Derivative Securities

QUESTION 1

As a valued member of the UTS alumni you have been asked to be a guest lecturer for Derivative Securities (25620). Vinay has asked you to focus your lecture on interest rate futures contracts. In addition, to prepare yourself for this lecture Vinay has strongly recommended that you read Chapters 4 and 6 of the prescribed textbook authored by John Hull.

a) The m-zero rate is the interest rate earned on an investment starting today and lasting for m periods. From Bloomberg you identify the following information from actively traded Treasury bonds which pay semi-annual coupons and have a face value of $100:

Demonstrate to students using the bootstrap method how to calculate the continuously compounded zero rates for the 6 month and 12 month maturities.

b) Zero-rates can be used to calculate the forward rate to borrow or invest at during a future time period. Using your answer to part a) to calculate the six-month continuously compounded forward rate.

c) Describe one use from calculating zero-rates and the zero curve (other than the fact that zero- rates can be used to calculate forward rates)?

d) Using the following information calculate the price of the bond: Annual coupon (with semi-annual payments) = $10
Yield to maturity = 4% p.a. with continuous compounding Time-to-maturity = 2 years
Face value = $100.

e) Using the following information calculate the price of the bond: Annual coupon (with semi-annual payments) = $10
Yield to maturity = 2% p.a. with continuous compounding Time-to-maturity = 2 years
Face value = $100

f) Based on your answers to part d) and part e), is there a direct or inverse relation between bond prices and bond yields?

g) A very simple strategy for financial institutions/investors to hedge themselves from interest rate risk is to match the duration of their assets to the duration of their liabilities. Duration captures the amount of time taken to recover an asset/liabilities cash flows. Hint: From Lecture 2 you should first understand how to hedge an equity portfolio using equity index futures, once you understand how this works, hedging the value of a bond portfolio uses a similar intuition.

I. Today is September 30 2018. You are a portfolio manager of the bank overseeing an $80 million investment portfolio consisting of a 75%/25% split between equities and government bonds. Going forward you predict that equity values will decrease and interest rates will increase over the next four months. You decide to use equity index futures to hedge the equity component of your portfolio and Treasury bond futures to hedge the bond part of your portfolio. Currently the price for a six-month Treasury bond futures contract which matures in April is $94,062.50. The cheapest-to-deliver bonds (these are the bonds underlying the Treasury bond futures contract) will have a duration of 9.2 years in six months. The duration associated with your bond portfolio is 7.1 years in six months. Determine the optimal number of Treasury bond futures contracts that you should take a position in to hedge the value of your bond portfolio against rising interest rates over the next four months. Is a long or short position in Treasury bond futures contracts required?

II. Assume that the bond yields on your bond portfolio have decreased over the four months to February 1 2019, resulting in your bond portfolio value increasing to $20,500,000 on February 1 2019. On February 1 2019 the Treasury bond futures price is $96,704.80. What is the gain or loss on your Treasury bond futures position? You need to use the information and your solution from part i) for your answer in part ii).

III. What is the value of the total hedged bond position (including bond portfolio value and gain or loss in Treasury bond futures contracts)? Comment on the outcome of the bond portfolio hedge? You need to use the information and your solutions from part i) and part ii) for your answer in part iii).

IV. If instead on February 1 2019 the bond yields on your bond portfolio increased from September 30 2018 and the Treasury bond futures price decreased from September 30 2018, qualitatively explain the following:

Would a gain or loss have been made on the Treasury bond futures position? Would a gain or loss have been made on the bond portfolio?
Comment on the outcome of the hedge?

V. Does the duration matching approach to hedging the value of a bond portfolio work for both a parallel shift in interest rates and a non-parallel shift in interest rates? Explain this concept to the person sitting next to you.

QUESTION 2

Two months ago, five clients of yours (Harrison Chen, Jack Li, Kieran Nguyen, Hoang Tang, Zhang Wang) each constructed an equity portfolio consisting of 300 of the largest US stocks. This portfolio has a beta of 1.30 and dividend yield of 2.80% per annum with annual compounding. Each client purchased a total of 1,500,000 shares at an average price of $50.

