Due Date: 5pm Friday 17th October

This assignment is to be done individually. This means that you are required to answer the questions in the assignment on your own. The purpose of the assignment is to help you become more familiar with pricing options, using a specifically designed piece of software. The pricing of options is a very technical area, and most of us do not have the technical expertise to price options from first principals. Therefore in your working lives, if you do need to price options, then it will most likely be done for you via software.

The software we will be using is called ‘DerivaGem’, which has been specifically developed for the Hull et al (2014) textbook. You can download the software from the Moodle unit site under the ‘Additional Resources’ section.

Software Installation

Installation instructions for the DerivaGem software are quite straight forward. Please follow these simple steps.

Step 1. Go to the BFF5915 Moodle site, under ‘Additional Resources’ Section, click on the DerivaGem software and download the file ‘DerivaGem.zip’.

Step 2. Double-Click on the DerivaGem.zip file to bring up the Winzip utility program.

Step 3. In the Winzip utility program, on the ‘Home’ tab, there should be a button labelled ‘1-Click Unzip’. Press on the ‘1-Click Unzip’ button. The compressed files directories and folders should then be extract. Once Winzip has completed the extraction, then close the Winzip program.

Step 4. There should now be a folder in your directory named ‘DerivaGem’. Drill down through the DerivaGem folder until you come to the list of files;

The only file you need to be concerned with for the Assignment is ‘DG201.xls’.

Step 5. Double-click on file ‘DG201.xls’. This should bring up your Excel application program with the DG201.xls spreadsheet. If you look in the ‘readme.txt’ file that is located in the same directory as the DG201.xls file, it will tell you the versions of the Excel program that DerivaGem will work with. If you are not using any of these versions of Excel then you will need to get access to one of them in order to do the assignment.

Step 6. Once you have opened the ‘DG201.xls’ program in Excel, that final step you need to perform is to ensure that Excel has ‘Enabled Macros’. This will allow the Excel program to use the Visual Basic for Applications (VBA) code on which DerivaGem is based. If you are using Excel 2010, the enable macros should appear as a ‘strip’ across the top of your spreadsheet when you first enter the DG201.xls program. You just need to press the enable button. If you a later version of Excel you will need to go to the macro security settings and enable the macros. Once you have enabled the macros, DerivaGem will be available for use.

Step 7: Click on the Excel spreadsheet tab ‘Equity_FX_Index_Futures_Options’. This is the only spreadsheet you will need to complete the assignment.

Format of Assignment and submission requirements:

The assignment consists of seven questions. You should attempt all questions. Responses to the questions should be presented according to the Q Manual. The Q Manual provides guidelines on the presentation of submitted materials. Submission of the assignment will require both an electronic submission AND a hard copy submission:

The submission of an electronic copy can be done via Moodle (further technical details will be provided prior to the submission date.)

The submission of a physical hard copy will be done via a designated assignment box located in the 3rd Floor Foyer of Building H. (further details to be provided closer to the submission date.)

ASSIGNMENT QUESTIONS

Please note: Most, but not all of the material needed to answer these questions will be covered in lectures. You are postgraduate students so it is expected that you will be able to seek out the material necessary to complete the questions using your own initiative. All information is available within the textbook, even though it may not all be covered in lectures.

Q1. Use DerivaGem to calculate the value of an American put option on a non-dividend-paying stock when the stock price is USD 30, the strike price is USD 32, the risk-free rate is 5%, the volatility is 30% and the time to maturity is 1.5 years. (Choose ‘Binomial American’ for the option type and 50 time steps.) (See Chapter 9 for supporting theory and materials)

a) What is the option’s intrinsic value?

b) What is the option’s time value? c) What would a time value of zero indicate? What is the value of an option with zero time value? d) Using a trial and error approach, calculate how low the stock price would have to be for the time value of the option to be zero. (use 50 and 500 time steps)

Q2. Consider an option on a stock when the stock price is $41, the strike price is $40, the risk-free rate is 6%, the volatility is 35% and the time to maturity is one year. Assume that a dividend of $0.50 is expected after six months. (See Chapter 10 for supporting theory and materials)

a) Use DerivaGem to value the option assuming it is a European call.

b) Use DerivaGem to value the option assuming it is a European put.

c) Verify that put–call parity holds.

d) Explore, using DerivaGem, what happens to the price of the options as the time to maturity becomes very large. For this purpose, assume there are no dividends. Explain the results you get.

Q3. Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30% and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the following positions. In each case provide a table showing the relationship between profit and final stock price. Ignore the impact of discounting. Each table should have two columns, ‘Stock Price Range’ and ‘Profit’ (See Chapter 11 for supporting theory and materials)

a) a bull spread using European call options with strike prices of $25 and $30 and a maturity of six months b) a bear spread using European put options with strike prices of $25 and $30 and a maturity of six months

c) a butterfly spread using European call options with strike prices of $25, $30 and $35 and a maturity of one year d) a butterfly spread using European put options with strike prices of $25, $30 and $35 and a maturity of one year

e) a straddle using options with a strike price of $30 and a six-month maturity f) a strangle using options with strike prices of $25 and $35 and a six-month maturity.

Q4. Consider a European call option on a non-dividend-paying stock where the stock price is AUD 40, the strike price is AUD 40, the risk-free rate is 4% per annum, the volatility is 30% per annum and the time to maturity is six months. (See Chapter 12 for supporting theory and materials)

a) Calculate , and for a two-step tree

b) Value the option using a two-step tree.

c) Verify that DerivaGem gives the same answer. d) Use DerivaGem to value the option with 5, 50, 100 and 500 time steps.

Q5. Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is six months, the volatility is 30% per annum and the risk-free interest rate is 10% per annum. Two equal dividends of 40 cents are expected during the life of the option, with ex-dividend dates at the end of two months and five months. (See Chapter 13 for supporting theory and materials)

a) Use Black’s approximation and the DerivaGem software to value the option.

b) Compare your approximate answer in a) against the American option price calculated using the Binomial model with 100 time steps.

Q6. It is 4 February. July call options on corn futures with strike prices of 260, 270, 280, 290 and 300 cost 26.75, 21.25, 17.25, 14.00 and 11.375, respectively. July put options with these strike prices cost 8.50, 13.50, 19.00, 25.625 and 32.625, respectively. The options mature on 19 June, the current July corn futures price is 278.25 and the risk-free interest rate is 1.1%. There are 135 days to maturity (assuming this is not a leap year, then 365 days in the year). Calculate implied volatilities for the options using DerivaGem using 500 time steps. Comment on the results you get. (hint: these options are American) (See Chapter16 for supporting theory and materials)

a) Complete the table for the calculated implied volatilities

Strike Price

Call Price

Put Price

Call Implied Vol

Put Implied Vol

260

26.75

8.50

270

21.25

13.50

280

17.25

19.00

290

14.00

25.625

300

11.375

32.625

Q7. Consider a one-year European call option on a stock when the stock price is $30, the strike price is $30, the risk-free rate is 5% and the volatility is 25% per annum. (See Chapter 17 for supporting theory and materials)

a) Use the DerivaGem software to calculate the price, delta, gamma, vega, theta and rho of the option.

b) Verify that delta is correct by changing the stock price to $30.1 and re-computing the option price.

c) Verify that gamma is correct by re-computing the delta for the situation where the stock price is $30.1.

d) Carry out similar calculations to verify that vega, theta and rho are correct.