**Assignment Task**

**Task **

**Derivative Securities – Group Assignment**

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Get Help Now!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 still 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 behaviour of the WTI crude 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 derivatives, including Bitcoin options. All in a day’s work for a UTS graduate.

**Question 1**

The COVID-19 pandemic and the war in Ukraine has seen a large shift in various commodity markets, particularly in the crude oil markets. These large price movements have left many of your clients wondering what’s next. Your first task, therefore, is to analyse the historical (and current) state of the crude oil market and its forward curves. Such forward curves provide important information about market conditions for traders and investors. NYMEX WTI light sweet crude oil futures prices (traded on NYME) are presented in Tables 1, 2 and 3 below, where prices are given as of three dates.

**(a)** A client asks you to plot the historical forward curves of NYMEX WTI light sweet crude oil as would have been observed at three specific dates:

i. February 23, 2015,

ii. April 21, 2020,

iii. March 29, 2022.

**(b)** Provide an explanation of the patterns of each forward curve and discuss the key difference between them. News announcements and online articles on economic market conditions should be used to support your discussion. Another client is particularly interested in computing the convenience yields implied by the most recent WTI light sweet crude oil futures prices (March 29, 2022). In the following, you should assume that the spot price of WTI crude oil on March 29, 2022 was USD106.98 (per barrel) and that the monthly storage cost for WTI crude oil is USD0.55 (per barrel), payable in advance. The risk-free interest rate should also be assumed flat at 2.0% per annum with continuous compounding.

**(c) **Using the futures price data on March 29, 2022 (Table 3), calculate the convenience yield of WTI crude oil implied by each futures contract from May 2022 to Dec 2022 (i.e., 8 numbers in total). Please round off the time-to-delivery to the nearest month (e.g. 2 months for the May ’22 contract, etc.). 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 curren oil market. You might want to reflect on the current market conditions and/or the current forward curve in your discussion. Again, news announcements and online articles can be used to support your discussion.

Next, an industrial manufacturer on the East Coast of the USA needs to purchase 30,000 barrels of crude oil sometime around mid-October 2022 and they are concerned that the crude oil price will rise before then. They are considering locking in a purchase price for their oil now using the NYMEX WTI light sweet crude oil futures and they have approached you to advise them on the hedge.

Assume that today is March 29, 2022 (and the current spot oil price is USD106.98 (per barrel)).

Also assume that you have estimated the quarterly standard deviation of the crude oil futures price changes to be 0.824, the quarterly standard deviation of the spot crude oil price changes to be 0.738, and the correlation coefficient between these price changes to be 0.912.

**(e) **Calculate the optimal number of contracts required (by tailing the hedge) and recommend an effective hedge for the manufacturer. Use the appropriate futures contract from Table 3 and recall that one NYMEX futures contract is written on 1,000 barrels of oil.

**(f) **Next, assume that the industrial manufacturer is ready to purchase the crude oil in midOctober. At this time, the crude oil spot price has decreased to USD59.50 per barrel and the crude oil futures price for delivery in one month is USD59.34 per barrel. Calculate the outcome with and without the hedge. What is the company’s effective purchase price with and without the hedge? Did the industrial manufacturer 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**

It’s not just commodity prices that have been on the move lately. The differential effects of COVID-19 on 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 crosscurrency interest rate swap and another GBP (British pounds) currency futures position. Your boss has asked you to evaluate these two positions.

**The Swap Position**

24 months ago, your institution entered into a two-year cross-currency interest rate swap with a British aviation company. The swap agreement was over-the-counter with the following terms: your institution is to pay 2.55% per annum (with semi-annual compounding) in GBP and receive 6-month LIBOR + 0.65% per annum in AUD. Payments are semi-annual and on a notional principal of AUD30 million. The 6-month LIBOR rate and the spot exchange rate at various dates over the last 24 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, 18 and 24 months).

**(b) **Unfortunately for you and your institution, the counterparty to the swap (the British aviation company) has just filed for bankruptcy with 10 months remaining on the swap agreement. Determine the value of the swap agreement (and ultimately the cost) to your institution at the time of the bankruptcy. You should assume that at the time of the bankruptcy the interest rate was 2.15% per annum in AUD and 1.72% per annum in GBP (with continuous compounding) for all maturities and the exchange rate was 1.7194 AUD for 1 GBP.

