Financial Derivatives 600 words + tables+ excels sheet

This is a group essay and my part is Implied volatility. That is what i need done.

I have submitted the coursework description, I only need my part done (IMPLIED VOLATILITY).

I have submitted a sample implied volatility essay so you can see how it is meant to be done

One of my members finished his part- PART 1 ”put call parity” you can look at that. I have submitted that,

The tables and graphs made on excel can u also send that too me.

SCHOOL OF SOCIAL SCIENces 8

Student Coursework Instructions

Course Title: Financial Derivatives

Course Code: C39SN

Author: Dr Boulis Ibrahim

2

COURSEwork instructions_ STUDENT C39SN Financial derivatives

INDEX OPTIONS

Group Project (in groups of five each)

Caution

Plagiarism is a serious offense that, in extreme cases, may lead to expulsion from the University. You should interact only within your group, and groups should work independently. You may not seek any help from anyone outside your group. Any evidence of accessing such help, copying from other peer groups or any other form of collusion or plagiarism will be raised to the Disciplinary Committee for investigation and your mark withheld until the issue is resolved.

Coursework Submission Instructions (please follow these carefully)

Deadline for submission is provided by course instructor/tutor.

You need to submit a hard copy of your project that has:

a. A cover page that contains the following information:

(i) Title: C39SN2 Coursework: Financial Derivatives Project

(ii) Names or IDs of all group participants;

(iii) Group number (the number of your group on the Vision sign-up list)

(iv) Word count

b. A filled-in and signed Coursework Group Self-Assessment Form (included with the course package and obtainable from Vision).

c. A CD or USB that contains a copy of your Excel caclualtions (this can be put in an envelope and attached to the report by a sticky tape).

Coursework

An index option is a financial derivative that gives the holder the right, but not the obligation, to buy or sell 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 options (CBOE symbol code: XEO) and American-style (CBOE symbol code: OEX) options on the S&P 100.

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 and are posted under Links in Vision).

This is a demanding coursework. You are advised to start as early as possible, and follow the following 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, 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 (from CBOE);

c) the dividend yield on the S&P100 index (from 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 if you download the data during different 15-minute intervals they 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 ‘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 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 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, but you are required to show how you performed the detailed calculations by attaching your Excel sheet in a USB or a CD to the hardcopy submission, or by attaching snapshots to Excel sheets in an Appendix to your report (in this latter case, please highlight, reveal or type the equations used in each cell).

STEP 1

Downloading Option Data

You will need to download prices for puts and calls as in the matrix in the table below (under Choice of Options for Analysis). First read the section below and when you are ready to download the data login to the CBOE website www.cboe.com. Choose the delayed quotes tool under the Quotes & Data tab. When the Delayed Quotes window opens, enter on the left panel the symbol codes OEX or XEO and press submit. You will see some data with a Filters by ribbon at the top. Choose Filters: Volume all; expiration type all; options range all; and choose the expiration month that will determine the maturity of all the options you download. Once these filter values are set press View Chain. A table of information on options available will then display. It is best to just copy and paste in Excel the entire table that comes up from the website (including the information right at the top line above the table, which pertains to the level of the S&P 100 index itself, not the options). You need to follow this procedure twice, once to extract a table of OEX options, and another for XEO options. Once you have all the data in Excel you can then pick and choose the options that you want t o focus on in the analyses (specific exercise prices and preferably options that have some volume you can use the filters too to help you narrow down this choice).

Beside prices, you will need to also 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.

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 month of each contract. (Please check the full contract specifications are available under the ‘Products’ tab in the CBOE website, http://www.cboe.com/products/indexopts/oex_spec.aspx ).

Choice of Options for Analysis

Consider pairs of put and call options with each pair having 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 a pair that are at-the-money, and another that are out-of-the-money, so 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 fit the description of the table below.

American (OEX)

European (XEO)

Same or close

ATM call

ATM call

Exercise price and time to maturity

All options should be active (i.e., showing some volume and/or open interest, but preferably both). But this is not a strict requirement.

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 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 (i.e., ones that are showing some volume of trade and/or open interest, but preferably both, although this is preferable it is not a strict requirement). 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. 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 are: 8:30 a.m. – 3:15 p.m. Central Time (Chicago time).

