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management exam ( include caculation )

management exam 19th Nov Sydney time 10amBFC5915 Options, Futures and Risk ManagementFinal Exam Workshop1. Futures and Forward1.1 Difference between OTC and exchanged traded marketsQ1. Outline differences between a forward contract and a futures contract11.2 Margin CallQ2. Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. Thesize of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is$3,000. What change in the futures price will lead to a margin call? What happens if you do not meetthe margin call?There will be a margin call when $1,000 has been lost from the margin account. This will occur whenthe price of silver increases by 1,000/5,000 $0.20. The price of silver must therefore rise to$17.40 per ounce for there to be a margin call. If the margin call is not met, your broker closes outyour position.

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BFC5915 Options, Futures and Risk Management

Final Exam Workshop

1. Futures and Forward

1.1 Difference between OTC and exchanged traded markets

Q1. Outline differences between a forward contract and a futures contract

1.2 Margin Call

Q2. Suppose that you enter into a short futures contract to sell July silver for $17.20 per ounce. The

size of the contract is 5,000 ounces. The initial margin is $4,000, and the maintenance margin is

$3,000. What change in the futures price will lead to a margin call? What happens if you do not meet

the margin call?

There will be a margin call when $1,000 has been lost from the margin account. This will occur when

the price of silver increases by 1,000/5,000

=

$0.20. The price of silver must therefore rise to

$17.40 per ounce for there to be a margin call. If the margin call is not met, your broker closes out

your position.

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1.3 Futures and Forward Price

Q3. The SFE has just introduced a futures contract on shares in Battery Company Limited (ASX:

BAT). BAT currently pays no dividends. Each futures contract is for delivery of 1,000 shares in BAT

in one year, the risk free rate is 4% and the BAT share price is currently $20. Answer the following

questions:

i. What should the futures price be?

ii. BAT makes an announcement that their first dividend of $2 will be paid in six months.

What should the futures price be?

Q4. A one-year long forward contract on a non-dividend-paying stock is entered into when the stock

price is $40 and the risk-free rate of interest is 10% per annum with continuous compounding.

a) What are the forward price and the initial value of the forward contract?

b) Six months later, the price of the stock is $45 and the risk-free interest rate is still 10%. What are

the forward price and the value of the forward contract?

a) The forward price,

F0

, is given by equation (5.1) as:

0 1 1 F e 0 40 44 21

= =

or $44.21. The initial value of the forward contract is zero.

b) The delivery price

K

in the contract is $44.21. The value of the contract,

f

, after six

months is given by equation (5.5) as:

0 1 0 5 f e 45 44 21 −

= − = 2 95

i.e., it is $2.95. The forward price is:

0 1 0 5 45 47 31 e

=

Q5.

1) Identify the fundamental difference between a futures contract and an option contract, and

briefly explain the difference between the ways that they modify portfolio risk.

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2) Assume that the spot price of the euro to Australian dollar is $1.40. If the one year forward

price is $1.41, and the one year interest rate in Australia is 3%, what is the interest rate in

Europe?

1.4 Optimal Hedge Ratio and Number of Contracts

Q6. A company has a $20 million portfolio with a beta of 1.2. It would like to use futures contracts

on the S&P 500 to hedge its risk. The index futures is currently standing at 1080, and each contract is

for delivery of $250 times the index. What is the hedge that minimizes risk? What should the

company do if it wants to reduce the beta of the portfolio to 0.6?

The formula for the number of contracts that should be shorted gives

20 000 000 1 2 88 9

1080 250

=

Rounding to the nearest whole number, 89 contracts should be shorted. To reduce the beta to 0.6, half

of this position, or a short position in 44 contracts, is required.

Similar Question. The standard deviation of monthly changes in the spot price of live cattle is (in

cents per pound) 1.2. The standard deviation of monthly changes in the futures price of live cattle for

the closest contract is 1.4. The correlation between the futures price changes and the spot price

changes is 0.7. It is now October 15. A beef producer is committed to purchasing 200,000 pounds of

live cattle on November 15. The producer wants to use the December live-cattle futures contracts to

hedge its risk. Each contract is for the delivery of 40,000 pounds of cattle. What strategy should the

beef producer follow?

