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JANUARY EXAMINATIONS 2023

ACFI310 Derivative Securities

TIME ALLOWED: 24 Hours

INSTRUCTIONS TO CANDIDATES

 This is a 24-hour assessment. You are allowed to use a calculator. You should not use the internet to ‘google’ your answers.

 For questions in Part A of the final exam, all parts are compulsory. Please type your answers using Microsoft Word.

 For questions in Part B, you should answer TWO questions out of THREE. For those two questions you choose, you should answer all parts.

 Please include your student ID but not your name in your answer as your work is marked anonymously.

Student declaration:

I confirm that I have read and understood the University’s Academic Integrity policy. I confirm that I have acted honestly, ethically and professionally in conduct leading to assessment for the programme of study. I confirm that I have not copied material from another source nor committed plagiarism nor fabricated data when completing the attached piece of work. I confirm that I have not previously presented the work or part thereof for assessment for another University of Liverpool module. I confirm that I have not colluded with any other student in the preparation and production of this work. I confirm that I have not incorporated into this assignment material that has been submitted by me or any other person in support of a successful application for a degree of this or any other University or degree awarding body. Students who require sympathetic marking should ensure that they attach the Sympathetic Marking Indicator to the first page of the document prior to submission

Section A: This section is COMPULSORY. Students have to answer ALL the parts to Question 1.

Question 1

a) Explain the relationship between volatility and European call option prices. (6 marks)

b) Distinguish between the forward price and initial set up value of a forward contract. (6 marks)

c) Distinguish Value-at-Risk with Expected Shortfall. (6 marks)

d) Explain why Delta hedging a written option involves a “buy high, sell low” trading rule. (6 marks)

e) What is the volatility curve if the left tail is heavier than the lognormal distribution and the right tail is less heavy than the lognormal distribution? (6 marks)

[Total marks: 30]

Section B: This section has THREE questions. Students should answer only any TWO questions in Section B.

Question 2

a) The current price of a stock is $25. The continuous compounded risk-free rate is 10% per annum. The stock pays continuous dividend yield of 2% per annum. An investor enters into a long position in a six-month forward contract on this stock today.

i. Calculate the forward price and the value of the forward contract today.

ii. Three months later, the price of the stock is $26. The risk-free rate is still 10% per annum and the dividend yield is still 2% per annum. What is the forward price three months later? What is the value of the long position in the forward contract three months later?

(10 Marks)

b) The current stock price is $35.00 and a six-month European call option with a strike price of $37.00 costs $1. An investor has $7000 to invest.

i. What are two alternative trading strategies for this investor?

ii. In which situation can both strategies make the same profits?

iii. Distinguish different situations and compare the profit or loss of each strategy in each situation.

(15 Marks)

c) A stock price is currently $75. It is known that at the end of three months, it will be either $70 or $80. The risk-free rate is 8% per annum with continuous compounding. A put option on this stock has a strike price of $72 and it will expire three months later. What is the value of this put option today? Use the no arbitrage argument.

(10 Marks)

[Total marks: 35]

Question 3

a) Suppose that a portfolio is worth $20 million and the S&P 500 is at 1000. The portfolio has a beta of 3.0, the risk-free interest rate is 8% per annum, and the dividend yield on both the portfolio and the index is 1% per annum. What options should be purchased to provide protection against the value of the portfolio falling below $16 million in one year’s time?

(15 Marks)

b) Consider a position consisting of a $1,200 investment in gold and a $1,500 investment in silver. Suppose that volatilities of these two assets are 12% p.a. and 18% p.a., respectively. The coefficient of correlation between their returns is 0.5. What is the 10-day 99% value at risk for the portfolio? What is the diversification benefit for the portfolio?

(10 Marks)

c) The spot price of the stock is $30. The volatility of the stock is 20% p.a. The continuous compounded risk-free rate is 5% p.a.

i. Calculate the value of a European call option to buy this stock at $28 in 3 months if the underlying stock pays no dividend.

ii. Calculate the value of a European call option to buy this stock at $28 in 3 months if the underlying stock pays a continuous dividend yield at 2% p.a.

iii. Explain the relationship between dividend yield and European call and put options.

(10 marks)

[Total marks: 35]

Question 4

a) The current stock price is $50. Over each of the next two three-month periods, it is expected that the stock price will increase by 15% or decrease by 10%. The continuously compounded risk-free rate of interest is 8% per annum.

i. What is the risk-neutral probability of an up state?

ii. What is the price for a six-month American put option with a strike price of $51? Please show the two-step binomial tree.

iii. Calculate delta at time 0. What does delta indicate in terms of hedging?

(15 Marks)

b) A European call option and put option on a stock both have a strike price of $25 and an expiration date in 6 months. Both sell for $2. The risk-free rate is 6% per annum. The current stock price is $21, and it pays continuous dividend yield at 2% per annum. Identify whether there is any arbitrage opportunity.

(10 Marks)

c) A financial institution has the following portfolio of over the counter options on sterling:

A traded option is available with a delta of 0.5, a gamma of 2.5 and a vega of 1.6.

i. What position in the traded option and in sterling would make the portfolio both gamma neutral and delta neutral?

ii. What position in the traded option and in sterling would make the portfolio both vega neutral and delta neutral?

(10 marks)

[Total marks: 35]