25620 Derivative Securities – Group Assignment
[15 + 12 + 13 + 20 = 60 marks]
Due by 5pm on Friday 25 October 2024
Please submit your assignment (as a single Word or PDF file) via Canvas.
Congratulations, you have been hired as a financial analyst at a leading investment bank following your studies at the University of Technology Sydney. It is a significant honour to work for such a prestigious institution and you have fought off tough competition for the job. The recruitment team selected you for your personable character, your analytical mind, your ability to solve problems, work in teams and your ability to get the job done. It is your second month on the job, and you must rely, now more than ever, on your knowledge from studying Derivative Securities. In what follows you will be asked, amongst other things, to investigate the behaviour of the current commodity futures market, to evaluate currency swaps and future positions, to advise concerned clients on the different hedging strategies available to them, and to explain the pricing of complex derivatives, including cryptocurrency options. All in a day’s work for a UTS graduate.
Submission requirements:
. Be succinct and concise.
. Part marks may be awarded, therefore, show your working out for each question.
. For each calculation question provide all workings. Make it obvious what your final answer is (i.e., before showing your working out, please show your final answer for each question).
. Hand-written answers will not be accepted.
. Include a reference list and in-text citations where necessary.
Statement on plagiarism:
. Refer to the subject outline to familiarise yourself with UTS’s statement on plagiarism. In a nutshell, do not copy another team’s solutions or another resource, do not share your assignment with another group, and do not submit work which is not your own.
. Please note that when you submit the assignment you are declaring that your assignment submission is plagiarism-free. Note that even if one member of the team demonstrates plagiarism (without the knowledge of the other team members), this will lead to problems for all members. You do not want to be responsible for causing issues to your teammates.
. Last semester several teams were caught for plagiarism. All of these students did not think they would be caught, and they seriously regret their actions. Some examples of penalties for plagiarism include sanctions, having to redo the subject, fees incurred to retake the subject, and so on.
Question 1
[3 + 2 + 3 + 2 + 2 + 2 + 1 = 15 marks]
Recent geo-political events and inflation concerns have seen large swings in commodity prices in recent years. Unsurprisingly, many of your clients are trying to understand these markets and your first task, should you choose to accept it, is to analyse the current oil and aluminium (aka aluminum) markets and their forward curves. Such forward curves provide important information about market conditions for traders and investors. Prices for NYMEX Crude Oil futures and COMEX Aluminum futures contacts are presented in Table 1 below. We also know that on 3 September 2024 the spot price of crude oil was USD73.57 per barrel and the spot price of aluminum was USD2,349.50 per metric ton.
Table 1: NYMEX Crude Oil and Aluminum futures prices observed on 3 Sep 2024
(Source: https://www.barchart.com/futures/quotes/)
(a) First, a client asks you to plot the forward curves (as of 3 September 2024) for both the NYMEX Crude Oil futures and the COMEX Aluminum futures contracts. You should plot these on the same set of axes with ‘time-to-delivery’ on the x-axis rather than the delivery date itself. So you can compare, you should also scale the futures prices by the current spot price, so that both forward curves start at 100 at time-to-delivery equals zero. For simplicity you should assume that the time-to-delivery is rounded off to the nearest month; e.g. 1 month for Oct ‘24 contracts, 2 months for Nov ’24 contracts, etc.
(b) Provide an explanation of the patterns of each forward curve and discuss any differences/similarities between them. News announcements and online articles on economic market conditions that may have had an impact should be used to support your discussion.
Another client is particularly interested in analysing the convenience yields implied in the crude oil and alumimum prices. In the following, you should assume that, on 3 September 2024, the estimated monthly storage costs of oil and aluminum (payable in advance each month) were USD0.75 per barrel, and USD15.00 per ton, respectively. You should also assume that the yield curve is flat, and the risk-free rate is 4.5% per annum with continuous compounding.
(c) Using the futures price data on 3 September 2024 in Table 1, calculate the convenience yield of oil and aluminum implied by each available futures contract (i.e., 2 × 18 numbers in total). Again, please round off the time-to-delivery to the nearest month. Plot these implied convenience yields with the time-to-delivery on the x-axis.
(d) Comment on any patterns you observe in the convenience yields calculated in part (c) and explain to your client what the convenience yield is and what it tells us about the current oil and aluminum markets. You might want to reflect on the current market conditions and/or the current forward curves in your discussion. Again, news announcements and online articles can be used to support your discussion.