The optimal hedge ratio is

1 2 0 7 0 6

1 4

=

The beef producer requires a long position in

200000 0 6 120 000 =

lbs of cattle. The beef producer

should therefore take a long position in 3 December contracts closing out the position on November

15.

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1.5 FRA Forward Rate Agreement

Q7. A bank can borrow or lend at LIBOR. Suppose that the six-month rate is 5% and the nine-month

rate is 6%. The rate that can be locked in for the period between six months and nine months using an

FRA is 7%. What arbitrage opportunities are open to the bank? All rates are continuously

compounded

Q8. The yield curve is flat at 5% per annum. What is the value of an FRA where the holder receives

interest at 7% per annum for a six-month period on a principal of $1,000 starting in two years? All

rates are compounded semi-annually.

2. Swap

Swap 解题五步法

1. 判断比较优势 固定利率与浮动利率

2. 求出 Swap Margin 互换前后进行对比

3. 画图 带金融中介记得画中间

4. 输出值为比较优势的值

5. 列式计算求解 X 固定利率并校验

2.1 Interest Rate Swap

Q9. Companies A and B have been offered the following rates per annum on a $20 million five-year

loan:

Fixed Rate Floating Rate

Company A 5.0% LIBOR+0.1%

Company B 6.4% LIBOR+0.6%

Company A requires a floating-rate loan; company B requires a fixed-rate loan. Design a swap that

will net a bank, acting as intermediary, 0.1% per annum and that will appear equally attractive to

both companies.

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Similar Question.

AAACorp and BBBCorp, both wish to borrow $10 million for five years, and are offered following

rates.

Fixed Floating

AAA 4.0% 6-month LIBOR – 0.1%

BBB 5.2% 6-month LIBOR + 0.6%

Spread

2.2 Valuation of Swap

Q10. A $100 million interest rate swap has a remaining life of 10 months. Under the terms of the

swap, six-month LIBOR is exchanged for 7% per annum (compounded semiannually). The average

of the bid–offer rate being exchanged for six-month LIBOR in swaps of all maturities is currently 5%

per annum with continuous compounding. The six-month LIBOR rate was 4.6% per annum two

months ago. What is the current value of the swap to the party paying floating? What is its value to

the party paying fixed?

3. Options

3.1 Basics of Option payoff and profit

Q11. An investor buys a European put on a share for $3. The stock price is $42 and the strike price is

$40. Under what circumstances does the investor make a profit? Under what circumstances will the

option be exercised? Draw a diagram showing the variation of the investor’s profit with the stock

price at the maturity of the option.

The investor makes a profit if the price of the stock on the expiration date is less than $37. In these

circumstances the gain from exercising the option is greater than $3. The option will be exercised if

the stock price is less than $40 at the maturity of the option.

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Similar Question. An investor sells a European call on a share for $4. The stock price is $47 and the

strike price is $50. Under what circumstances does the investor make a profit? Under what

circumstances will the option be exercised? Draw a diagram showing the variation of the investor’s

profit with the stock price at the maturity of the option.

The investor makes a profit if the price of the stock is below $54 on the expiration date. If the stock

price is below $50, the option will not be exercised, and the investor makes a profit of $4. If the stock

price is between $50 and $54, the option is exercised and the investor makes a profit between $0 and

$4.

Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity.

Under what circumstances will the holder of the option make a profit? Under what circumstances will

the option be exercised? Draw a diagram illustrating how the profit from a long position in the option

depends on the stock price at maturity of the option.

Ignoring the time value of money, the holder of the option will make a profit if the stock price at

maturity of the option is greater than $105. This is because the payoff to the holder of the option is, in

these circumstances, greater than the $5 paid for the option. The option will be exercised if the stock

-5

0

5

10

30 35 40 45 50

Profit ($)

Stock Price ($)

-10

-5

0

5

40 45 50 55 60

Profit ($)

Stock Price ($)

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price at maturity is greater than $100. Note that if the stock price is between $100 and $105 the

option is exercised, but the holder of the option takes a loss overall.

