# What are the positions in the stock and 55-strike call in order to make the portfolio both delta and gamma neutral?

Long 200 shares of stock

• Long 200 puts with a strike of 50 and maturity of three months

• Short 200 calls with a strike of 60 and maturity of three months.

All of the options are European options and each option is on 1 share.

This portfolio information and information on interest rate and dividend are contained in the attached Excel file (rows 1-4).

Prices of various options (including the ones held in your portfolio) are listed in the Excel file (see rows 6-12).

Requirements:

a. Based on the option prices, compute their implied volatility using the Black-Scholes model. Record your answers (in sheet Part1) in range D7:D12 (6×0.25=1.5 marks).

b. Based on the computed volatilities and information provided, compute delta, gamma and vega of the 6 options. Record your answers in range E7:G12. (18×0.25=4.5 marks).

c. What is the objective of the strategy employed in your portfolio? Write the answer (A, B, C or D) in cell B14 (0.25 mark).

A. Income

B. Insurance

C. Long volatility

D. Short volatility

d. Compute the portfolios value and write the answer in cell B15 (0.25 mark).

e. Compute the portfolios delta and write the answer in cell B16 (0.25 mark).

f. Compute the portfolios gamma and write the answer in cell B17 (0.25 mark).

g. Compute the portfolios vega and write the answer in cell B18 (0.25 mark).

h. Use delta and gamma to approximate the portfolios value if the stock price suddenly increases by $3. Write the answer in cell B19 (0.25 mark).

i. What is the additional share position in order to make the portfolio delta neutral? E.g. -10 means short 10 shares, +15 means long 15 shares. Write the answer in cell B20 (0.25 mark).

j. If the stock price a week later changes to 54.55 (from 55), what would be the additional share position (compared to 1 week ago) to make the portfolio delta neutral again? Assume volatility, interest rate and dividend do not change. Write the answer in cell B21 (0.25 mark).

Now back to the current date.

k. What are the positions in the stock and 55-strike call in order to make the portfolio both delta and gamma neutral? Write the answers in cells B23 and B24 respectively (2×0.25=0.5 mark).

l. What is the net cash flow of achieving delta and gamma neutrality for the portfolio? Write the answer in cell B25 (0.25 mark).

m. What are the positions in the stock and 55-strike put in order to make the portfolio both delta and gamma neutral? Write the answers in cells B27 and B28 respectively (2×0.25=0.5 mark).

n. What is the net cash flow of achieving delta and gamma neutrality for the portfolio using the strategy in (m)? Write the answer in cell B29 (0.25 mark).

o. Compute the delta of a bull spread using calls with strikes of 55 and 60. Write the answer in cell B30 (0.25 mark).

p. Compute the gamma of a butterfly spread using calls with strikes of 50, 55 and 60. Write the answer in cell B31 (0.25 mark).

Note:

– After opening the Excel file for the first time, widen column A of sheet “Part1” to completely see the questions.

– Use 4 decimal places for delta, gamma, vega, volatility.

– Use 2 decimal places for portfolio values.

– Round the number of shares and options to the nearest 1.

– Use “- “ for short positions. Example: -10 means short 10 shares/options. 10 means long 10 shares/options.

– Make sure your answer is worksheet “Part1” in the file. Do not use this worksheet for anything else apart from recording the answer. Do not insert rows or columns in this worksheet.