Next, a small aluminum smelter (producer) located in Kentucky, USA has estimated that it will have approximately 35,000 metric tons of aluminum ready for sale sometime around mid- March 2025. Naturally, given the volatility in the market, they are concerned that prices will fall before then and so they are considering locking in a selling price for their aluminum now using the COMEX futures contract analysed above. They have approached you and your company to advise them on the hedge.
Assume that today is 3 September 2024 and that you have estimated the following standard deviations of price changes:
The correlation coefficient between the spot and futures price changes has also been estimated to be 0.8175 for aluminum.
(e) Calculate the optimal number of futures contracts required (by tailing the hedge) and recommend an effective hedge for the aluminum producer. Use the appropriate futures contracts from Table 1 and recall that one COMEX aluminum futures contract is written on 25 metric tons of aluminum.
Finally, assume that the aluminum producer is ready to sell their metal in mid-March. At this time, the spot price of aluminum has changed to USD2,079.25 per ton and the futures price for delivery in one month is USD2,151.00 per ton.
(f) Calculate the outcome with and without the hedge. What is the producer’s effective selling price with and without the hedge? Did they benefit from this hedge?
(g) Explain the main objective of the recommended hedge and the reasons why it cannot be a perfect hedge.
Question 2
[4 + 6 + 3 = 13 marks]
It’s not just commodity prices that have been on the move lately. The effects of differing inflation rates in different countries has caused significant movements in exchange rates. Such volatile FX markets increase the need for financial derivatives to hedge such volatility, but it can also increase the counterparty risk involved in such derivative trades.
To this end, the financial institution you are working for has a current position in a cross- currency interest rate swap and another CHF (Swiss Franc) currency futures position. Your boss has asked you to evaluate the two positions.
The Swap Position
21 months ago, your institution entered into a three-year cross-currency interest rate swap with a watchmaking company based in Switzerland. The swap agreement was over-the- counter with the following terms: your institution is to pay LIBOR+0.50% per annum in CHF and receive 4.75% per annum in AUD (both rates are semi-annually compounding). Payments are semi-annual and on a notional principal of CHF30 million. The 6-month LIBOR rate and the spot exchange rate at various dates over the last 21 months are shown in the table below:
(a) Compute the cash flow paid and received by your financial institution on each payment date of the swap (i.e., at t = 0, 6, 12, and 18 months).
(b) Unfortunately for you and your institution, the counterparty to the swap (the Swiss watchmaker) has just filed for bankruptcy with 15 months remaining on the swap agreement. Determine the current value of the swap agreement (and ultimately the cost) to your institution. You should assume that the current risk-free interest rate is 5.05% per annum in AUD and 1.75% per annum in CHF (with continuous compounding) for all maturities.
The Futures Position
(c) Worried about a further change in the FX rate, 9 months ago your institution also entered into a long position in 2-year currency futures contracts on CHF30 million. At the time, the interest rate was 5.43% per annum in AUD and 1.84% per annum in CHF (with continuous compounding) for all maturities. Your boss asks you the following questions:
i. What was the value of the futures position 9 months ago?
ii. If we closed out the position today, what would be the profit/loss on the futures transaction?
3 Hint: You need the spot exchange rate 9 months ago, the current spot exchange rate today, and the current interest rates in AUD and CHF, all stated above.
Question 3
[4 + 3 + 2 + 3 = 12 marks]
Word gets out that you are doing such a great job in your new role and so it is not long before you are asked back to UTS to give a guest lecture. The subject coordinator has asked you to help with some of the trickier aspects of lectures 7, 8, and 9 of Derivative Securities (25620). Specifically, you have been asked to go through the calculation of various option prices using both binomial trees and the Black-Scholes model.
The example to be used is as follows: The FTSE MIB (which covers the 40 largest stocks traded on the Borsa Italiana in Italy) is currently trading at 33,682 (index points). The dividend yield of the index is 3.4% per annum and the risk-free interest rate in Italy is 4.1% per annum (both with continuous compounding for all maturities). The volatility of the index over the next six months is also estimated to be 32% per annum. Given your expertise in option valuation you decide to use a five-step binomial tree to calculate the following derivative values (in units of index points, to two decimal places):
(a) A 10-month European put option on the index with a strike of 35,000. Calculate also the value of the option by using the Black-Scholes formula. Compare the values and comment.
(b) A 10-month American put option on the index with a strike of 35,000. Is the answer different to the answer from (a)? Is so, explain why, making explicit reference to the dividend yield and risk-free rate.