Based on analyst predictions and articles in the Wall Street Journal, today these five clients approach you and they want to protect the value of their equity portfolio from falling below $60 million over a nine-month period, however they want this strategy to begin three months from today. From Bloomberg you identify that today the risk-free rate is 1% per annum with annual compounding, the S&P 500 index value is 2,700 with a dividend yield of 4.20% per annum with annual compounding and volatility of 18% annually.

a) Detail the insurance strategy required to meet each client's requirements. Assume that the risk-free rate, dividend yields on the portfolio/index, volatility of the index remain constant overtime. Assume that the following information is true:

b) How much money does each client have to transfer to us (the bank) to setup the index insurance strategy at the initiation date? For the purposes of part (b) only, simplicity and calculating the cost of the insurance strategy, assume the dividend yield on the index/portfolio and the risk-free rate when expressed with annual compounding are approximately the same as the continuously compounded rates.

c) Using simulation analysis how much money does each client gain or lose if the S&P 500 decreases by 25% from the insurance initiation date until the conclusion of the strategy. Please explain the outcome of the strategy to each client.

d) How does the outcome of the strategy differ to the outcome in part c) if the S&P 500 decreases to 1,500 at the conclusion of the strategy?

e) Using simulation analysis how much money does each client gain or lose if the S&P 500 increases by 15% from the insurance initiation date until the conclusion of the strategy. Please explain the outcome of the strategy to each client.

QUESTION 3

Due to a sequence of significant global events (e.g., increased tariffs on natural resource imports) the consensus across different investment banks suggests that there will be high volatility in the resources sector over the next six months. Six high-net clients of the bank: Ksenjia Yang, Emilio Paula, Holly Kelly, Eleni Eric and Chris Tarun-Shami trade BHPEE (a resource stock). From Bloomberg you identify the following current information about BHPEE and BHPEE stock options with six-months to maturity:

Based on the current stock price and using the six-month European BHPEE stock options, construct the profit-loss tables and the profit-loss diagrams showing the profit-loss (as a function of the terminal stock price) of each component position as well as of the combined position of the strategies listed below to show your clients how they could profit from these strategies. Similar to what we have done in class label all important features of your graph (breakeven points, minimum/maximum points, etc).
a) A strip
b) A reverse butterfly spread using call options
c) A straddle
d) A strap
e) Eleni thinks it is more likely that the stock price was going to increase (than decrease) over the next six months, which of the strategies above is the most profitable. Provide numerical support for your answer?

QUESTION 4

Following your successful interest rate futures lecture you have been invited back to do a lecture on option pricing methods (including binomial trees and the Black-Scholes model). Vinay has asked you to:

a) Show students how to estimate option prices using the binomial tree model. Using a live demonstration on Bloomberg you identify the following information: 1 USD = 1.30 AUD, the volatility of the USD/AUD currency pair is 40% per annum, the US risk-free rate is 1.50% per annum with continuous compounding and the AUD risk-free rate is 1.75% per annum with continuous compounding. Using a four-step binomial tree calculate the price of a European six month put option to sell 1 USD for 1.40 AUD.

b) Using the same information in part a) estimate the value of the European six-month put option using the Black-Scholes model. In addition, explain to students what happens when the number of steps in the binomial tree increases to a very large number.

c) Using the same information as part a) estimate the price of an American six month put option to sell 1 USD for 1.40 AUD. In addition, explain why the American option price differs from the European option price.

d) For the final part of the lecture show using Microsoft Excel how students can estimate implied volatility using option information. From Bloomberg you identify that the price of a nine- month silver futures contract is 16.50, the risk-free rate is 1.50% per annum with continuous compounding, and the dividend yield is 3.80% per annum with continuous compounding. Calculate implied volatility for the following futures options on silver with six-months to maturity and explain the importance of implied volatility.

Attachment:- Assignment.rar