## 15.1 – Background

Having understood Delta, Gamma, and Theta we are now at all set to explore one of the most interesting Option Greeks – The Vega. Vega, as most of you might have guessed is the rate of change of option premium with respect to change in volatility. But the question is – What is volatility? I have asked this question to quite a few traders and the most common answer is “Volatility is the up down movement of the stock market”. If you have a similar opinion on volatility, then it is about time we fixed that ☺.

So here is the agenda, I suppose this topic will spill over a few chapters –

- We will understand what volatility really means
- Understand how to measure volatility
- Practical Application of volatility
- Understand different types of volatility
- Understand Vega

So let’s get started.

## 15.2 – Moneyball

Have you watched this Hollywood movie called ‘Moneyball’? It’s a real life story Billy Beane – manager of a base ball team in US. The movie is about Billy Beane and his young colleague, and how they leverage the power of statistics to identify relatively low profile but extremely talented baseball players. A method that was unheard of during his time, and a method that proved to be both innovative and disruptive.

You can watch the trailer of Moneyball here.

I love this movie, not just for Brad Pitt, but for the message it drives across on topics related to life and business. I will not get into the details now, however let me draw some inspiration from the Moneyball method, to help explain volatility :).

The discussion below may appear unrelated to stock markets, but please don’t get discouraged. I can assure you that it is relevant and helps you relate better to the term ‘Volatility’.

Consider 2 batsmen and the number of runs they have scored over 6 consecutive matches –

Match | Billy | Mike |
---|---|---|

1 | 20 | 45 |

2 | 23 | 13 |

3 | 21 | 18 |

4 | 24 | 12 |

5 | 19 | 26 |

6 | 23 | 19 |

You are the captain of the team, and you need to choose either Billy or Mike for the 7^{th} match. The batsman should be dependable – in the sense that the batsman you choose should be in a position to score at least 20 runs. Whom would you choose? From my experience I have noticed that people approach this problem in one of the two ways –

- Calculate the total score (also called ‘
**Sigma**’) of both the batsman – pick the batsman with the highest score for next game. Or.. - Calculate the average (also called ‘
**Mean**’) number of scores per game – pick the batsman with better average.

Let us calculate the same and see what numbers we get –

- Billy’s Sigma = 20 + 23 + 21 + 24 + 19 + 23 = 130
- Mike’s Sigma = 45 + 13 + 18 + 12 + 26 + 19 = 133

So based on the sigma you are likely to select Mike. Let us calculate the mean or average for both the players and figure out who stands better –

- Billy = 130/6 = 21.67
- Mike = 133/6 = 22.16

So it seems from both the mean and sigma perspective, Mike deserves to be selected. But let us not conclude that yet. Remember the idea is to select a player who can score at least 20 runs and with the information that we have now (mean and sigma) there is no way we can conclude who can score at least 20 runs. Therefore, let’s do some further investigation.

To begin with, for each match played we will calculate the deviation from the mean. For example, we know Billy’s mean is 21.67 and in his first match Billy scored 20 runs. Therefore deviation from mean form the 1st match is 20 – 21.67 = **– 1.67**. In other words, he scored 1.67 runs lesser than his average score. For the 2^{nd} match it was 23 – 21.67 = **+1.33, **meaning he scored 1.33 runs more than his average score.

Here is the diagram representing the same (for Billy) –

The middle black line represents the average score of Billy, and the double arrowed vertical line represents the the deviation from mean, for each of the match played. We will now go ahead and calculate another variable called ‘Variance’.

Variance is simply the ‘**sum of the squares of the deviation divided by the total number of observations’**. This may sound scary, but its not. We know the total number of observations in this case happens to be equivalent to the total number of matches played, hence 6.

So variance can be calculated as –

Variance = [(-1.67) ^2 + (1.33) ^2 + (-0.67) ^2 + (+2.33) ^2 + (-2.67) ^2 + (1.33) ^2] / 6

= 19.33 / 6

**= 3.22**

Further we will define another variable called ‘**Standard Deviation**’ **(SD) **which is calculated as –

std deviation = √ variance

So standard deviation for Billy is –

= SQRT (3.22)

= 1.79

Likewise Mike’s standard deviation works out to be 11.18.