Suppose that a European put option to sell a share for $60 costs $8 and is held until maturity. Under

what circumstances will the seller of the option (the party with the short position) make a profit?

Under what circumstances will the option be exercised? Draw a diagram illustrating how the profit

from a short position in the option depends on the stock price at maturity of the option.

Ignoring the time value of money, the seller of the option will make a profit if the stock price at

maturity is greater than $52.00. This is because the cost to the seller of the option is in these

circumstances less than the price received for the option. The option will be exercised if the stock

price at maturity is less than $60.00. Note that if the stock price is between $52.00 and $60.00 the

seller of the option makes a profit even though the option is exercised.

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3.2 Put-Call Parity

Q12. According to the Put-Call Parity, how to effectively synthesize a call option and a put option?

Q13. Assume stock price is $31, a call option with exercise price K=30 is $3 and the risk free rate is

10% p.a., the maturity is 3 months.

▪ What is the equilibrium price for a put option according to the put call parity?

▪ Construct a table shows an arbitrage opportunity if the premium for this put option is $2.25.

3.3 Option Trading Strategies

Three alternatives strategies:

➢ A single option and the underlying asset

➢ Two or more options of the same type

➢ A mixture of calls and puts

Combination strategies involve a mixture of calls and puts, such as straddles, strips, straps, and

strangles

Spread strategies involve two or more options of the same type, such as bull spreads, bear spreads

and butterfly spreads

Q14. Call options on a stock are available with strike prices of $15,

1

2

$17

, and $20 and expiration

dates in three months. Their prices are $4, $2, and

1

2

$

, respectively. Explain how the options can be

used to create a butterfly spread. Construct a table showing how profit varies with stock price for the

butterfly spread.

An investor can create a butterfly spread by buying call options with strike prices of $15 and $20 and

selling two call options with strike prices of $17

1

2

. The initial investment is

1 1

2 2 4 2 2 $ + − =

. The

following table shows the variation of profit with the final stock price:

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Stock Price, ST Profit

15 T

S

1

2 −

1

2

15 17 T S

1

2

( 15) T

S − −

1

2

17 20 T S

1

2

(20 ) T − − S

20 T S

1

2 −

Q15. Suppose that put options on a stock with strike prices $30 and $35 cost $4 and $7, respectively.

How can the options be used to create (a) a bull spread and (b) a bear spread? Construct a table that

shows the profit and payoff for both spreads.

A bull spread is created by buying the $30 put and selling the $35 put. This strategy gives rise to an

initial cash inflow of $3. The outcome is as follows:

Stock Price Payoff Profit

35 T

S

0 3

30 35 T S 35 T

S − 32 T

S −

30 T

S −5 −2

A bear spread is created by selling the $30 put and buying the $35 put. This strategy costs $3 initially.

The outcome is as follows:

Stock Price Payoff Profit

35 T

S

0 −3

30 35 T S 35 T − S 32 T − S

30 T

S

5 2

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Q16.

3.4 Binomial Tree

Q17. A stock price is currently $40. It is known that at the end of one month it will be either $42 or

$38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of

a one-month European call option with a strike price of $39?

Consider a portfolio consisting of

− 1

Call option

+

Shares

If the stock price rises to $42, the portfolio is worth

42 3 −

. If the stock price falls to $38, it is

worth

38

. These are the same when

42 3 38 − =

or

= 0 75

. The value of the portfolio in one month is 28.5 for both stock prices. Its value today

must be the present value of 28.5, or

0 08 0 08333 28 5 28 31 e

−

=

. This means that

− + = f 40 28 31

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where

f

is the call price. Because

= 0 75

, the call price is

40 0 75 28 31 $1 69 − =

. As an

alternative approach, we can calculate the probability,

p

, of an up movement in a risk-neutral world.

This must satisfy:

0 08 0 08333 42 38(1 ) 40 p p e

+ − =

so that

0 08 0 08333 4 40 38 p e

= −

or

p = 0 5669

. The value of the option is then its expected payoff discounted at the risk-free rate:

0 08 0 08333 [3 0 5669 0 0 4331] 1 69 e

− + =

or $1.69. This agrees with the previous calculation.