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 (http://www.bloomberg.com/markets/rates/index.html). 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 Treasurys then you may need to interpolate the yield to end up with one that matches the maturity of the options.

Alternative sources of interest rate data (the first is more official) are:

1. http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/Historic-LongTerm-Rate-Data-Visualization.aspx

2. If you have access to Datastream use the following mnemonics (instrument code): FRTB3M for 3-months Treasury Bills or FRTB6M for 6-months treasury bills. Datastream is available in the pc labs in Mary Burton Building only (for Edinburgh Campus Students). It is found under icon names Avance4 or Advance5. But it takes a bit of getting used to.

Proper calculations of theoretical option prices

Note that in calculating theoretical option prices you will also need a value for the annual dividend yield on the S&P100 index and an estimate of volatility of the index. Read the relevant sections below on how to obtain these.

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. (Enter VXO on the right hand side box that appears when you click the ‘Quotes&Data’ tab and choose ‘Delayed Quotes’ in the CBOE website). Note, this must be obtained within the same 15-minute interval during which you download the option data.

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.

If you can’t find dividend yield try:

http://markets.on.nytimes.com/research/markets/mutualfunds/snapshot.asp?symbol=OEF

Enriching Your Analysis?

As a base case, do your analysis using mid prices = (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 from those using the ask prices should be a reflection of the effect of transaction costs, and hence you can discuss these effects.

Step 1 Coursework Requirements

The requirements are:

a. Investigate (calculating and checking) 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).

STEP 2

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 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.

Step 2 Course Requirements

Binomial Pricing

Using the VXO estimate of volatility 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 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 their exercise price. Compare these values of implied volatility with each other. Also compare them with the value obtained from the VXO index. Discuss the reasons for any differences from each other and from the VXO value. (More emphasis and marks will be given to discussion.)

Notes

CBOE symbol explanation

In CBOE each option has a symbol, and here is an example of how to read these symbols.

Example: OEX1327A600 – E (2013 Jan 600.00 Call)

OEX is the CBOE symbol of the instrument (in this case S&P 100 American option).

The next two digits, 13, stand for the year 2013.

The next two digits and letter stand for the expiration date and expiration month

For call options A stands for January, B for February, C for March,..etc. to L.

For put options M stands for January, N for February, O for March,..etc. to X.

So 27A is 27th of January, and since the letter A is used then it indicates a call option.

The following number, 600, stands for the strike price of the option.

The letter E after the hyphen indicates the market in which the option is traded, in this case CBOE.

Alternatively, if you click on the symbol the website will provide you with more detail.

Full description of the contract details, including the exact maturity day can be found by clicking the ‘Products’ tab in the CBOE website,

http://www.cboe.com/products/indexopts/oex_spec.aspx

Marking Scheme

In general the overall course requirements and the marking scheme are as follows:

1. Discussions of whether the put call parity holds for actual market prices of European and American options? (10 marks)

2. Interpret differences between American and European option prices (5)

3. Interpret differences between Black and Scholes and actual prices (5)

4. Construct 3-step binomial (10)

5. Compare binomial and actual prices interpret any differences (10)

6. Calculate B&S prices. (5)

7. Compare B&S with binomial prices discuss differences,

B&S vs Binomial (6)

American vs European (6)

ATM vs OTM (8)

8. Calculate implied volatility (5)

9. Discussion of the plot of implied volatility against moneyness or X. (10)

10. Compare implied vol with each other and with VXO interpret differences (10)

Spreadsheets (10) 5.

