seem to be attempting to solve for two unknowns price AND volatility! B&S can be used to solve for one unknown! AndrewS is correct, that B&S can be used to derive the volatility IFF you have the price (as well as all other inputs)! Seems you have neither from your question. Note: the price is typically estimated from the MID, which is avail from your broker or BID/ASK values from other services. IMHO: anytime the Theo Price (via B&S) differs from the MID price, you have provided at least one incorrect value to B&S.

#### Marcas

##### Active member
what Gary suggested is very (VERY!!!) important. You have to clearly specify your goal. I understand it may not be easy task if you don't play with iv often. I will (also) try to answer your question assuming that:
- you deal with single, liquid underlying
-you just want to see how calculation is done without any further goals.

So, let's look at today's options chain.
We randomly pick 11 Nov 2019 expiration and 3020 strike, Put.

You see that MID is 22.75.

I assume that 'risk free' interest rate is .25% (that is not important for that short dte).
We have all necessary data. Let's do our math.
I use Python, so I did find some BSM formula for Python. You probably use Excel, so you should find BSM formula for Excel.
You may also want to code formula by yourself - all depends what you aim to achieve.
Here is how it looks on my screen:

Mind that this is very simplified approach.
BS is my formula, in green are model inputs, second line calculates iv number.

You see that we get 9.52%. Now compare this to TOS number which is 9.68%. If you wish you can compare this to ONE, OV, IB and other calculations and you, most likely, will see different numbers.
The differences in iv calculations come from different inputs that were used by each party. I used simple formula which didn't take dividends into account, I didn't use optimized interest rate. I also could use better number for underlying value (forward). I didn't bother putting more exact dte number - just used what TOS displayed. I didn't verify if Put price is in line with neighbor prices. Changing those inputs will produce different outputs.
Now, which of those numbers is correct?
Hmmm,
- none is, as it was discussed in posts above
or...
- all are correct. All numbers are correct in a sense that calculations were done properly from math point of view.

Let's plug my calculated iv to BS model and see what put price will pop out.

Is it close enough? You have to decide.

Now, you have your calculation. What do you want to do with it?

#### AndrewS

##### Member
Further complications arise when the MID is not really the MID because a limit order is posted inside the market maker bid/ask spread moving the MID away from the true middle of that spread. Example: market maker bid 1.10 1.30 ask, MID is 1.20. Now someone enters an inside order that is not filled: market maker bid is still 1.10 but (limit order) ask is now 1.20 and MID is now 1.15 instead of 1.20 giving you a low IV.

#### status1

##### Well-known member
- you deal with single, liquid underlying
-you just want to see how calculation is done without any further goals.

I see that you used the price to calculate the iv and than used that iv to calculate the price
Now my question can you do it in reverse ? Or is that not possible ?
As AndrewS was saying " If you have the price then you can calculate the volatility; if you have the volatility then you can calculate the price "
From that I assume you can do it either way independently from each other or am I wrong on that assumption ?
Perhaps he meant to say once you calculate for iv ONLY THEN you can calculate the price ?
If the second if statement is dependent on the first one in other words if you need the price to calculate the iv and than use that iv to calculate the price it seems to me like unnecessary calculation since you already have the price in the first place

garyw could be correct I am trying to solve for price provided that I have the iv which is an unknown at the moment so I was wondering if the iv could be obtained by other means than by using the calculation using the price

I am guessing the VIX would not be good to use for iv

#### AndrewS

##### Member
One of the inputs of the B&S formula to obtain the option price is volatility. The other inputs are all known: strike, expiration, underlying price etc. Volatility is the wild card since you need to supply the expected volatility over the life of the option; something which is empirically unknowable. All we are able to do is give our best guess based on the recent movement of the underlying. Being a mathematical formula, it also works in reverse. If you have the option price then you can use that price to obtain the volatility implied by that price. You can also use the prices or volatilities of surrounding options, if known, to infer the price or volatility of the option in question which is where skew modeling comes in. It's also why all this is more art than science.

#### Marcas

##### Active member
I see that you used the price to calculate the iv and than used that iv to calculate the price
Now my question can you do it in reverse ? Or is that not possible ?

As AndrewS was saying " If you have the price then you can calculate the volatility; if you have the volatility then you can calculate the price "
From that I assume you can do it either way independently from each other or am I wrong on that assumption ?

As Andrew said above: you do need volatility to calculate price via BSM model.
Not sure what you mean by 'doing it either way independently'. You do need IV number to calculate any price, no way around (within BSM model).
Not so important where this number come from... sort off..

If the second if statement is dependent on the first one in other words if you need the price to calculate the iv and than use that iv to calculate the price it seems to me like unnecessary calculation since you already have the price in the first place

Correct! You need a price to calculate iv to calculate price. And know what? If you calculate iv by yourself and use the same parameters to calculate original price you always be close enough. Problems start when you try to derive t+n lines. You may get results all over the place and one of them can be 'correct'. Please remember that we are talking about SPX like underlings.

