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 since you are working from home due to COVID‐19, you have to 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 strange behaviour of the current oil futures market, to evaluate currency swaps and futures positions, to advise concerned clients on the different hedging strategies available to them, and to explain the pricing of complex equity derivatives. All in a day’s work for a UTS graduate.
[2 + 4 + 4 + 3 + 2 = 15 marks]
Oil prices have collapsed and many of the bank’s clients are scrambling to understand what the $%#! is going on. Your first task is to analyse the crude oil forward curve, past and present. Forward curves provide important information about market conditions to traders and investors. NYMEX WTI light sweet crude oil futures for a day this April and a day last April are presented in the tables below.
Table 1: NYMEX WTI light sweet crude oil futures prices on 22 April 2019
Table 2: NYMEX WTI light sweet crude oil futures prices on 21 April 2020
- First, a client asks you to plot the NYMEX WTI light sweet crude oil forward curve on 22 April 2019 and on 21 April 2020. (Be sure to label you axes and include a legend.)
- Discuss the key differences of the forward curves between these two dates and provide possible explanations of the observed patterns. News announcements and online articles on economic market conditions that may have had an impact should be used to support your discussion.
- Notice that the price of the May 2020 contract on 21 April 2020 was negative! Discuss what a negative price actually means and also its likely cause. Again, news announcements and online articles should be used to support your discussion.
A few months later oil has rebounded significantly and Table 3 shows crude oil futures for a day in early August 2020.
Table 3: NYMEX WTI light sweet crude oil futures prices on 4 August 2020
Another client is particularly interested in analysing the convenience yields implied in the new oil futures prices. In the following, you should assume that on 4 August 2020, the spot crude oil price was USD40.27 per barrel, the monthly storage (freight) cost for crude oil was estimated at 35 cents per barrel payable in advance, and the risk‐free rate (zero curve) was flat at 1.40% per annum with continuous compounding.
- Using Table 3, calculate the convenience yield of crude oil implied by each futures contract from September 2020 to September 2021 (i.e., thirteen numbers in total). Round off the time to maturity to the nearest month (e.g., 1 month for the September 2020 contract, 2 months for the October 2020 contract, etc). Plot these implied convenience yields with the time to maturity on the x‐axis. Comment on any patterns you observe.
- Provide an interpretation of the convenience yields calculated in part (d). You might also want to reflect on the current market conditions (as of 4 August 2020) and/or the current forward curve in your discussion. Again, news announcements and online articles should be used to support your discussion.
[5 + 6 + 4 = 15 marks]
It’s not just oil prices that have been varying wildly. The COVID‐19 pandemic caused an initial ‘flight to quality’ with investors seeking shelter in the US dollar, seeing it strengthen considerably. The effect was short lived however as the uncertain US economic outlook subsequently caused the US dollar to weaken considerably. Combined, this has caused large movements in exchange rates over the last few months. In this regard, the financial institution you are working for has a current position in a cross‐currency interest rate swap and another USD currency futures position. Your boss has asked you to evaluate these two positions.
The Swap Position
27 months (2.25 years) ago, your institution entered into a three‐year cross‐currency interest rate swap with an Australian retail chain. The swap agreement was over‐the‐counter with the following terms: your institution is to pay 6‐month LIBOR + 0.55% per annum in USD and to receive 2.25% per annum (with semi‐annual compounding) in AUD. Payments are semiannual and on a notional principal of AUD20 million. The 6‐month LIBOR rate and the spot exchange rate at various dates over the last 27 months are shown in the table below:
|Date of observation||6‐month LIBOR rate observed||Spot exchange rate observed (AUD for 1 USD)|
|t = 0 (contract initiation)||1.74%||1.3152|
|t = 6 months||2.06%||1.3949|
|t = 12 months||2.31%||1.4156|
|t = 18 months||1.88%||1.4562|
|t = 24 months||1.36%||1.5908|
|t = 27 months (today)||1.12%||1.4080|
- 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, 18, and 24 months).
- Unfortunately for you and your institution, the counterparty to the swap (the Australian retail company) has just filed for bankruptcy with 9 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 interest rate is 0.36% per annum in AUD and 0.25% per annum in USD (with continuous compounding) for all maturities.
