EXAMINATION PAPER
Examination Session:
August
Year:
2020
Exam Code: ECON41415/WE01
Title:
Derivative Markets
Time Allowed:
2 hours
Additional Material provided:
None
Materials Permitted:
None
Calculators Permitted:
Yes
Models Permitted:
Only the following list of calculators is permitted in this examination: Casio FX83; Casio FX85; Texas TI-30XS; Sharp EL-531. Calculators with additional text after model designations are acceptable (e.g. Casio FX85GT). Other models of calculator will not be considered on the day of the examination. If your calculator is not permitted, it will be confiscated.
Visiting Students may use dictionaries: No
Instructions to Candidates:
Answer one question from Section A which accounts for 50% of the total marks.
Answer one question from Section B which accounts for 50% of the total marks.
Revision:
Page 2 of 5
ECON41415/WE01
Question 1
Section A
The following table gives the current prices of bonds. Half the stated coupon is assumed to be paid every six months.
a) Define the term “zero rate” and explain how zero rates are determined using the bootstrap method. Calculate the zero rates with continuous compounding for the maturities of 0.5, 1.0 and 1.5 years. Present the theories of the term structure of interest rates. Discuss which theory can best explain the term structure implied by the current example.
(30 marks)
b) Explain the concept “forward rate” and calculate the forward rate for the period between 0.5 years and 1.5 years. Discuss also the term “par yield” and explain how the par yield is calculated. Determine the par yield for the maturity of 1.5 years. Report the par yield with continuous compounding.
(30 marks)
c) Explain the term “bond yield” (YTM). Determine the YTM of corporate bond D. Explain the concept of zero-volatility spread (Z-spread). Check which of the following rates better approximates the Z-spread of corporate bond D: 1.18% or 1.98%. Explain the term “Macaulay duration” and determine the Macaulay duration of the bond. Explain how duration is used in portfolio risk management.
(40 marks)
Bond
Bond principal ($)
Time to maturity (years)
Annual coupon ($)
Bond price ($)
Treasury bond A Treasury bond B Treasury bond C
100 100 100
0.5 1.0 1.5
2.0 4.0 4.0
100 98 98
Corporate bond D
100
1.0
6.0
98
Page 3 of 5 ECON41415/WE01 Question 2
Company Alpha, a British manufacturer, is required to borrow US dollars at a fixed rate of interest. Company Beta, a US multinational, is required to borrow sterling at a fixed rate of interest. The two companies have been offered the following borrowing rates per annum with semi-annual compounding on £100 million and $122 million for two years. Interest payments are to be made every six months.
a) Explain the term “comparative advantage” and discuss why currency swap can be motivated by comparative advantage. Assume that a swap is arranged in which a financial institution acts as an intermediary between the two companies. Design a currency swap that is equally beneficial for the two companies assuming that the financial institution earns 20 basis points per annum and bears the entire foreign exchange risk. Show your calculations and use a diagram to explain how the swap is arranged.
(30 marks)
b) Assume that the swap has a remaining life of 15 months to maturity and the term structure of interest is flat in both currencies. The USD interest rate is 3% per annum and the GBP interest rate is 2% per annum, both reported with continuous compounding. The spot exchange rate GBP/USD is 1.10. Determine the value of the currency swap for Company Alpha.
(30 marks)
c) Discuss the risks for the counterparties in a currency swap. Discuss the scenarios in which these risks arise and whether these risks can be hedged. Suppose that six months before the final exchange of payments in the swap company Alpha declares bankruptcy and defaults on its current and future swap payments. Calculate the losses that arise as a result of the default and explain which party bears these losses. Use the same interest rates and exchange rate in your calculations as in Part b).
(40 marks)
GBP
USD
Company Alpha
3.0%
5.6%
Company Beta
2.0%
3.0%
Page 4 of 5
ECON41415/WE01
Section B
Question 1
a) Critically discuss the assumptions about asset price dynamics in the standard option pricing models. Explain the factors that these models consider and the effect of these factors on the prices of European call and put options.
(30 marks)
b) Assume that the current price of a non-dividend paying stock is 𝑆0 =£40. In the next six months, the stock price can either increase by 20% or decrease by 10%. Assume that the risk-free interest rate is r=0% per annum. Consider a 6-month European call option with a strike price of K=£40. Explain the “no-arbitrage” argument for pricing options. Determine the option price using the no-arbitrage approach. Explain all your calculation steps.
(30 marks)
c) Assume that the current exchange rate GBP/USD is 𝑆0=1.23 and in the next six months it can either move up to 1.30 or down to 1.10. The risk-free interest rate in GBP is 1.5% and in USD is 2.5% per annum (reported with continuous compounding). Determine the value of a six-month European exchange rate put option (i.e. to sell GBP) with a strike price of K=1.25 and a contract size of 10,000 GBP.
(40 marks)
Page 5 of 5 ECON41415/WE01 Question 2
A stock price 𝑆 follows a geometric Brownian motion with expected return 𝜇 and a volatility 𝜎: 𝑑𝑆 = 𝜇𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧
Itô’s lemma states that if 𝐺 is a function of 𝑆 and 𝑡, then
𝑑𝐺 = (𝜕𝐺 𝜇𝑆 + 𝜕𝐺 + 1 𝜕2𝐺 𝜎2𝑆2) 𝑑𝑡 + 𝜕𝐺 𝜎𝑆𝑑𝑧
The Black-Scholes formula for the price of a European call option is given by
𝜕𝑆 𝜕𝑡2𝜕𝑆2 𝜕𝑆
𝑐 = 𝑆0 ∙ 𝑁(𝑑1) − 𝐾 ∙ 𝑒−𝑟𝑇 ∙ 𝑁(𝑑2)
where
a) Two derivatives on the stock are proposed with values 𝐺(𝑆, 𝑡) = √𝑆 and 𝐺(𝑆, 𝑡) = 𝑡 ∙ 𝑆. Use Itô’s lemma to examine whether these new derivatives also follow a geometric Brownian motion.
(30 marks)
b) Critically discuss the assumptions on the return dynamics of the underlying asset in the Black-Scholes model. Explain the effect of the model parameters on option prices.
(30 marks)
c) Consider a stock with a current price of $20 and a volatility of 20% per annum. Assume that the current risk-free interest rate is 5% per annum reported with continuous compounding. Determine the price of a European put option on the stock with a strike price of $15 and an expiration date in six months. Express your answers in terms of the cumulative normal distribution 𝑁(𝑥).
𝑙𝑛(𝑆0)+(𝑟+𝜎2)∙𝑇
𝑙𝑛(𝑆0)+(𝑟−𝜎2)∙𝑇
𝑑1= 𝐾 2 𝜎∙√𝑇
and𝑑2= 𝐾 2 𝜎∙√𝑇
END OF EXAMINATION
(40 marks)