Lets stack up all the numbers (or statistics) here –

Statistics | Billy | Mike |
---|---|---|

Sigma | 130 | 133 |

Mean | 21.6 | 22.16 |

SD | 1.79 | 11.18 |

We know what ‘Mean’ and ‘Sigma’ signifies, but what about the SD? Standard Deviation simply generalizes and represents the deviation from the average.

Here is the text book definition of SD “*In statistics, the standard deviation (SD, also represented by the Greek letter sigma, σ) is a measure that is used to quantify the amount of variation or dispersion of a set of data values*”.

Please don’t get confused between the two sigma’s – the total is also called sigma represented by the Greek symbol ∑ and standard deviation is also sometimes referred to as sigma represented by the Greek symbol *σ. *

One way to use SD is to make a projection on how many runs Billy and Mike are likely to score in the next match. To get this projected score, you simply need to add and subtract the SD from their average.

Player | Lower Estimate | Upper Estimate |
---|---|---|

Billy | 21.6 – 1.79 = 19.81 | 21.6 + 1.79 = 23.39 |

Mike | 22.16 – 11.18 = 10.98 | 22.16 + 11.18 = 33.34 |

These numbers suggest that in the upcoming 7^{th} match Billy is likely to get a score anywhere in between 19.81 and 23.39 while Mike stands to score anywhere between 10.98 and 33.34. Because Mike has a wide range, it is difficult to figure out if he is going to score at least 20 runs. He can either score 10 or 34 or anything in between.

However Billy seems to be more consistent. His range is smaller, which means he will neither be a big hitter nor a lousy player. He is expected to be a consistent and is likely to score anywhere between 19 and 23. In other words – selecting Mike over Billy for the 7^{th} match can be **risky**.

Going back to our original question, which player do you think is more likely to score at least 20 runs? By now, the answer must be clear; it has to be Billy. Billy is consistent and less risky compared to Mike.

So in principal, we assessed the riskiness of these players by using “**Standard Deviation**”. Hence ‘Standard Deviation’ must represent ‘**Risk**’. In the stock market world, we define ‘Volatility’ as the riskiness of the stock or an index. Volatility is a % number as measured by **standard deviation**.

I’ve picked the definition of Volatility from Investopedia for you – *“**A statistical measure of the dispersion of returns for a given security or market index. Volatility can either be measured by using the standard deviation or variance between returns from that same security or market index. Commonly higher the standard deviation, higher is the risk”*.

Going by the above definition, if Infosys and TCS have volatility of 25% and 45% respectively, then clearly Infosys has less risky price movements when compared to TCS.

## 15.3 – Some food for thought

Before I wrap this chapter, let’s do some prediction –

Today’s Date = 15^{th} July 2015

Nifty Spot = 8547

Nifty Volatility = 16.5%

TCS Spot = 2585

TCS Volatility = 27%

Given this information, can you predict the likely range within which Nifty and TCS will trade 1 year from now?

Of course we can, let us put the numbers to good use –

Asset | Lower Estimate | Upper Estimate |
---|---|---|

Nifty | 8547 – (16.5% * 8547) = 7136 | 8547 + (16.5% * 8547) = 9957 |

TCS | 2585 – (27% * 2585) = 1887 | 2585 + (27% * 2585) = 3282 |

So the above calculations suggest that in the next 1 year, given Nifty’s volatility, Nifty is likely to trade anywhere between **7136 and 9957** with all values in between having varying probability of occurrence. This means to say on 15^{th} July 2016 the probability of Nifty to be around 7500 could be 25%, while 8600 could be around 40%.

This leads us to a very interesting platform –

- We estimated the range for Nifty for 1 year; similarly can we estimate the range Nifty is likely to trade over the next few days or the range within which Nifty is likely to trade upto the series expiry?
- If we can do this, then we will be in a better position to identify options that are likely to expire worthless, meaning we could sell them today and pocket the premiums.