Q18. A stock price is currently $100. Over each of the next two six-month periods it is expected to

go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous

compounding. What is the value of a one-year European call option with a strike price of $100?

In this case

u = 1 10 , d = 0 90 , = t 0 5

, and

r = 0 08

, so that

0 08 0 5 0 90 0 7041

1 10 0 90

e

p

−

= =

−

The tree for stock price movements is shown in Figure S12.1. We can work back from the end of the

tree to the beginning, as indicated in the diagram, to give the value of the option as $9.61. The option

value can also be calculated directly from equation (12.10):

2 2 2 0 08 0 5 [0 7041 21 2 0 7041 0 2959 0 0 2959 0] 9 61 e

− + + =

3.5 Upper bound and lower bound

Q19. A one-month European put option on a non-dividend-paying stock is selling for $2.50. The

stock price is $47, the strike price is $50, and the risk-free interest rate is 6 percent per annum.

What opportunities are there for an arbitrageur?

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3.6 Black Scholes Merton Model

Q20. What is the price of a European call option on a non-dividend-paying stock when the stock

price is $52, the strike price is $50, the risk-free interest rate is 12% per annum, the volatility is 30%

per annum, and the time to maturity is three months?

In this case

0

S = 52 , K = 50, r = 0 12, = 0 30

and

T = 0 25 .

2

1

2 1

ln(52 50) (0 12 0 3 2)0 25 0 5365

0 30 0 25

0 30 0 25 0 3865

d

d d

+ +

= =

= − =

The price of the European call is

0 12 0 25 52 (0 5365) 50 (0 3865) N e N −

−

0 03 52 0 7042 50 0 6504 e

− = − = 5 06

Similar Question. What is the price of a European put option on a non-dividend-paying stock when

the stock price is $69, the strike price is $70, the risk-free interest rate is 5% per annum, the volatility

is 35% per annum, and the time to maturity is six months?

In this case

0

S = 69 , K = 70, r = 0 05, = 0 35

and

T = 05 .

2

1

2 1

ln(69 70) (0 05 0 35 2) 0 5 0 1666

0 35 0 5

0 35 0 5 0 0809

d

d d

+ +

= =

= − = −

The price of the European put is

0 05 0 5 70 (0 0809) 69 ( 0 1666) e N N −

− −

0 025 70 0 5323 69 0 4338 e

− = − = 6 40

Q21. What is implied volatility? How can it be calculated?

The implied volatility is the volatility that makes the Black–Scholes-Merton price of an option equal

to its market price. The implied volatility is calculated using an iterative procedure. A simple

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approach is the following. Suppose we have two volatilities one too high (i.e., giving an option price

greater than the market price) and the other too low (i.e., giving an option price lower than the market

price). By testing the volatility that is half way between the two, we get a new too-high volatility or a

new too-low volatility. If we search initially for two volatilities, one too high and the other too low

we can use this procedure repeatedly to bisect the range and converge on the correct implied

volatility. Other more sophisticated approaches (e.g., involving the Newton-Raphson procedure) are

used in practice.

Q22. Calculate the price of a three-month European put option on a non-dividend-paying stock with

a strike price of $50 when the current stock price is $50, the risk-free interest rate is 10% per annum,

and the volatility is 30% per annum.

In this case

0

S = 50 , K = 50, r = 01, = 03, T = 0 25

, and

1

2 1

ln(50 50) (0 1 0 09 2)0 25 0 2417

0 3 0 25

0 3 0 25 0 0917

d

d d

+ +

= =

= − =

The European put price is

0 1 0 25 50 ( 0 0917) 50 ( 0 2417) N e N −

− − −

0 1 0 25 50 0 4634 50 0 4045 2 37 e

− = − =

3.7 Option hedging and Risk Management

Q23. A financial institution has just sold 1,000 seven-month European call options on the Japanese

yen. Suppose that the spot exchange rate is 0.80 cent per yen, the exercise price is 0.81 cent per yen,

the risk-free interest rate in the United States is 8% per annum, the risk-free interest rate in Japan is 5%

per annum, and the volatility of the yen is 15% per annum. Calculate the delta, gamma, vega, theta,

and rho of the financial institution’s position. Interpret each number.