The Futures Position
(c) Worried about a volatile exchange rate, three months ago your institution also entered into a short position in one‐year currency futures contracts on USD15 million. At the time, the interest rate was 0.63% per annum in AUD and 0.45% per annum in USD (with continuous compounding) for all maturities. Your boss asks you the following questions:
- What was the value of the futures position three months ago?
- If we closed out the position today, what would be the profit/loss on the futures transaction?
Note: you will need the spot exchange rate three months ago, the current spot exchange rate today, and the current interest rates in AUD and USD, all stated above, to answer.
[4 + 4 + 2 + 5 = 15 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 (via Zoom). The subject coordinator has asked you to help with some of the trickier aspects of lectures 8, 9, and 10 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: A stock is currently trading at $14.56 and has a volatility of 32% per annum and a continuous dividend yield of 1.30% per annum with continuous compounding. The risk‐free interest rate is 2.51% per annum with continuous compounding for all maturities. Given your expertise in stock option valuation you decide to use a four‐step
binomial tree to calculate the following derivative prices (to four decimal places):
- A one‐year European put option with a strike of $15.00. Calculate also the value of the option by using the Black‐Scholes formula. Compare and comment.
- A one‐year American put option with a strike of $15.00. Is the answer different to the answer from (a)? Explain.
- A short position in a forward contract on the stock for delivery in one year at a price of $15.00. Calculate also the theoretical value of the forward contract. Compare and comment. (Hint: the fair forward price that would be agreed for new contracts today is around $16.71 ≠ $15.00, hence the forward contract in question has a non‐zero value.)
- A European up‐and‐out barrier put option with a strike of $15.00 and knockout barrier of $17.00 maturing in one year. An up‐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 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 price of this option relative to the option in (a) and explain any differences.
[3 + 4 + 8 = 15 marks]
A UK based client holds a well‐diversified equity portfolio which has finally regained all the value it lost during the recent COVID‐19 related crash. The portfolio is currently worth £75 million and has a market beta of 1.5. Moreover, the expected dividend yield on the equity portfolio is 3.3% per annum with simple compounding. The client is concerned about a further downturn in the market as she thinks the recent bull market rally does not truly reflect the underlying economic reality. She has therefore requested your advice on how best to protect the value of her portfolio going forward, while still benefitting if the bull market continues. The FTSE100 index is currently at 5,969 and the dividend yield of the FTSE100 index is 2.1% per annum with simple compounding. The risk‐free interest rate is 0.3% per annum with continuous compounding for all maturities. Available index options are quoted in increments of 25 index points and the multiplier (per point) for each option contract is equal to £10.
- Describe the options portfolio insurance strategy that would insure against your client’s portfolio falling below £70 million over the next three months. Explain why this strategy fulfils the client’s request and why hedging with index futures does not suffice.
- Calculate the gains/losses on the strategy if the level of the FTSE100 index in three months is either (i) 5,250, or (ii) 6,250, and prepare a short summary for your client to discuss the outcome of the insurance strategy in these two scenarios. (Note you do not need to calculate the insurance premium).
Your client is impressed by the proposed strategy, but she is surprised by how expensive the insurance premium is. You state that this is due to an increased market volatility brought about by the uncertainty over the future impact of COVID‐19. To explain things further you decide to investigate the implied volatility of the FTSE100 index using market option prices.
- Use Excel’s GoalSeek (or otherwise) to estimate the implied volatility of the index, based on market prices of three‐month European call and put options on the index. Specifically, complete the following table with the estimated implied volatilities (to 3 significant figures). Note you will need information stated earlier in the question to do this:
|Strike||Call price||Call implied volatility||Put price||Put implied volatility|
|K = 5,400||589||112|
|K = 5,600||432||153|
|K = 5,800||292||213|
|K = 6,000||176||297|
|K = 6,200||91||412|
|K = 6,400||38||552|
You should also answer the following questions:
- Plot the implied volatility as a function of the strike price.
- Are the option prices consistent with the assumptions underlying the Black‐Scholes model? Does the implied volatility depend on the moneyness of the option? Explain. iii. Is the implied volatility of one option class higher than the other? If so, explain why.