- We figured the range in which Nifty is likely to trade in the next 1 year as 7136 and 9957 – but how sure are we? Is there any degree of confidence while expressing this range?
- How do we calculate Volatility? I know we discussed the same earlier in the chapter, but is there an easier way? Hint – we could use MS Excel!
- We calculated Nifty’s range estimating its volatility as 16.5% , what if the volatility changes?

Over the next few chapters we will answer all these questions and more!

### Key takeaways from this chapter

- Vega measures the rate of change of premium with respect to change in volatility
- Volatility is not just the up down movement of markets
- Volatility is a measure of risk
- Volatility is estimated by standard deviation
- Standard Deviation is the square root of variance
- We can estimate the range of the stock price given its volatility
- Larger the range of a stock, higher is its volatility aka risk.

I have asked this question to quite a few traders and the most common answer is “Volatility is the up down movement of the stock market”. If you have a similar opinion on volatility, then it is about time we fixed that: Nice play with words – I saw what you did there with ‘ fixed and volatile’ 😛

You ended this chapter as they ended the movie Inception – wanting for more as soon as possible 🙂

Saurabh, glad you noticed my little joke 🙂

Next chapter coming soon !

Its amazing. How in a simple way you can not only explain but put (fix) things in our mind so that we will never forget it.

Just one question about Infy. Today Infosys results came so naturally yesterday i.e. on 20th July volatility of infosys must have been very high compared to today. Is it right and how it has affected the premium of strikes say at ITM, OTM?

Is it not similar to the Bollinger bands theory which also i suppose works on SD of 2%. Can you explain here meaning of 2%.

Volatility has a huge impact on option premiums – we will address all your questions in the upcoming chapters. Request you to please stay tuned till then!

This chapter ended soon:(

Trying to break it into bite size chapters to make the learning easy 🙂

great work, good going. Keep it up

Thanks Suresh 🙂

Sir, in the above example on what basis you mentioned the nifty probability range as 25% & 40% in 2016?

Its based on a statistical function called ‘

Cumulative Distribution Function‘ abbreviated as CDF. I will touch upon this topic soon.“Volatility is the up down movement of the stock market” – ROTFL this is exactly what I used to say before reading this chapter :)).

Also, nice example of moneyball movie, when I watched it few years ago I had a thought “Wish I find someone who uses this method is stock market”..and today it was like ….Voila! …

cannot wait for next chapters…. 🙂

Next chapter is up – http://zerodha.com/varsity/chapter/volatility-calculation-historical/

Thanks for the chapters. They r great.

When r the options strategies chapter gonna release ?

We have another 4 or 5 chapters in Options Theory…once it is through we will take up Options Strategies. Some maybe sometime in August we should get started on that module.

my friend, thank god NNT (Taleb) has not read this or else… he would label this as LUDIC fallacy, very high on B.S. let me try and explain wat I am saying. To predict the price of any extremistan product like stock/index based on a gaussian curve tools like standard deviation is waiting naked and bent over for a black swan to drill you.

black scholes and other jokers invariably price the OTM options very very cheap and cannot account for black swan. (Availability bias, what they have not seen, does not exist).

Now if you add a disclaimer that we will always create a spread (i.e buy further OTM option everytime we write one) then may be, just may be you can earn a living from that. Otherwise this Standard deviation tool to measure volatility is like collecting pennies in front of a steam roller. you will collect 99 cents in ur 99 attempts but that 1 black mother f:*king swan will take everything from ur family coffers in 01 go.

Manish Dhawan

Well, thanks for your comment Manish. Looks like you’ve read Mr.Lowenstein books besides NNT’s. I’ve ended the chapter with a line “

We figured the range in which Nifty is likely to trade in the next 1 year as 7136 and 9957 – but how sure are we? Is there any degree of confidence while expressing this range“.I hope you appreciate and recognize where this is leading to.

