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Examination Session:

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

Section A

Question 1

The following table gives the current prices of bonds. Half the stated coupon is assumed to be paid

every six months.

Bond Bond principal

($)

Time to maturity

(years)

Annual coupon

($)

Bond price

($)

Treasury bond A 100 0.5 2.0 100

Treasury bond B 100 1.0 4.0 98

Treasury bond C 100 1.5 4.0 98

Corporate bond D 100 1.0 6.0 98

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)

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.

GBP USD

Company Alpha 3.0% 5.6%

Company Beta 2.0% 3.0%

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)

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,

The Black-Scholes formula for the price of a European call option is given by

� 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).

(40 marks)

END OF EXAMINATION

Examination Session:

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

Section A

Question 1

The following table gives the current prices of bonds. Half the stated coupon is assumed to be paid

every six months.

Bond Bond principal

($)

Time to maturity

(years)

Annual coupon

($)

Bond price

($)

Treasury bond A 100 0.5 2.0 100

Treasury bond B 100 1.0 4.0 98

Treasury bond C 100 1.5 4.0 98

Corporate bond D 100 1.0 6.0 98

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)

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.

GBP USD

Company Alpha 3.0% 5.6%

Company Beta 2.0% 3.0%

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)

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,

The Black-Scholes formula for the price of a European call option is given by

� 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).

(40 marks)

END OF EXAMINATION