I am guessing the VIX would not be good to use for iv

No, it wont. If you trade VIX dependent underlings then very little, but generally for any particular trade VIX is pretty much useless. I think some people will disagree. Remember that I hold opinions that some (most) people don't agree with. I'm not stubborn, I will change them if proven wrong. So rather do your own thinking/research if you want put \$\$ behind actions based on what we are talking here.

#### status1

##### Well-known member
It's clear now that the price cannot be calculated without the proper iv
I have seen online in the past all kinds of calculators claiming to calculate the price of the options but they just have one example that may have worked on that particular underlying but as soon as you change those values particularly the iv the prices no longer makes sense I thought maybe I gave it another try when I saw Bruno's B+S spreadsheet but of course that's not going to work either
Thanks everyone

#### Marcas

##### Active member
I realized that there is one more thing that wasn't spelled out clear enough. In our talk so far we were considering calculation for single expiration, single strike and single type. This is quite important. If you forget about that you may end up with wrong impression about skew calculations.

It was mentioned before that you can look up and down options chain to verify that MID you want to plug into your model is not an oddball. If your MID and MID of next strike is the same and you are relatively close to ATM you may suspect that there is something 'not right'. There are constant fluctuations in prices that makes hard to capture 'correct MID'. If you have your goals clearly expressed you may want to smooth such distortions away or do quite opposite - if you are hunting for 'miss-priced' options you want to expose this difference even more.

But you may also want to look at other side of options chain, at Calls. Notice that in our chain example IV for puts is different from IV for calls. We are not focusing on single strike/single type calculations anymore, but looking at bit broader picture. There is no such a thing as separate skew for put side and Call side. There is (should be) single IV number for given expiration and given strike that fits Calls and Puts. If you see difference you have indication that 'something is going on, and that 'something' means 'not right'. At this point inputs become important, if you manage to do them right (not so easy) you should have the same IV as output regardless if you are calculating IV from Put price or Call price. In this context you do can talk about 'correct IV".

To make things more fun: even if you manage to obtain the same IV for Call and Put for single strike it doesn't necessary mean that your quest is completed. This IV still may require further modifications... Depends on your goals.

I made it brief but didn't want to leave subject of IV with last post because of possible misguidance. Now, you should have an idea about calculations of single IV and about constructing of the whole skew. If at some point you decide to purse skew from practical side - there are gents here who know much about that and likely will help you. If you are trading solely SPX, and if one or few strategies only, then, imo, you don't need to bother with skew as a whole at all. Just look at relevant parts and TOS, or any other, calculations are fine for that.

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#### AndrewS

##### Member
I enjoy a good academic debate but if you are a retail trader and not a market maker then I think your time is much better spent learning the characteristics of whatever you are using: butterfly, condor, credit spread, etc. What is the price and Greeks and what does the T+0 line look like at various DTE under different market conditions. What is normal, what is abnormal? Is the market telling you this is a good or bad time to enter, stay with or exit that trade? At a certain point, trying to obtain the "perfect" model becomes like the quest to count the angels dancing on the head of a pin.

#### Bruno

##### Member
Gold Member
@status1: B&S is a closed equation however there are other statistical approaches e.g. binomial or Cox-Rubinstein where you don't have the same dilemma. Also my spreadsheet proposes to use ATM IV and a slope to estimate IV on OTM strikes. It is not 100% correct but it's not a bad approximation. In addition, it generally works better on Moneyness than Strikes on the X axis. A higher polynomial degree would also improve the IV approximation.
I personally don't spend much time on whether the Mid is a true Mid. However the Mid of a combo must always be evaluated leg by leg as the combo spread becomes larger and larger. Marcas is absolutely right saying that on account of Put-Call Parity, near ATM strikes (the most trades ones) must have the same IV. It is a bit more difficult to generalise on American options.
As far as learning is concerned, you will over time find a balance of "theory" and practice that suits you. I put quotes around "theory" because this discussion is still very much a practitioner's approach. Actual theory is far far more complex. We're barely scratching the surface here and that should be more than sufficient for us retail traders.

#### Marcas

##### Active member
No disagreement here. Bruno, you are doing good job explaining how things work. I wish you have those classes when I started looking into IV issue with modeling. Some mentors/gurus convey message that deep understanding of IV and modeling is critical for options traders. I don't agree with that but reached this conclusion only after dipping my toes a little. Learning wasn't waste of time - now I can shrug modeling issues with confidence
I'm of the same opinion as Andrew about what trader should focus at. I'm mot saying that proper modeling is useless, absolutely not. It is highly important for many trading parties, but for small/mid retails... not necessary. I can live with my delta positions given me by TOS or by my own very imperfect model. I only question Andrew's theory that market makers relay on tip-top modeling - I think not.

#### garyw

##### Active member
Gold Member
I enjoy a good academic debate but if you are a retail trader and not a market maker then I think your time is much better spent learning the characteristics of whatever you are using: butterfly, condor, credit spread, etc. What is the price and Greeks and what does the T+0 line look like at various DTE under different market conditions. What is normal, what is abnormal? Is the market telling you this is a good or bad time to enter, stay with or exit that trade? At a certain point, trying to obtain the "perfect" model becomes like the quest to count the angels dancing on the head of a pin.

