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Project: Volatility Smiles in Options Markets

This project is about volatility smiles in options markets. I need to collect recent two years' daily or weekly data and generate the volatility smiles, discuss the changes and explain why those changes have taken place.

I have done some reading and writing on the literature side of the volatility smiles but I still have some question marks in my mind on the empirical side of the subject, data analysis, graphs etc. I'd like to be 100% clear before I collect data and start the analysis.

1. My supervisor at the university asked me to generate the volatility smiles (construct the curve) on options of stocks on a particular sector e.g. tobacco or metal or chemical companies and discuss the changes over the time

1a) Which sector is the easiest to use, or, which one do you recommend, why?

1b) Are these companies listed on the S&P 500 index or on any other index, which? I think options on S&P 500 are European. Are the options on these sectors mentioned above European or American?

1c) How do I sort out which companies are in chemical, or metal, or in tobacco sector? Are they listed or categorized in any paper, book or web site, or on an index?

1d) Or, do these sectors have their own index, if so, what is the index called?

2. I will be collecting the data at my university. As far as I understand from my readings I will first need to calculate the implied volatilities using Black-Scholes for each option on the same underlying with the same time to expiration but different strike prices and plot them against strike prices?

2a) Am I right? If not, would you please write your thoughts?

2b) Do I plot implied vols against normal strike prices or percentage of strikes prices? (In some studies they used percentage of strike prices, I did not quite understand what percentage they used and why? Is it x/s (exercise price/ stock price)? But stock price may change over time where exercise price remains the same until expiry? How and why do they do that? If it was not important I'd rather go for the easiest way and use simple strike prices. What do you think?

2c) Some studies used percentage of spot. What is spot or percentage of spot or forward price?

2d) Shall I use daily or weekly data? Which one is the easiest? How do I use this data?

3e) Do I need to categorise the options i.e. according to their moneyness level before I calculate the implied volatilities or before I construct the smile curve?
If so, how do I do that?

3f) Shall I use put or call options? Why?

3. DATA
Given that I will be using Black/Scholes model to calculate the implied vols I think I will need the following data. Would you please make it very clear in what form I will be using this data set and correct my thoughts if any of them is wrong?

3a) Option prices (they may change every day, I think I'd better use weekly Friday's closing prices. What do you think?) Do I use all the options or do I exclude any of them such as at the money options? Do I need only in the money and out of the money options?

3b) Current stock prices (this may also change everyday, how do I sort out all these changes?)

3c) Exercise price of the options. I guess they remain the same once quoted or do they change as well?

3d) Risk free interest rate (this is also not constant, what do I do?) As these options are written on US index I guess I need to use the interest rate applied in US don't I?

3e) Standard deviation of return on stock (this is not constant either how do I deal with this?) is it annualised historic volatility going into Black/Scholes formula?

3f) Dividends: I just do not know how to use them would you please explain?

3g) Time remaining till expiration (I think I'd better start from the 1st day the option is written. Do I put time to expiration in days in the formula? Such as 90 days or in any other way, would you please explain? Are expiration dates standard for all the companies/options, or does each company have different length to expiration how does this system work? What is the minimum and maximum expiration time? How do they verify expiration dates and how often do they write options, is there a certain rule or a certain date these companies list the options? Maximum how many options a company can write? I feel completely lost about how this system works.

3h) Do I need any other data?

3i) Is there any sources that contain implied volatilities, which is already calculated for above-mentioned sectors so that I do not have to go through the hassle of collecting all above-mentioned data.

Could you please make very clear in what form/ format how do I use the above mentioned data? If possible can you please demonstrate the curve with an example by using false data ?

4. CONSTRACTING THE VOLATILITY SMILE CURVE:

4a) Can you please explain step by step how do I construct the curve in the computer? Do I do it in excel? What functions do I use? Step by step please.

4b) Is there any web site (i.e. prof. john hull's site) or software that can automatically calculate implied vols and draw volatility smile curves?

