Posted: October 29th, 2022

Fianance

You need to make submissions of two files for your coursework, in two separate links on Canvas:

1. A Word file of the full written project report.
This file should be named according to the following convention:
“C39SN_CW_Word_Group X’
where you replace ‘X’ with the group number you signed up to on Canvas.

This Word file should contain the following:
a. A cover page that contains the following information:
(i) Title: ‘C39SN2 Coursework: Financial Derivatives Project’
(ii) Group number (the number of your group on the Canvas sign-up list)
(iii) Names or IDs of all group participants;
(iv) Word count
b. A filled-in and signed ‘Coursework Group Self-Assessment Form’ (included with the course package and obtainable from Canvas). You can fill this in, scan it or photograph it, and append the scan or photograph at the end of your Word file.

2. An Excel file of your Excel calculations. Name this file:
“C39SN_CW_Excel_Group X”
where you replace ‘X’ with your group number, as on the Canvas sign-up list. Make sure you do not forget to identify your group number in the file name

Coursework Description and Objectives
An index option is a financial derivative that gives the holder the right, but not the obligation, to buy (Call) or sell (Put) a basket of stocks (e.g., S&P 100 or FTSE 100) at an agreed-upon price (strike price) on a certain future date (European), or any time up to that date (American).

Index Options on US stock indices are traded at the Chicago Board Options Exchange (CBOE, see www.cboe.com).

This coursework is an investigation of the pricing of options on the S&P 100 index.
There are European-style and American-style options on the S&P100. CBOE identifies:
European-style S&P100 options with the symbol code: XEO, and
American-style S&P100 options with the symbol code: OEX.

The full contract specifications can be found at
https://www.cboe.com/tradable_products/sp_100/sp_100_options

Approach
(You need to read Chapter 4 of Chance and Brooks (Option Pricing: The Binomial Model) prior to attempting the coursework. Links to all websites relevant to this coursework are given below.)

This is a demanding and challenging numerical coursework. You are advised to start as early as possible, and follow these steps in order:

1. Read Chapter 4 of Chance and Brooks (Option Pricing: The Binomial Model).
2. Read the instructions below to understand the requirements.
3. Read the instructions again and extract a list of ALL the data that you need to download (from CBOE, Bloomberg and etfdb.com, see below). This data will consist of
a) Prices (option premiums), exercise prices and time to maturity of European and American call and put options on the S&P100 index (from CBOE);
b) the level and volatility of the S&P100 index or of NASDAQ 100 (from CBOE);
c) the dividend yield on the S&P100 index (from www.etfdb.com); and
d) interest rate data (from Bloomberg).
Note that you will need to download all this data within the same 15-minute interval, since CBOE’s website provides data with a 15-minute delay, and updates its data every 15 minutes. If you download different pieces of the data during different 15-minute intervals then the data will not match with each other.
4. Prior to downloading the data logon to the relevant websites, including the CBOE, and familiarise yourself with them, the option symbols (also explained below), and with the number of ‘mouse clicks’ that you need to perform in order to download the data. This will prepare you well to act fast enough to download all the data you require within the same 15-minute interval window.
5. Once the data is downloaded, you will need to do calculations based on this data and, therefore, will require extensive use of Excel. You will also need to repeat these calculations for different options, and hence automation of the calculations would help a lot. To automate the calculations, you are strongly advised to think carefully about the structure and organisation of your spreadsheet prior to implementation.
6. Assume a 365-day year throughout.

Main Requirement:
A report of 1500 to 2000 words in length (excluding figures, tables, references and appendix) that contains a response to the questions that feature under the two STEPS below. Your report should be stand alone and contain all what you want to show the reader and all what you are going to be assessed upon.

However, and for verification purposes (including the External Examiner as well as the identification of collusion or plagiarism) you are required to show how you performed the detailed calculations by submitting your Excel spreadsheet(s). The Excel sheet is to verify that you have done the calculations, but do not use it as a compliment to your Word report. Your Word report should be stand alone and should not refer the reader to the Excel sheet for further detail.

As noted above, there will be two separate submission links on Canvas: one for the Word file of the report, and one for the Excel Sheet. Please do not submit your Word report to the link for the Excel sheets, or vice versa, and make sure you name these files as instructed.

STEP 1

Step 1 Coursework Requirements
The requirements are:
a. Investigate (by calculating, checking and interpreting) whether the put-call parity holds for the actual market prices of both the European and the American options. Interpret the results (more emphasis and marks will be given to interpretation).

b. Calculate the difference between the actual market prices of the American and European options that have the same exercise price and maturity. Interpret the results (more emphasis and marks will be given to interpretation).

c. Calculate the theoretical Black and Scholes prices of all the selected European and American options and compare these theoretical values with the actual market prices of the options. Interpret any differences (more emphasis and marks will be given to the interpretation).

Guide to carrying out Step 1 Requirements

Downloading Option Data
(PLEASE access cboe during Chicago trading times – 8:30am to 3:15pm CET Chicago Time)

You will need to download prices and exercise prices for puts and calls. First read the section below and when you are ready to download the data login to the CBOE website www.cboe.com.