My Post Versalog is suggesting the number is 2.14. (I think that was for angels ...) (a little humor... perhaps very little)

#### AndrewS

##### Member
My Post Versalog is suggesting the number is 2.14. (I think that was for angels ...) (a little humor... perhaps very little)
View attachment 2267
Gary, I think that's how many can dance on the crust of a pie.

#### Bruno

##### Member
Gold Member
I'll get back to volatility; it is an endless topic
Next session will be on American versus European options.

#### status1

##### Well-known member
I finally found the answer to my own question
The answer is to use the Implvol from TOS for each individual strike
I also changed the formula to use the trading days instead of calendar days
Just changing those 2 things made the calculations of the put price very close
Probably if TOS had more than 2 precision digits for the Implvol it would make it even closer but it's good enough for me to know that this works

Interestingly I see that the put prices are moving even on the weekend when I am testing this
When I first tested it out I got the price of the SPX dec 13 3025 strike at 1.612 but than a few minutes later as I am writing this post I see it going lower to 1.602 and 1.599
I guess it's sensing the time decay or maybe it's something else ?
The ask on TOS is at 1.60 for that strike

#### Marcas

##### Active member
Which question did you find answer for? Just curious.
I don't know what your standard is but ImVols by TOS are not perfect.
Remember what Gary said: always have a clear goal when doing such a things, if not, you can _e a s i l y_ drawn in models, theories, papers and one day you ask yourself: what, th, I'm doing here? Maybe your goal is to mimic TOS then it is different than trying to get 'more perfect' iv values. Thus comments you receive can be not in line with your goal if it is not revealed. That's why you may feel that your question wasn't answered before.

I know TOS prices are moving over the weekend, although not always. Not sure about source of this behaviour, you can ask TOS but I doubt you will get satisfactory answer.

Wonder what tools are you using for your iv investigation; excel, py, r, c or something else?

#### status1

##### Well-known member
This question was in regard to the Black Scholes spreadsheet that Bruno attached at the beginning of this post
He was using the option prices to calculate the IV
My question was about doing the opposite using the same formula so the question was what IV to use for the formula
I know a lot of the members were trying to answer my question and I appreciate it I was just not getting the answer that I was looking for
I think AndrewS came in closest to answering it I just did not quite pick up on it when he said
"IV is the volatility implied by the price of the option at this moment " meaning to use the ImpVol from TOS

The goal was simply to see if the formula works by entering the IV and the result being the option price no more than that I am not planing on using it for any modeling or anything like that

I have not seen TOS option prices moving on the weekend while the prices were moving in the spreadsheet
I think this may be a question for Bruno than TOS since he made the spreadsheet and he knows better how it works but it's not important it's just a strange behavior that I observed

I think because he is using the NOW() function to calculate the DTE so that function gets refreshed as I go back and forth between the spreadsheet pages so every time I go back to the same page the function changes the time which is used in the formula so therefore it affects the price value by a small amount

#### Bruno

##### Member
Gold Member
Sorry guys, I admit I don't spend much time on this forum.
I agree that we're small fish in a big ocean that is the market. I also agree that if you have strong command over Greeks, you're in a better position to manage your positions well. That includes understanding interactions or cross derivatives like for instance how IV also impacts Delta.
My questioning turned into something deeper when OV started acting weird in low IV even though I've been looking at options theory since the early 90s.
It is interesting to know market makers quote IV internally, taking for granted the model is Black & Scholes or a close variation of it. Then there may be all sorts of participants who look for arbitrage opportunities or other edges here or there making an overall market picture vertically (same chain) and horizontally (across chains): IV equates option price for a given model.
My point has been that most of us rely on black boxes to figure out how our trades would behave in various environments. We admittedly mostly have current (and possibly historical) prices to work with. Prices i.e. are discrete so we always have to extrapolate to draw curves and they have no projection capabilities, in other words, one can use E.I.O.I.O. (OV jargon) that is individual volatilities extrapolated to form a T+0 line, and it still tells us nothing beyond T+1 maybe T+2. Most traders only follow blindly what they've got...
To come back to @status1: you're definitely on the right track, i.e. whatever you do to build a satisfactory IV picture, considering all other variables can be known, Black & Scholes will churn out prices, with of course due consideration for its anomalies and limitations (GBM etc.).
The spreadsheet is no rocket science. It's only B&S back and forth (with Raphson-Newton) and at the end of the day, it is essentially a matter of figuring out a good looking volatility surface.
Let's say that you have that vol surface equation worked out, or you have a very comprehensive data set available in which case you only need local approximation to fill the gaps, that continuous non linear function IV(X,t) tells you what IV to use in volatility surface mode.
At the end of the day, I know BWB trading backward (Rhino and many extensions on the same theme), I know a few other trades and overall I am not so good at the myriad of other ways to trade options. All I am saying is that beyond this exercise we have to fight the routinal complacency of doing the same things over and over. Markets will always surprise us and it's essential to make our brain tick in order to be better prepared for the unknown.