IMPLIED VOLATILITY

XEO Calls

E

Model

Actual

Lower/ Raise

ITM CALL XEO

1145

42.95752053

37.25

Lower

16.03%

ATM CALL XEO

1170

24.11158

19.8

Lower

14.09%

OTM CALL XEO

1245

5.808925074

0.55

Lower

11.58%

XEO PUT

E

Model

Actual

Lower/Raise

ITM PUT XEO

1245

79.1626283

75.20

Lower

14.89%

ATM PUT XEO

1170

22.4982

19.55

Lower

15.10%

OTM PUT XEO

1145

16.35510301

11.90

Lower

17.0812%

15

OEX Calls

E

Model

Actual

Lower/ Raise

(%)

OEX ITM CALL

1145

42.47860321

38.00

Lower

17.10%

OEX ATM CALL

1170

28.508307

20.5

Lower

14.90%

OEX OTM CALL

1245

5.741794078

0.60

Lower

11.86%

OEX PUT

E

Model

Actual

Lower/Raise

ITM OEX PUT

1245

79.85495443

78.20

Lower

18.62%

ATM OEX PUT

1170

27.654376

21.05

Lower

15.94%

OTM OEX PUT

1145

16.6356427

13.05

Lower

17.8355%

Black and Scholes Model, defines volatility as the “standard deviation of the continuously compounded return on an underlying asset”. Volatility may be estimated using two approaches: historical volatility and implied volatility. The former assumes that volatility prevailing over recent past holds true in the future. In accordance to implied volatility, the price of a option is a reflection of the current volatility of the underlying asset. The implied volatility is “the “volatility that makes the theoretical value of the option equal to the actual value of the option”” (Chance and Brooks, 2013).

The implied volatility is calculated using a trial and error approach, however Excels Goal Seek method, allows easy computation of accurate volatility.

As shown above, the implied volatility amongst all the options are lower than the VXO value. It is inferred that options with high implied volatility are costlier as against options with lower implied volatility. Differences, in these volatilities may be viewed as differences in the relative cost of the options (Chance and Brooks, 2013).

Volatility is a representation of volatilities of underlying assets across option expirations. Thus, it is plausible, that across varying time periods, volatility could be different. Usually,” longer the time to maturity of an option, the higher the volatility, however this may not always hold true” (Chance and Brooks, 2013).

5.1.Implied Volatility amongst options and VXO

In both American (OEX) options and European (XEO) options, the implied volatility is seen to be lower than the VXO, which indicates that these options are undervalued. The implied volatility, as discussed in the previous section, offers investors insight into an options cost. The general rule being, options with higher implied volatility are more expensive. Thus, there proves to be an opportunity for arbitrage; by buying at a lower price and selling them at higher prices.

16

In the American (OEX) put option we can see that ITM and OTM options have the highest implied volatility in comparison to the lower implied volatility of the ATM option. This is because these options (ITM and OTM) represent higher risk of large movements on the underlying asset and are thus are more expensive.

5.2.Smiles

“For a specified exercise price, the relationship between implied volatility and option expiration is recognised as structure of volatility (Don and Robert)”. The implied volatility smile is the correlation between implied volatility and the exercise price for a given option. Due to the 1989 market crash the implied volatility surface has become more skewed. Risk variables like Delta and Gamma can be approximated using the non-flat implied volatility surface (Don and Robert).

XEO CALL

17.00%

Volatility(%)

16.00%

15.00%

14.00%

Implied

13.00%

12.00%

11.00%

10.00%

1100

1150

1200

1250

Excercise Price

Implied Volatility

XEO PUT

17.50%

17.00%

16.50%

16.00%

15.50%

15.00%

14.50%

1100 1150 1200 1250

Excercise Price

OEX CALL

Implied Volatility

18.00%

17.00%

16.00%

15.00%

14.00%

13.00%

12.00%

11.00%

10.00%

1100 1150 1200 1250

Excercise Price

The above graphs represent a reverse skew or otherwise known as a volatility smirk; the lower the strike price higher the implied volatility. Investors in such a scenario heavily trade on ITM call options and OTM put options. It occurs when investors are concerned about the market i.e. they are expecting a crash in the market. When such a scenario is anticipated, investors are often pressured to buy OTM puts to hedge their investment portfolios (Bamford, 2014).

17

OEX PUT

Implied Volatility

19.00%

18.50%

18.00%

17.50%

17.00%

16.50%

16.00%

15.50%

1140 1160 1180 1200 1220 1240 1260

Excercise Price

In the above graph, implied volatility is clearly higher in ITM and OTM options but is lower at the ATM option. This implies there is a higher demand for ITM and OTM options. Investors tend to buy ITM options as these have a higher intrinsic value, whereas OTM options similarly have higher extrinsic values than ATM options. Speculative markets often see the presence of volatility smiles. Upon occurrence of large volatility shifts, investors buy ITM options to steady gains whereas they buy OTM options for speculative reasons (Bamford, 2014).

