**MODULE: BK7023: Financial Engineering **

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**DURATION: 24 Hours**

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Instructions to Candidates

This paper contains **FOUR** questions

Answer ALL **FOUR** questions.

Each question carries **25** marks

**Formula sheet **and** Z-table** is at the end of the paper

Please type your answer in a **Word document** and **submit via Canvas**

If you do not have laptop/PC/Mac, please write down answers **on paper** and take photos of your answers using smart phone. Then upload the photos via Canvas.

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**Number of Pages: 7**

**Plus front page**

**Question 1**

- Explain what is basis. Discuss how basis risk can be minimised in a hedging strategy.

**(4 marks)**

- On March 1 a commodity’s spot price is $60 and its August futures price is $59. On July 1 the spot price is $64 and the August futures price is $63.50. A company entered into futures contracts on March 1 to hedge its purchase of the commodity on July 1. It closed out its position on July 1. What is the effective price (after taking account of hedging) paid by the company?

**(4 marks)**

- Explain what is a stack and roll strategy. Discuss the advantages and disadvantages of using stack and roll as a hedging strategy.

**(5 marks)**

- A short forward contract that was negotiated some time ago will expire in six months and has a delivery price of $52. The current forward price for the six-month forward contract is $55. The six month risk-free interest rate (with continuous compounding) is 2.5%. What is the value of the short forward contract?

**(4 marks)**

- A speculator takes a long position in a futures contract on a commodity on November 1, 2019 to hedge an exposure on March 1, 2020. The initial futures price is $60. On December 31, 2019 the futures price is $61. On March 1, 2020 it is $64. The contract is closed out on March 1, 2020. Each contract is on 1000 units of the commodity. What is the gain of the trade in 2020?

**(4 marks)**

- Explain the difference between forward and futures contracts. Discuss why a futures contract is considered less risky than an equivalent forward contract.

**(4 marks)**

** [Total 25 marks]
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**Question 2:**

- Companies A and B have been offered the following rates per annum on a $40 million five-year loan:

Fixed Rate | Floating Rate | |

Company A | 2.5% | LIBOR+0.1% |

Company B | 3.2% | LIBOR+0.3% |

Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a swap that will net a bank, acting as intermediary, 0.02% per annum and that will appear equally attractive to both companies. Please demonstrate your results on a diagram.

**(4 marks)**

- An interest rate swap has three years of remaining life. Payments are exchanged annually. Interest at 3% is paid and 12-month LIBOR is received. An exchange of payments has just taken place. The one-year, two-year and three-year LIBOR/swap rates are 2%, 3% and 4%. All rates an annually compounded. What is the value of the swap as a percentage of the principal when swap and LIBOR rates are the same?

** (5 marks)**

- Explain the comparative advantage of using swap contracts. What are the criticisms of the comparative advantage argument?

**(4 marks)**

- A deposit account pays 2.4% per annum with continuous compounding.
- If Interest is actually paid quarterly, how much interest will be paid in the first quarter on a $4,200 deposit?
- If Interest is actually paid monthly, how much interest will be paid in the first month on a $4,200 deposit?

- If Interest is actually paid daily, how much interest will be paid in the first day on a $4,200 deposit?

**(4 marks)**

- Should an American call option on a stock that pays no dividend be exercised early? Explain you answer.

**(4 marks)**

- Briefly explain what a volatility smile is. Also briefly discuss what causes volatility smiles in equity options.

**(4 marks)**

** [Total 25 marks]**

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**Question 3**:

- Call options on a stock that expire in three months are available with strike prices of $10, $15, and $20. Their prices are $4.4, $2.1, and1, respectively. Explain how the options can be used to create a butterfly spread. Construct a table showing how profit varies with the stock price for the butterfly spread. You should also demonstrate the strategy’s profits and losses on a diagram.

**(5 marks)**

- A five-month call option with a strike price of $12 costs $1.6. A put option with the same underlying, the same maturity and the same strike price costs $2.1. The risk free rate is 2.8%, and the underlying price is $9. The dividend yield is 3.0%. What are the arbitrage possibilities? Which securities (or combinations of securities) should you trade? Demonstrate the arbitrage profit that is obtainable.

** (5 marks)**

- A stock price has an expected return of 9% and a volatility of 40%. The current price is $10. Assuming the stock price is lognormally distributed, what is the probability that a European call option on the stock with an exercise price of $12 and a maturity date of seven months will be exercised?

**(4 marks)**

- Calculate the delta of an at-the-money six-month European call option on a non-dividend-paying stock when the risk-free interest rate is 1.5% per annum and the stock price volatility is 32% per annum (All rates are continuously compounded).

**(4 marks)**

- An interest rate is 6% per annum with continuous compounding. What is the equivalent rate with semiannual compounding?

**(3 marks)**

- Despite its popularity in the financial industry, the Value at Risk (VaR) methodology has been criticised for underestimating potential risk. Briefly explain the major drawbacks of using VaR.

**(4 marks)**

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** [Total 25 marks]**

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**Question 4**

- A company has a $35 million portfolio with a beta of 1.25. The one-month futures price for a contract on the S&P index is 950. Futures contracts on $250 times the index can be traded. What trade is necessary to achieve the following (Indicate the number of contracts that should be traded and whether the position is long or short)?

- (i) Eliminate all systematic risk in the portfolio
- (ii) Reduce the beta to 0.3
- (iii) Increase the beta to 1.8

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**(5 marks)**

- Calculate the value of a six-month at-the-money European call option on a stock index when the index is at 400, the risk-free interest rate is 2% per annum, the volatility of the index is 25% per annum, and the dividend yield on the index is 0.5% per annum. All rates are continuously compounded.

**(5 marks)**

- A stock’s price is currently $15. It is known that at the end of six months it will be either $16.5 or $13.5. The risk-free interest rate is 1.5% per annum with continuous compounding. What is the risk-neutral probability that the stock price will be $16.5?

**(4 marks)**

- An investor has $2,000 invested in stock A and $5,000 in stock B. The daily volatilities of A and B are 1.5% and 1% respectively and the coefficient of correlation is 0.8. What is the one day 99% VaR of the portfolio?

**(6 marks)**

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- Explain what is the gamma of an option? How does gamma impact the hedging of a portfolio?

**(5 marks)**

** [Total 25 marks]**

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**Formula Sheet**

Formulas for European Options on Assets with a Known Yield

Put-call parity:

*c* + *Ke ^{ -rT }*=

*p*+

*S*

_{0}

*e*

^{ –qT}Optimal hedge ratio:

Number of contract for hedging equity portfolio:

Call option lower bound:

*c*≥

*S*

_{0}

*– Ke*

^{-rT}Put option lower bound:

*p*≥

*Ke*

^{-rT}–S_{0}

Risk neutral probability of an up movement in a binomial tree:

**END OF EXAMINATION PAPER**