(c) A short position in a forward contract on the index for delivery in 10 months at a price of 35,000. Calculate also the theoretical value of this forward position. Compare these values (the theoretical and binomial tree values) and comment. Hint: the fair forward price that would be agreed for new contracts today is different to 35,000 and hence the forward contract in question has a non-zero value.
(d) An American up-and-out barrier put option with a strike of 35,000 and knockout barrier of 38,000 maturing in ten months. An American up-and-out put option gives the holder the right to sell the underlying asset at the strike price at anytime on or prior to the expiration date so long as the price of that asset did not go above a pre-determined barrier during the option’s lifetime. When the price of the underlying asset rises above the barrier, the option is "knocked-out" and no longer carries any value. Comment on the value of this option relative to the option in (b) and explain any differences. Would the value of this option change if the knockout barrier was increased from 38,000 to 39,000? Is so, comment on the reason why.
Question 4
[5 + 5 + 5 + 5 = 20 marks]
Vitalik Buterin, the co-founder and inventor of the decentralized open-source blockchain Ethereum, currently has a portfolio of cryptocurrency worth USD1.4 billion (including about 244,000 Ether, or ETH, the cryptocurrency generated by the Ethereum protocol). The portfolio also has an income yield of 3.25% per annum with simple compounding. Vitalik has seen significant fluctuations in the value of his portfolio over the last few years and, while he is still bullish on the crypto market in the long run, he is worried that there will be further falls before the year is out. As such, he has approached your institution for advice on how best to protect the value of his existing crypto portfolio over the next three months. He also does not want to miss out on any potential upside if the crypto market starts to rise sooner than he expects.
You know just the derivative for the job and suggest that a position in Ether (ETH) options will provide the desired protection. Since Vitalik’s portfolio is highly correlated with Ether prices, options on Ether can be used as a portfolio hedge (in the same way that stock index options can be used to hedge an equity portfolio). To this end, you estimate from historical price data that the ‘Ether beta’ of Vitalik’s portfolio is 0.9. You also note that Ether has an average income yield of 2.5% and that the risk-free interest rate in the US is currently 4.4% per annum with simple compounding for all maturities.
Assume that today is 27 September 2024, and the current spot price of Ether is USD2,410. Moreover, the following table provides the current market prices (in USD as of 27 September) for the available exchange traded European call and put options written on Ether with different strikes and two different maturities (27 November and 27 December 2024):
(Source: https://www.deribit.com)
(a) Describe the options portfolio insurance strategy that would insure against Vitalik’s cryptocurrency portfolio falling below USD1 billion between now and 27 December 2024 (in exactly 3 months). Explain why this strategy fulfils his request and why hedging with Ether futures does not suffice. Also, highlight some of the potential downsides of the strategy if implemented in practice.
(b) Calculate the gains/losses on the strategy if the price of Ether on 27 December 2024 is either (i) $1,500, or (ii) $3,500, and prepare a short summary for Vitalik to discuss the outcome of the insurance strategy in these two scenarios. Note that here you should include the cost of the options purchase (inferred from the table above) in your calculation of the P&L in each scenario (ignoring time value in the P&L as usual).
Vitalik is somewhat interested in the proposed strategy, but he is surprised by how expensive the options are. You state that the high prices are due to the very high volatility in ETH and the wider crypto market. To explain things further you decide to investigate the implied volatility of ETH using the option prices given above.
(c) Use Excel’s GoalSeek (or otherwise) to estimate the implied volatility of the spot price of ETH, based on the market prices of all the ETH options above. Specifically, complete the following table with the estimated implied volatilities (to 3 significant figures):
You should also perform. the following tasks and write a short email to Vitalik summarising your answer to the following questions:
i. Plot the implied volatility from each maturity and for each option type (put/call) as a function of the strike. In other words, four lines on the same graph;
ii. Is the implied volatility of one option class higher than the other? If so, explain why;
iii. Does the implied volatility depend on the moneyness of the option and/or on the maturity of the option? Is so, explain;
iv. What do your results tell you about the observed market prices and their consistency with the assumptions underlying the Black-Scholes model?
(d) Vitalik has decided to go ahead with the options strategy you suggested. However, he does not want to use exchange-traded derivatives, but instead wants your institution to sell him OTC options with the precise strike price you calculated previously that would protect his portfolio from dropping below USD1 billion, and also with a bespoke maturity of 15 December 2024. Use linear interpolation on the implied volatilities of the exchange-traded options you have calculated above to estimate the implied volatility and hence price (in dollars) of an OTC option on one Ether with the strike and maturity requested by Vitalik. Write an email to your boss with your price for this option and describe the methodology used (so that your boss can understand).