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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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