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Derivative Securities
Section A: This section is COMPULSORY. Students have to answer ALL the parts to Question 1.
Question 1
a) Discuss how margin account works in futures markets. (6 marks)
b) Explain the relationship between dividend and American call and put options (6 marks)
c) Discuss how the Black-Scholes option pricing formulas can be used in pricing American call options. (12 marks)
d) What is the volatility curve if the left tail is less heavy than the lognormal distribution and the right tail is heavier 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) An investor owns 500 shares of stock A. The current stock price is $40. A three-month put option with a strike price of $36 costs $3. Explain how investor can hedge using the put option, and compare investor’s portfolio value with or without hedging three months later in different situations.
(15 marks)
b) The GBP/USD exchange rate is 1.3900 (i.e., the price of 1 GBP is 1.3900 USD). The exchange rate annualized volatility is 20%. The GBP and USD risk-free rates are 2.5% per annum and 2% per annum, respectively.
i. What is the price of a three-month European call option, which enables the buyer to buy GBP at the strike price equal to 1.4000 (i.e., the price of 1 GBP is 1.4000 USD)?
ii. What is the price of a three-month European put option, which enables the buyer to buy GBP at the strike price equal to 1.4000 (i.e., 1 GBP can be exchanged to 1.4000 USD)? Use the put-call parity for calculation.
(10 marks)
c) Verify the put-call parity relationship under no-arbitrage assumption for European call and put options on stock with continuous dividend yield.
(10 marks)
[Total marks: 35]
Question 3
a) The current stock price is $15. Over each of the next two six-month periods, it is expected that the stock price will increase by 10% or decrease by 12%. The continuously compounded risk-free rate of interest is 5% per annum.
i. What is the risk-neutral probability of an up state?
ii. What is the price for a one-year American put option with a strike price of $13.5? Please show the two-step binomial tree.
iii. When should this one-year American put option be exercised?
iv. Calculate delta at time 0.
(15 Marks)
b) Consider a position consisting of a $2,500 investment in asset A and a $3,000 investment in asset B. Suppose that daily volatilities of these two assets are 1.8% and 2.2%, respectively. The coefficient of correlation between their returns is 0.65. What is the 10-day 95% value at risk for the portfolio? What is the diversification benefit for the portfolio?
(10 Marks)
c) A European call option and put option on a stock both have a strike price of $12 and an expiration date in 3 months. Both sell for $2. The risk-free rate is 5% per annum. The current stock price is $11, and it pays a dividend of $0.5 one month later. Identify whether there is any arbitrage opportunity.
(10 Marks)
[Total marks: 35]
Question 4
a) A stock price is currently $24. It is known that, at the end of six months, the stock price will either increase to $27 or decrease to $21. The risk-free rate is 12% per annum with continuous compounding. What is the value of a six-month European call option with the strike price of $25? Use the no-arbitrage argument.
(10 marks)
b) A financial institution has the following portfolio of over the counter options on sterling:
| Type | Position | Delta of option | Gamma of Option | Vega of option |
| Call | -800 | 0.8 | 1.5 | 1.7 |
| Call | -1000 | 0.5 | 1.2 | 1.3 |
| Put | -600 | -0.6 | 2.2 | 0.5 |
| Call | -200 | -0.3 | 1.9 | 0.7 |
A traded option is available with a delta of 0.8, a gamma of 2.0 and a vega of 1.55.
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)
c) A stock is expected to pay a dividend of $0.5 per share in two months and in five months. The stock price is $40, and the risk-free rate of interest is 8% per annum with continuous compounding for all maturities. An investor has just taken a short position in a six-month forward contract on the stock.
i. What are the forward price and the initial value of the forward contract?
ii. Three months later, the price of the stock is $43 and the risk-free rate of interest is still 8% per annum. What are the forward price and the value of the short position in the forward contract three months later?
iii. Discuss differences between forward contracts and futures contracts.
(15 marks)
[Total marks: 35]
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