25620 DERIVATIVE SECURITIES - ASSIGNMENT
QUESTION 1- Amelia Francis (a high net worth client) has a well-diversified stock portfolio worth $200,000,000. The portfolio has a beta of 1.2 and the dividend yield on the portfolio is 1.85% per annum with simple compounding. The S&P 500 index is currently at 2100 and the dividend yield of the index is 2.10% per annum with simple compounding. The risk-free interest rate is 4.7% per annum with simple compounding.
(a) Describe the strategy that provides insurance against the portfolio declining below $170,000,000 in three months. Please round the strike price to the nearest five index points.
(b) Calculate the insurance premium. Assume that the volatility of the index is 22% per annum. For the purposes of part (b) only, assume that the dividend yield on the index and the risk-free rate when expressed as simple rates are approximately the same as continuously compounded rates.
(c) Calculate the gain or loss of the strategy, if the level of the market in three months is 1600. Discuss the outcome of the insurance strategy.
(d) Calculate the gain or loss of the strategy, if the level of the market in three months is 2600. Discuss the outcome of the insurance strategy.
QUESTION 2- On April 9, the theoretical futures prices in AUD per EUR and the AUD continuously compounded interest rate for different maturities are given in the table. Also, 1 EUR is currently worth AUD 1.5699.
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Maturity
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Futures exchange rate in AUD per 1 EUR
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AUD interest rate
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EUR interest rate
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3 months
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1.5721
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2.04%
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|
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6 months
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1.5692
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2.21%
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|
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9 months
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1.5673
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2.69%
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12 months
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1.5670
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2.98%
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(a) Calculate the EUR interest rate implied by these prices and complete the table (correct to four decimal places).
(b) On April 9, a financial institution offers a nine-month currency forward contract on the Euro at 1.5815 AUD/EUR. Identify the arbitrage opportunity available, provide a detailed description of the arbitrage strategy and calculate the arbitrage profit made on AUD 1,000,000.
(c) On April 9, a well-known investor (Hugh Nguyen) has just entered into a long position on a twelve-month currency futures contract on the Euro. What is the initial value of his position? On July 9, he decides to close out his position. Calculate the profit/loss made on his futures transactions, given that the spot exchange rate on July 9 is 1.5024 AUD per 1 EUR, the AUD interest rate is 2.30% and the Euro interest rate is 2.97% both continuously compounded.
(d) Under the terms of a cross-currency interest rate swap, a financial institution has agreed to receive 5.7% per annum (annual compounding) in EUR with semi-annual payments and to pay 6-month LIBOR+2% per annum (annual compounding) in AUD on a notional principal of AUD60 million for three years. At the time of the contract initiation, 1 EUR was worth AUD1.4528.
On September 30, 1 EUR is worth AUD1.3811 and the swap has a remaining life of twenty months. Assume that the AUD interest rate is 2.55% per annum and the EUR interest rate is 3.12% per annum with continuous compounding for all maturities. The 6-month LIBOR rate four month ago was 3.45% per annum. Calculate the value of the swap to the financial institution. Discuss the exposure of the financial institution to credit risk in the occasion of bankruptcy of the counterparty company.
QUESTION 3- A stock index with a dividend yield of 3.3% per annum with continuous compounding is currently standing at 1995.10 and has a volatility of 18% per annum. The risk-free interest rate is 4.9% per annum with continuous compounding. You are interested in calculating the theoretical option price to see whether there is mispricing in the market. Use a four-step binomial tree to calculate the price of:
(a) A European nine-month put option with a strike of 1850. Calculate also the value of the option by using the Black-Scholes formula. Compare and comment.
(b) An American nine-month put option with a strike of 1850.
(c) A European down-and-out barrier put option with a strike of 1850 and knockout barrier of 1700 maturing in nine months. A down-and-out put option gives the holder the right to sell the underlying asset at the strike price on the expiration date so long as the price of that asset did not go below a pre-determined barrier during the option's lifetime. When the price of the underlying asset falls below the barrier, the option is "knocked-out" and no longer carries any value.
(d) As a valued member of the UTS alumni, you have been asked to be a guest lecturer in Derivative Securities (25620). Vinay remembered how much you enjoyed 25620, and has asked you to spend 30 minutes to come and discuss the role of credit derivatives in causing the global financial crisis to his cohort of students. Please provide a summary brief of your lecture. Your summary brief must be no longer than half-an-A4 page.
QUESTION 4- The gold futures price of a twelve-month futures contract is currently trading at 1217.30. The risk-free rate of interest is 4.5% per annum with continuous compounding. Consider the following market prices of nine-month options on gold futures. The price value of an option point is $100.
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Call price ($)
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Implied Volatility
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Call (K=1190)
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7670
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Call (K=1220)
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6060
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Call (K=1250)
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4730
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Call (K=1280)
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3640
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Put price ($)
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Implied Volatility
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Put (K=1190)
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4950
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Put (K=1220)
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6330
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Put (K=1250)
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7990
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Put (K=1280)
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10200
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(a) Use Excel's GoalSeek Tool function to calculate the implied volatility of these options (correct to 4 decimal places).
(b) You have spent a day shadowing a colleague (James Zhang) on the proprietary trading desk. James showed you his techniques for identifying arbitrage opportunities in the gold futures market. Use European put-call parity to identify the maximum arbitrage profit. Provide a detailed description of the strategy which will allow you to lock in the maximum arbitrage profit. Hint: in your arbitrage strategy assume you close out your futures position after nine months.
(c) Your manager (Charlene White) is retiring and next week will put you in charge of managing three clients (Mr Bond, Mr Holmes and Miss Watson) stock portfolios. From your knowledge of derivative securities prepare a brief which you will pitch to your clients to explain the benefits of trading in the derivatives market. Your brief must be no longer than half-an-A4 page. You may refer to a recent working paper which can be downloaded from: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2480870.