**The Futures Position****(c) **Worried about a volatile exchange rate, 12 months ago your institution also entered into a short position in 2-year currency futures contracts on GBP40 million. At the time, the interest rate was 2.25% per annum in AUD and 1.55% per annum in GBP (with continuous compounding) for all maturities. Your boss asks you the following questions:

i. What was the value of the futures position 12 months ago?

ii. If we closed out the position today, what would be the profit/loss on the futures transaction? The interest rate today is 2.05% per annum in AUD and 1.15% per annum in GBP (with continuous compounding) for all maturities.

**Question 3**

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 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 S&P 500, which covers the 500 largest companies on the U.S. stock market, is currently trading at 4,070 (index points). The dividend yield of the index is 3.8% per annum and the risk-free interest rate in U.S. is 1.6% per annum (both with continuous compounding for all maturities). The volatility of the index over the next 8 months is also estimated to be 22.5% per annum. Given your expertise in option valuation you decide to use a four-step binomial tree to calculate the following derivative values (in units of index points, to two decimal places):**(a)** A 8-month European put option on the index with a strike of 4,500. Calculate also the value of the option by using the Black-Scholes formula. Compare the values and comment.

**(b) **A 8-month American put option with a strike of 4,500. Is the answer different to the answer from (a)? Explain why if so.

**(c) **A short position in a forward contract on the index for delivery in 8 months at a price of 4,500. Calculate also the theoretical value of this forward position. Compare these values and comment. (Hint: the fair forward price that would be agreed for new contracts today is different to 4,500 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 4,500 and knockout barrier of 4,750 maturing in 8 months. An American up-and-out put option gives the holder the right to sell the underlying asset at the strike price at any time on or prior to the expiration date so long as the price of that asset did not rise above a pre-determined barrier during the option’s lifetime. When the price of the underlying asset is 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 4,750 to 4,900? Comment on the reason why.

**Question 4**

A US based client has been investing in cryptocurrencies and cryptocurrency related assets over the last couple of years. They currently have a portfolio of these assets worth $15 million with an income yield of 0.25% per annum with simple compounding. However, the client is about to embark on a 5-month trip to trek the entire length of the Pacific Coast Trail in North America.1 The client is worried that in their absence the ‘crypto’ market may plummet, and so they have requested your advice on how best to protect the value of their portfolio over the next five months. The client also does not want to miss out on potential upside if the crypto market continue to rise during their absence.

You know just the derivatives for the job and suggest that a position in Bitcoin (BTC) options will provide the desired protection. Since the client’s portfolio is highly correlated with Bitcoin prices, options on Bitcoin 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 ‘Bitcoin beta’ 2 of your client’s portfolio is 1.6. You also note that Bitcoin does not pay any income (and hence has a zero-income yield) and that the risk-free interest rate in the US is 1.6% per annum with simple compounding for all maturities. The current spot price of Bitcoin is $47,208 and the following table provides the market prices (in USD) for various European call and put options written on Bitcoin with different strikes and maturing in five months’ time:

**(a) **Describe the options portfolio insurance strategy that would insure against your client’s cryptocurrency portfolio falling below $8 million over the next five months. Explain why this strategy fulfils the client’s request and why hedging with Bitcoin futures does not suffice. (Note that the $8 million value should not include the cost of the options used for insurance and each option contract is written on exactly one unit of BTC.)

**(b) **Calculate the gains/losses on the strategy if the price of Bitcoin in five months is either (i) $30,000, or (ii) $60,000, and prepare a short summary for your client to discuss the outcome of the insurance strategy in these two scenarios. Note that you should include the cost of the options purchase (inferred from the table above) in your calculation of the P&L in each scenario.

Your client is impressed by the proposed strategy, but she is surprised by how expensive the options are. You state that this is due to the very high volatility in BTC and the wider crypto market. To explain things further you decide to investigate the implied volatility of BTC using the option prices given above.

**(c) **Use Excel’s GoalSeek (or otherwise) to estimate the implied volatility of the spot price of BTC, based on the market prices of the BTC options above. Specifically, complete the following table with the estimated implied volatilities (to 3 significant figures):

**You should also perform the following tasks and answer the following questions:**

i. Plot the implied volatility from each option type (put/call) as a function of the strike.

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? 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?

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