Downloading steps (see picture as a guide):

1. In www.cboe.com, click on ‘Data’ then ‘Quotes Dashboard’ and make sure you press the tab ‘Options’. The page should look like the picture above.
2. In the entry panel next to ‘Search’ enter the symbol OEX for American options or XEO for European options, and press ‘Search’.
3. There are two main ways to download the data. The first is to download all the data they have on OEX or XEO options by using the ‘View Chain’ green button on the same page. At the bottom of the table you will be able to see a ‘Download CSV’ link that will download the entire table into an Excel readable file. The second method of downloading parts of the data for specific options only is to use the filters that appear in a ribbon to the left of the ‘View Chain’ button. You can choose the appropriate filer settings. For example, you can choose to select options of a certain volume, expiration type, option range (ITM, ATM or OTM), size, or expiration date. It is probably best to just download all options using ‘View Chain’ and then choose whichever ones you want to do the analysis on. But the filters are useful to investigate and to download ITM, ATM or OTM active options (see further instructions below on how and what options to choose).

You need to follow the above more than once: once to extract a table of OEX options, another to extract the same for XEO options, and yet again to extract the level of volatility of the S&P 100 index (by entering the symbol VXO or ^VXN in the Search button – see instructions below).

Once you have all the data in Excel you can then pick and choose the options that you want to focus on in the analyses (specific exercise prices and preferably options that have some volume or open interest – here you can use the filters to help you narrow down this choice). Choose your options by using the matrix in the table below (under Choice of Options for Analysis), which is not prescriptive but provided as a guide only.

Note that beside prices, you need to note the time you downloaded the data, in order to calculate the time left to maturity for these options, and the level of the S&P 100 index (i.e., the spot price of the index) at the time of your downloading the options data. These last two pieces of information appear towards the top of the table anyway, so it is good practice to copy the entire table together with the information that appears at the top of it, which also includes the value of the S&P100 index (spot price) at the time.

Choice of Options for Analysis
You need to consider pairs of put and call options. Each pair should have the same (or close) strike price and maturity. You need pairs of American-style (OEX) and pairs of European-style (XEO) options. For each exercise style choose at least three pairs: one in-the-money (ITM), one at-the-money (ATM), and one out-of-the-money (OTM). Do this such that American and European pairs that are at-the-money have the same (or close) time to maturity and exercise price. Do the same for in-the-money and out-of-the-money options. You should end up with a choice of options that roughly fit the description of the table below.

American (OEX) European (XEO) Same or close Comments
ATM call ATM call Exercise price and time to maturity All options should preferably be active (i.e., showing some volume and/or open interest, but preferably both). But this is not a strict requirement, and indeed is not always possible. In this case try to get options that loosely or closely fit the requirements. Should this be impossible, then choose from what is available.
OTM call OTM call Exercise price and time to maturity
ITM call ITM call Exercise price and time to maturity
ATM put ATM put Exercise price and time to maturity
OTM put OTM put Exercise price and time to maturity
ITM put

ITM put Exercise price and time to maturity

For ATM options choose options that are closest at-the-money (i.e., with a strike price closest to the level of the S&P100 index at the time of downloading the data). Then choose options with exercise price on either side of that of the ATM options such that a pair is ITM and a pair is OTM. Here the filter ‘Options Range’ can be very useful.

Choose a particular maturity between 1 week and 1 year for all options that you will use. In all your choices of strike price and maturity be guided by the table above (flexible) and by options that are most active (see ‘comments’ column in the table). You will quickly notice that options with specific time to maturity only are active (1 Week or 1 Month), choose a maturity that has active options.

If you do not see price, volume and open interest data next to an option it could be that you are either accessing the website when CBOE is not open, or the options are not heavily traded during the time you are downloading the data. In this case choose other options and access the website during Chicago opening times. Avoid lunch time in Chicago. Take a note of the exact time and date of your data download, or simply copy the screen information. The Chicago Trading Hours and holidays are posted at https://www.cboe.com/about/hours/us-options. For US options the main trading session times are 9:30am – 4:15pm Eastern Time, or 8:30am -3:15 Central Time, difference is due to changes in light saving hours. Chicago is usually one hour behind Eastern Time. So always download data between 8:30 and 3:15 Chicago local time, and try to avoid lunch hour.

Calculating time to maturity
With regard to calculations of the maturity dates of the options, bear in mind that the XEO and OEX option contracts at CBOE mature on the Saturday following the third Friday of the maturity/expiration month of each contract. (Please check the full contract specifications are available under the ‘Products’ tab in the CBOE website,
https://www.cboe.com/tradable_products/sp_100/sp_100_options).

Interest Rate Data
You also need an annualised risk-free rate to calculate ‘theoretical’ fair prices for the options using the Black and Scholes and the Binomial pricing models. Use the ‘interest rates’ market data provided by Bloomberg ( https://www.bloomberg.com/markets/rates-bonds/government-bonds/us ). Select an appropriate yield for the risk free interest rate. This usually is the yield of a Treasury Bill (or zero curve) that has a maturity closest to that of the option(s) you downloaded. You are advised to have the same maturity for all your options, but note that if you download options with different maturities you may need more than one interest rate to match. If the maturity of your options is in-between the maturities of two listed Treasury Bills, or has a maturity less than that of the shortest Treasury Bill, then you may need to ‘interpolate’ or ‘extrapolate’ the yield to end up with one that matches the maturity of the options.