In this case

0

S = 0 80 , K = 0 81, r = 0 08, rf = 0 05, = 0 15 , T = 0 5833 ( )

2

1

2 1

ln(0 80 0 81) 0 08 0 05 0 15 2 0 5833

0 1016

0 15 0 5833

0 15 0 5833 0 0130

d

d d

+ − +

= =

= − = −

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1 2 N d N d ( ) 0 5405 ( ) 0 4998 = =

The delta of one call option is

0 05 0 5833

1

( ) 0 5405 0 5250 f

r T e N d e − − = = .

2

1 2 0 00516

1

1 1 ( ) 0 3969

2 2

d N d e e − −

= = =

so that the gamma of one call option is

1

0

( ) 0 3969 0 9713 4 206

0 80 0 15 0 5833

f

r T N d e

S T

−

= =

The vega of one call option is

0 1 ( ) 0 80 0 5833 0 3969 0 9713 0 2355 f

r T S T N d e−

= =

The theta of one call option is

0 1

0 1 2

( ) ( ) ( )

2

0 8 0 3969 0 15 0 9713

2 0 5833

0 05 0 8 0 5405 0 9713 0 08 0 81 0 9544 0 4948

0 0399

f

f

r T

r T rT

f

S N d e r S N d e rKe N d

T

−

− −

− + −

= −

+ −

= −

The rho of one call option is

2

( )

0 81 0 5833 0 9544 0 4948

0 2231

rT KTe N d −

=

=

Delta can be interpreted as meaning that, when the spot price increases by a small amount (measured

in cents), the value of an option to buy one yen increases by 0.525 times that amount. Gamma can be

interpreted as meaning that, when the spot price increases by a small amount (measured in cents), the

delta increases by 4.206 times that amount. Vega can be interpreted as meaning that, when the

volatility (measured in decimal form) increases by a small amount, the option’s value increases by

0.2355 times that amount. When volatility increases by 1% (= 0.01) the option price increases by

0.002355. Theta can be interpreted as meaning that, when a small amount of time (measured in years)

passes, the option’s value decreases by 0.0399 times that amount. In particular when one calendar

day passes it decreases by

0 0399 365 0 000109 =

. Finally, rho can be interpreted as meaning that,

when the interest rate (measured in decimal form) increases by a small amount the option’s value

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increases by 0.2231 times that amount. When the interest rate increases by 1% (= 0.01), the options

value increases by 0.002231.

Q24. Delta Hedging

Suppose you are holding the following option portfolio on a given stock.

(a) What is the aggregate Delta, aggregate Gamma, and aggregate Vega of your portfolio?

(b) A traded option is available with a Delta of 0.2, a Gamma of 0.01, and a Vega of 0.02. What

position in the traded option and/or underlying stock would make the portfolio both Delta neutral and

Vega neutral?

(c) There are two traded options available:

What position in the traded option and/or underlying stock would make the portfolio both Gamma

neutral and Vega neutral?

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Exercise

1. An investor shorts a futures contract on an asset when the futures price is $1,000. Each contract is

on 100 units of the asset. The contract is closed out when the futures price is $1,040. Which of the

following is true?

Select one:

a) The investor has made a loss of $4,000

b) The investor has made a gain of $4,000

c) The investor has made a loss of $40

d) The investor has made a gain of $40

2. An investor buys a futures contract on the SPI futures when the index is at 5000. The multiplier on

the index is $25. The investor closes out the futures contract when the futures index is at 5500.

Which of the following is true?

Select one:

a) The investor has made a loss of $25,000

b) The investor has made a gain of $25,000

c) The investor has made a gain of $12,500

d) The investor has made a loss of $12,500

3. An investor shorts a futures contract on an asset when the futures price is $1,000. Each contract is

on 100 units of the asset. The contract is closed out when the futures price is $1,040. Which of the

following is true?