4c) As I will be arguing that the underlying asset distribution is not lognormal as it was assumed by black/scholes and explain that the lognormal distribution maybe one of the reasons for the existence of volatility smiles, I thought I'd better draw the lognormal and actual probability density (distribution figure) functions in a graph form (on the same curve) and compare them. I think I need to do this according to the implied volatility distribution. Would you please explain how do I do that?

4e) where can I get the above mentioned data apart from the university?

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1. My supervisor at the university asked me to generate the volatility smiles (construct the curve) on options of stocks on a particular sector e.g. tobacco or metal or chemical companies and discuss the changes over the time

1a) Which sector is the easiest to use, or, which one do you recommend, why?

For your research, I suggest that you pick a sector that contains many stock options, instead of restricting yourself to Tobacco, Chemical or Metal companies. Following is a list of top 4 sectors that make up the S&P 500 Index along with the number of stocks in each sector: Consumer Discretionary - 86, IT - 83, Financials - 82, Industrials - 60. If you pick such sectors that are widely constituted in an Index, finding information will be easier and you will have a better possibility of finding up-to-date information. Though you will find stock options in narrow sectors such as tobacco/chemical companies, most of them may not be widely traded resulting in stale data. Doing research on stale data may not be a good idea. You should give the same reasoning to your professor for picking up one of these widely followed sectors instead of chemical/metal/tobacco companies. For the rest of this explanation, I assume that you picked the IT sector.

1b) Are these companies listed on the S&P 500 index or on any other index, which? I think options on S&P 500 are European. Are the options on these sectors mentioned above European or American?

There is difference between Options on Index (Index Options) and Options on Stock (Stock Options) here. Index Options are usually European and the underlying for these options is an Index (such as the S&P 500 Index).

But I believe you are required to construct smiles for stock options. The underlying instruments for Stock options are stocks (the individual components that make an index). Stock options are usually American.

For the purposes of this study, you don't have to be concerned about Index Options (thus European Options) at all. You just need a resource that will give you data on stock options in the IT sector of S&P 500 Index. As I mentioned in the earlier answer, S&P 500 Index has 83 IT companies listed in it and finding data will be very easy. Most of these stocks are widely traded and you can lay your hands on the latest data.

1c) How do I sort out which companies are in chemical, or metal, or in tobacco sector? Are they listed or categorized in any paper, book or web site, or on an index?

I am giving you two links for this purpose.
LINK 1:
http://www2.standardandpoors.com/NASApp/cs/ContentServer?pagename=sp/Page/IndicesConstituentsPg&b=4&r=1&l=EN&s=6&ig=48&i=56&xcd=500&f=1&az=all&pg=1&bp=0&so=1&dt=20-JUN-2003&fs=45

LINK 2:
http://www.cboe.com/TradTool/Symbols/SymbolDirectory.asp

Link 1 has all the IT stocks in S&P 500 Index along with their symbols. Link 2 gives you a list of stock options based on these IT stocks. List 2 also has stock options on many other stocks, but you may want to pick options on stocks that constitute the Index only (i.e., those stocks present in link 1). Such stock options tend to be widely traded compared to others.

1d) or, do these sectors have their own index, if so, what is the index called?

You don't need a specific index as you will find the required stocks if you follow links 1 and 2 that I gave in the previous response.

2. I will be collecting the data at my university. As far as I understand from my readings I will first need to calculate the implied volatilities using Black-Scholes for each option on the same underlying with the same time to expiration but different strike prices and plot them against strike prices?

2a) am I right? If not, would you please write your thoughts?

Your understanding is right. Also, for each plot, it might not be a good idea to mix calls and puts. For a better understanding, you may do both separately and compare one against the other. (My guess is that they will be symmetric to each other, but you will learn the right thing yourself).

2b) Do I plot implied vols against normal strike prices or percentage of strikes prices? (In some studies they used percentage of strike prices, I did not quite understand what percentage they used and why? Is it x/s (exercise price/ stock price)? But stock price may change over time where exercise price remains the same until expiry? How and ...

Solution Summary

The solution presents very practical and understandable explanations dealing with the options markets asked by the student including suggestions for where to find more data.

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