18

23 PUT CALL PARITY

The Put Call Parity defines a price relationship between Call options, Put Options, and Underlying stock. The idea was first introduced by finance journalist Hans Stoll in 1969, in his paper The Relationship Between Put and Call Prices in 1969. It is considered as one of the most important principle in Option Pricing (Hecht, 2019).

There are two types of Options

1. XEO (European Options): the exercise of these options will only happen at the options expiry date.

2. OEX (American Options): the exercise of these options can be at any time during their life.

The Put Call Parity tends to work perfectly only with European options compared to the American Options (Hull, 2017).

The formula applied to check the relationship between European put and call options with same strike prices and expiry is

C + X = P + S

Here,

C: Call Premium

P: Put Premium

r: risk free interest rate

T: Time to maturity in terms of year.

X: Strike price of Call and Put Option.

S: Initial Price or Current Price of underlying.

Put Call Parity Testing

(XEO) European Put Call Parity

EUROPEAN PUT CALL PARITY

OPTIONS

ATM

ITM

OTM

X

640

560

1190

PV(X)

617.2154797

540.0635448

1147.635033

C

627.7

706.5

140.35

LHS

1244.91548

1246.563545

1287.985033

S

1271.2

1271.2

1271.2

P

2.375

63.75

1.45

RHS

1273.575

1334.95

1272.65

DIFFERENCE

-28.65952028

-88.38645524

15.33503261

OUTCOME

DOES NOT HOLD TRUE

DOES NOT HOLD TRUE

HOLDS TRUE

In the European Put Call Parity the options that hold true are the ones where the LHS is equal to the RHS, or they must be almost similar to each other with a small difference. In this scenario, from the above mentioned table we can conclude that OTM with X=1190 holds true. The other two figures were ITM with X=560 and ATM with X=640, since they have a slightly larger gap comparing to OTM. It does not hold true. This offers an arbitrage opportunity for the investors.

(OEX) American Put Call Parity

AMERICAN PUT CALL PARITY

OPTIONS

ATM

ITM

OTM

C

636.8

716.1

143.45

D

2.16%

2.16%

2.16%

PV(D)

0.020831022

0.020831022

0.020831022

X

640

1190

560

PV(X)

617.2154797

1147.635033

540.0635448

LHS

1254.036311

1863.755864

683.5343758

S

1271.2

1271.2

1271.2

P

2.375

64.85

1.475

RHS

1273.575

1336.05

1272.675

DIFFERENCE

-19.53868925

527.7058636

-589.1406242

OUTCOME

DOES NOT HOLD TRUE

HOLDS TRUE

DOES NOT HOLD TRUE

In the American Options the Put Call Parity holds true when LHS is greater than or equal to RHS. As from the table above we can see that LHS > RHS when X=1190 when the option is in the money (ITM).

The Put Call Parity is denoted by the equation,

P (So, T, X) = C (S0, T, X) So + X (1+ r)-T

Applying to the formula,

1.45 = 140.35 1271.2 + 1147.635033

2.375 < 16.785033
As we can notice here that 16.785033 is greater, the call is over-priced. The investor is advised to sell his over-priced call in this situation and buy a put of the same value, as it becomes a perfect hedge.
We could use an alternative equation i.e.,
S0 = C (So, T, X) P (So, T, X) + X (1 + r) -T
Applying to the formula,
1271.2 = 140.35 1.45 + 1147.635033
1271.2 < 1286.535033
When using the above formula, it indicates that the call, put and bonds are overpriced, so the investor should sell theses and buy the shares as they are underpriced. Therefore, to check arbitrage profits Put Call Parity variations can be used.
Hecht, A., 2019.Options: The Concept Of Put-Call Parity. [online] The Balance. Available at:

Hull, J., 2017.Fundamentals Of Futures And Options Markets. 10th ed. Harlow: Pearson Education Limited.