Alternative source of interest rate data:
1. https://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=billrates. Use the ‘Coupon Equivalent’ rates, and read the comments at the bottom of the table.

Dividends and Volatility and the proper calculation of theoretical option prices
Note that in calculating theoretical option prices by the Black and Scholes as well as the Binomial Model, you will also need a value for the annual dividend yield on the S&P100 index and an estimate of the annual volatility of the index. Read the relevant sections below on how to obtain these.

Estimating Annualised Dividend Yield (Downloading index dividend yield data)
To calculate theoretical option prices you also need an estimate of the annualised dividend yield of the S&P100 index at the time of downloading the price data. Search for a reasonable value, and although this can be difficult, try http://etfdb.com/index/sp-100-index/dividends/ which gives the dividend yield on an Exchange Traded Fund (ETF) that tracks the S&P100.

An alternative source of the dividend yield is to try:
https://www.dividend.com/etfs/oef-ishares-s-p-100-etf/

Estimating Volatility of the Index (Downloading index volatility data)
An estimate of sigma (volatility) for the index can be obtained by reading the value of the volatility index that has a symbol ^VXO but this has been decommissioned. Instead, Enter ^VXN on the ‘Search’ button in the Quote Dashboard in cboe (please also see the text in bold typeset below). Note, this must be obtained within the same 15-minute interval during which you download the option data.

CBoE has decommissioned VXO for the S&P100, so use ^VXN as an alternative symbol. It is a volatility index on the NASDAQ 100 stock index which is not the same as the S&P100 but close.

Enriching Your Analysis?
Bid and Ask Prices
As a base case, do your analysis using mid quotes = (bid+ask)/2. Or you can enrich your analysis by performing calculations on bid and on ask prices separately. The difference between the results using the bid prices from those using the ask prices should be a reflection of the effect of ‘transaction costs,’ and hence you can discuss these effects.

Maturity
You can also enrich your analysis further by redoing the calculations for options with different maturities. In this case you may choose three or four sets of options, with each set having a different maturity (in this sentence a ‘set’ refers to all the options in the matrix mentioned above). The idea is that you may see patterns across maturities as well as exercise prices, and by discussing these patterns across maturities and exercise prices your analysis will be enriched.

STEP 2

Step 2 Course Requirements

Binomial Pricing
Using the VXO estimate of volatility as input evaluate the call and put options using the three-step binomial tree previously constructed for the index level (here you need to have your tree calculations automated so you can evaluate all options). Compare the binomial option prices with the actual option prices observed in the market and discuss the reasons why you do, or do not, observe any differences. (More emphasis and marks will be given to discussion.)

Black and Scholes versus Binomial
Using the VXO estimate of volatility as input calculate the Black-Scholes prices of the put and call options. Compare these values with the actual market prices and with those obtained by the Binomial model. Discuss possible reasons for any differences (i.e., compare Black-Scholes versus Binomial, American versus European, puts versus calls, ATM versus OTM). (More emphasis and marks will be given to the discussion of each of these comparisons)

Implied volatility
By trial and error, find the value of the volatility parameter at which the Black and Scholes price equals the observed actual market price for each option. The value of volatility at which the observed actual market price equals the Black and Scholes price is known as ‘implied volatility’. Create plots of the implied volatility of the options against the exercise prices of these options. Compare these values of implied volatility with each other. Also compare them with the value obtained from the VXO or VXN index. Discuss the reasons for any differences from each other and from the VXO or VXN value. (More emphasis and marks will be given to discussion.)

Guide to Step 2 Requirements

Binomial Model Setup Features
Using the ‘binomial model’, build a binomial tree for the index level with three time steps, so that the overall time horizon is equal to the maturity of the options selected (i.e., divide the time remaining to maturity into three equal intervals).

With regard to the binomial calculations choose the upward and downward size of price movement as a function of the volatility of the index level (i.e., function of sigma of the index). You can use the equations provided by Chance and Brooks for up (u) and down (d) parameter movements as functions of sigma, also provided in the lecture material. For estimates of sigma see Estimating Volatility of the Index above. You will also need an estimate of the dividend yield on the index. For these see Estimating Annualised Dividend Yield above.

*** END OF INSTRUCTIONS

—————————–

You must submit two files for your coursework via two separate links on Canvas:

1. A Word document containing the whole written project report.

This file should be named according to the following convention:

“C39SN_CW_Word_Group X’

where you replace ‘X’ with the group number you signed up to on Canvas.

This Word file should contain the following:

a. A cover page that contains the following information:

(i) Title: ‘C39SN2 Coursework: Financial Derivatives Project’

(ii) Group number (the number of your group on the Canvas sign-up list)

(iii) Names or IDs of all group participants;

(iv) Word count

b. A filled-in and signed ‘Coursework Group Self-Assessment Form’ (included with the course package and obtainable from Canvas). You can fill this in, scan it

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