And just to add to the point Manish, I exactly know what you mean by “you will collect 99 cents in ur 99 attempts but that 1 black mother f:*king swan will take everything from ur family coffers in 01 go”. I’ve done those trades and I know how hard it can hit.

In the batsmen example, SD was added to and subtracted from average score to estimate the range. For nifty & TCS, the day’s price was used instead of average. Is it correct?

Yup….in fact chapter 17 (will upload next week) will give you exact details on how this can be done.

Really appreciate the efforts. Had to struggle a lot to put the information in right context and now everything is at click. Superb and Thank you

Good to know this Vishal! Please stay tuned we have lots more stuff coming up in Varsity!

If i need to trade nifty on 25th oct 2015 then to find standard deviation which data or data of nifty from which to what date is needed. and can you come up with a figure at which the nifty shall trade on 25th oct.2015

Use Nifty’s 6 months data to evaluate the SD, this will help.

can we use standard deviation to predict “DAY HIGH & DAY LOW ” purely for intraday trading ?

if yes , how ?

Tough call, but I do know ppl who use this.

can you share more details on the techniques used by those people , or connect me with any of them ?

does PI have any tool to calculate / use standard deviation function ?

PI does not have this utility. Btw, which people are you referring to?

i mean people who predict intraday high & low using math functions .

No one really reveals the strategy so it would be tough to identify who is doing what 🙂

Having said that you could check some of our successful traders here – https://zerodha.com/60-day-challenge/winners

“So the above calculations suggest that in the next 1 year, given Nifty’s volatility, Nifty is likely to trade anywhere between 7136 and 9957 with all values in between having varying probability of occurrence. This means to say on 15th July 2016 the probability of Nifty to be around 7500 could be 25%, while 8600 could be around 40%. ” —— how did we calculate the probability percentage of 25% & 40% respectively

This is using the distribution function, a slightly more complex topic at this stage. Hence decided not to develop on it further.

Can we use the same calculation for monthly.I mean taking the today’s closing price to Nifty as average and calculate the range for month too.Will it be simple returns or log.

Yes you can. Just multiply it with sqrt of 30. You can use log returns.

Sir We Are New Traders We are not Able To Guess When To Buy And When To Sell Please Guide Me, Iam Trading In Equity

Suggest you read through the module on Technical Analysis to develop a framework on when to buy and when to sell.

Hi Karthik,

“This means to say on 15th July 2016 the probability of Nifty to be around 7500 could be 25%, while 8600 could be around 40%.” The 25% and 40% what means (how to calculate % from which numbers)? how it comes? i can’t understand this line. Explain Please.

These are derived from the Standard Deviation numbers. For example 1 SD has the probability of 68%.

Dear Karthik,I have started reading your lessons since yesterday onwards and thank you very much for the simple, lucid, easily understandable style they are presented.

Thanks Kumar, I’m glad you liked the content.

Sir, for identifying the Volatility Can we use Bollinger band and ATR in option trading.

Yes you can as both these indicators are volatility derivatives.

How to find usdinr spot volatility for any particular date? i.e 29 th march 2016.

You will have to calculate this yourself.

Please tell me how to calculate “usdinr spot volatility for any particular date? i.e 29 th march 2016.”

Actually i want to forecast volatility using Garch (1,1) model but i can not understand how to calculate (n-1) th variance for forecast n th term. Please help.

Historical spot volatility can be calculated using normal STDEV function in excel. I’m afraid I wont be able to help you on the Garch (1,1) model, I’d suggest you post the query on willmot forums.

Thanks.

Karthik sir, you have used best and easy to understand examples to explain Volatility and Standard Deviation. Well explained. Thanks.

Cheers!

how did u arrive at d figure of 25% i.e. 7500 and 40% i.e.8600 in 2016 july … bcoz nifty 50 wasin the rangeof 8450 to 8650 most days of July 2016. how to do those calculations.

even the sd 1 is 68% which u mentioned in one of the above post … sorry but I too cud not get it.

Will it be revealed in next chapters.

will u pls explain. tks

I’d suggest you read the next chapter 🙂

Read d next two chapters on Std deviation .. got it right!! tks.

But still arriving at chances of 25% and 40% prices for nifty …still not clear how to calculate… though nifty was in that range of 8600 which is almost 95.5% accurate .. instead of 40%.

So v can say that volatility is diversion from the mean in percentage terms during the period or everyday when we talk about intraday.

Its based on a statistical Cumulative distribution function, dint discuss this as I thought it would be a bit over the top 🙂

I karthik,

I come with a strange situation today,while taking a positional option trade i was calculating the effect of theta.I entered 20(time left to expiry) in option calculator but i am confused here about weekends,holidays .Basically if it (20) will not take holidays and weekends into consideration so by this logic i would incur a loss of time value of 6 days(3 weekends). Am i missing something here? Help me out.

Its better to include weekends as well.

hello Sir, I have question related to IV related to call/put option.

I have observed lots of time when the spot price is going down then the overall ‘ IV ‘ of put option decrease but of call ‘IV’ option increased.Even i observed with the specific strike price this thing happened.

why its so happening sir. For ex: when Sunpharma is in bearish mode on 15-12-2016 then Put IV decrease but call IV increase..

Please explain me sir…Its great help for me.

Thanks In Advance.

Check this – http://www.thehindubusinessline.com/portfolio/technically/when-options-strike/article7596687.ece

Thanks alot sir for ur reply….I have one more doubt …plz clearify me…

I read article on investopedia related to option pricing…I see very interesting thing about the call/put price verses volatility…

Details:

its written there that when market risen then volatility decrease that the reason call buyer doesn’t get reward as price movement but in the case of put buyer ,the volatility increased when its moving in downward direction…its give good reward in that case….Even i have observed many times that i got good return on decline rather than upward…plz explain me sir why its happen like this..its very unusal..

Link: http://www.investopedia.com/articles/optioninvestor/05/020205.asp?lgl=bt1tn-baseline-below-textnote

Thanks in advance

Hi Avinash – all this and more has been explained in detail in this module 🙂

Sir I have just started reading your module on options theory…..its Awesome!!!!

Thanks a lot for such a great education initiative!!!

Happy you said so 🙂

Hi Karthik,

I wanted to get a better understanding of the option prices. As of today (8-Feb-2017) IOC (stock) is trading at 402. The IOC 17 Feb 370 CE (370 call options) premium is at 20.80. The bid and ask are 21.75/22.85 respectively. The options expire on 23 Feb 2017. The intrinsic value itself is around 30 Rs. I used the Black and Scholes calculator and even with zero volatility the calculated premium is around 33. I check the option chain and there is no IV values.

Is this happening because the call option is deep ITM? It is obvious that there is no time value but I was under the impression that at least it should trade close to its intrinsic value. The market depth shows that the vol:27000. That indicates that trades are taking place. In such a case, what does the trader need to do? Does he hold on till expiry? Will it be settled at intrinsic value?

On a quick note – I sincerely wanted to appreciate the fantastic content that you have created. It is solid and foundational and has really helped me grasp the big picture and the nuances at the same time. A big thank you!

Maybe I did not frame the question correctly. Why is the option trading below its intrinsic value in spite of being ITM? The difference is around 10 Rs. This is same with the other ITM options as well.

Like I mentioned earlier, it could be because of the dividends. Request you to kindly check.

Koustubh – Yes, the minimum option price for the 370 CE when spot is 402 is 32 or 33. However, I think IOC is giving out a big dividend (please double check this)…so whenever such a thing happens, the stock price goes down and hence the option prices.

Karthik,

Thank you for your reply. You are correct. There is announcement of 135% dividend by IOC and date is 9th Feb. What confused me was that the underlying price was cum-dividend and option premiums had already factored that in. I got it now. Thank you once again.

On a side note: when I was using the B&S calculator and trying to input the dividend field there was no change. I tried several ways like 135, 1.3 etc but the premium did not change. Do you know what is the issue? Sorry for bothering you.

Thanks for pointing, will check the B&S calculator.

Dear Karthik,

Could you please guide me how to calculate Implied volatility in excel I am doing on one project but not able to do that and I am halted because of that I know my project will add huge benefits to the option traders if anyone can help me how to calculate that IMPLIED VOLATILITY in excel would be grateful if you and any member in this forum help me out.

Thanks and Regards

It is a quant heavy topic. Have a look at this note put up by NSE – this will help you get started – https://www.nseindia.com/content/indices/white_paper_IndiaVIX.pdf

Hello karthik sir, How R you

Sir as told you i had encountered lots of misunderstandings about options some of them are i am writing today to ask you pls…..share some knowledge….it will be great help…..

In NSE option chain IV column incates volatility is this daily volatility or something else????

I’ve shared whatever I know on option in this and the next module 🙂

Btw, its IV is the implied volatility on the option chain.

Sir this is ANKIT above i forgot to mention my name sorry….

Cool, no problem.

Hello karthik sir….

For volatility caculation purpose i am assuming 5 days in a week , avg 21 days in a month, and 250 on an avg in a year

Is this right way to caculate volatility i am asking because in some place you put 365 days for a year and 1or 2 place you put around 250 days for a year

What to do and why….

Thanks for your valuable advice

I’d go with 252 days.

Hello karthik,

Thanks for these wonderful posts, very helpful really!

Require your help to understand the following related to volatility..

We have a volatility as given in NSE website for each stock/index, post applying the required formulas we can find the standard deviation likely for a one day.

Want to understand ideally these volatility calculations to arrive at the standard deviation for the day to be against the previous day close or previous day traded price or current day opening price ?

Thx!

Glad you liked them, Prasanna.

I guess volatility info is available only for F&O stocks. I’m assuming NSE updates this info on a daily basis considering the previous day;s close. Btw, standard deviation is the measure of volatility.

Glad to read this part in varsity, I have truly imbibed the standard deviation concept now after multiple failed attempts in school.

However, I would like to ask the real advantage for options in real world apart from making money from money. Is there a special purpose they fulfill?

You can use this concept in several places, including to figure out the likely expiry of the series.

Where did this come from? ” This means to say on 15th July 2016 the probability of Nifty to be around 7500 could be 25%, while 8600 could be around 40%.”

When you make predictive statements, it is also good to state the probability of occurrence. In statistics, it is called the degree of confidence.

What is the correlation of leverage in a portfolio with volatility and the return made / expected in a portfolio?

Sorry, I dont seem to understand your query. Can you kindly elaborate? Thanks.

Moneybaaallllllll <3 <3

*shuts varsity*

*starts watching moneyball*

P.S Are to hiring people who can work under you ???

Enjoy the flick!

No hiring anytime soon 🙂

Amazing content Karthik. I’m a proud Zerodha client since 3 years.

Just an observation – in math heavy chapters (where you have mentioned words like derivatives, calculus, SD), i see less number of people commenting as compared to other chapters. If i take the readers as a sample and extrapolate to participants in the stock mkt, this could be an indication of how many (less) people are actually trading options mathematically. I see a huge opportunity here.

i) One of the trainers once told us – maturity of Indian mkts is like European mkts of the 1980s. Many imbalances can be seen in our mkts. When asked for an example, he said – chances of making money selling far OTM options is higher here in Indian mkt than EU. 2-3 days after PNB fiasco was made public, i actually made 2k+ selling very far OTM PNB calls (IV was >100%). But i think our mkts too have huge algo participation and those algos should neutralise such imbalances. Do you agree that our mkts are immature?

ii) How to find % retail/FII/MF participation in our indices/stocks?

Many thanks

You are 100% right! Participation is highly skewed towards a speculative approach. However, this is should be a system based approach. Lots of HFTs already in place which fill random orders in the options markets. The objective of these HFTs is just to make the market for such orders.

why do people consider buying options when indiavix is high and selling options when it is low..? because if volatility is already high then option prices are high too..and if it is low then prices are cheap..so doesn’t it make more sense to buy when vix is low and sell when its high?

It does, but not many are aware of how to trade options 🙂