Select one:

a) The investor has made a loss of $4,000

b) The investor has made a gain of $4,000

c) The investor has made a loss of $40

d) The investor has made a gain of $40

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5. The spot price of an investment asset that provides no income is $54 and the risk-free rate for all

maturities (with continuous compounding) is 3%. What is the 4-year forward price? Give your

answer to 2 decimal places.

6. Today the spot rate between US dollars (USD) and Australian dollars (AUD) is USD1 =

AUD1.6290, and the current spot price of gold is USD1,500 per ounce. You are a large Australian

jewellery maker and expect to take delivery of 800 ounces of gold in 3 months time. You decide not

to hedge your exposure. When you are ready to buy the gold in 3 months, the spot price of gold is

USD1,300 per ounce, and the spot exchange rate is USD1 = AUD1.5000. Answer the following

questions.

a) The Australian jewellery maker is exposed to the risk of the gold price ___

b) The Australian jewellery maker is exposed to the risk of the USD ___

c) In three months time, when the Australian jewellery maker buys 800 ounces of gold at the

spot price for gold, the payment in AUD to buy 800 ounces of gold will be AUD ___

Give your answer in millions to two decimal places.

7. You will receive $132 in 11 years. If the discount rate is 12% per annum continuously

compounded, what is the present value of this future cash flow?

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8. When the stock price increases with all else remaining the same, which of the following is true?

Select one:

a) Both calls and puts decrease in value

b) Both calls and puts increase in value

c) Puts increase in value while calls decrease in value

d) Calls increase in value while puts decrease in value

9. We intend to value a European call option with a strike price X=$11 and T=1 year to expiry, using

a one step binomial model, with the following parameters: u=1.1, d=0.9091. The risk-free interest

rate (continuously compounded) is 8% per annum.

The value of the call option in the up state at time 1 is $?

Give your answer correct to two decimal places, or your answer will be incorrect.

The value of the call option in the down state at time 1 is $?

Give your answer correct to two decimal places, or your answer will be incorrect.

The portfolio that replicates the payoff on the call option is a ? position in ? units of shares and a ?

position of $ ? in a riskless bond (that is borrowing). Give your numerical answers to this part of the

question correct to three decimal places.

Hence the value of the call option at time 0 is $? Give your answer correct to two decimal places, or

your answer will be incorrect. Do not include the dollar sign “$” in any of your numerical answers.

10. When we apply the binomial option pricing model to calculate an option value, one approach is to

use risk neutral valuation, where we calculate the risk neutral probability.

We have constructed a 3 period binomial tree to price a 6 month put option. The tree parameters are u

=1.177, d = 0.849, r = 0.05.

The risk neutral probability correct to three decimal places is:

a) 0.486

b) 0.538

c) 0.527

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d) 0.617

11. Risk-neutral valuation and no-arbitrage arguments give the same option prices

Select one:

a) True

b) False

12. To value a European call option on a stock paying a single dividend, you should adjust the

Black-Scholes-Merton formula by adding the present value of the dividend to the stock price.

Select one:

a) True

b) False

13. An American-style call option written on ABC has one month to expiry and a strike price of

$35.00. It is currently trading at $5.40. ABC shares are trading at $40.00. The riskless rate of interest

is 4%.

To make arbitrage profits you should:

Select one:

a) Sell the call option immediately because it is overpriced.

b) Arbitrage profits cannot be made.

c) Sell the call option, immediately exercise the right to sell the ABC shares, immediately buy the

shares, for arbitrage profit of $0.40.

d) Buy the shares and sell the call option, using borrowed funds of $35.60. The arbitrage profit at

maturity is greater than $0.60.

14. When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate

(continuously compounded) is 6%, the volatility is 30% and the time to maturity is 3 months, which

of the following is the price of a European call option on the stock?

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Select one:

a) 19.70N(0.025) – 20N(0.175)

b) 19.70N(0.175) – 20N(0.025)

c) 20N(0.175) – 19.70N(0.025)

d) 20N(0.025) – 19.70N(0.175)

15. The price of a stock, which pays no dividends, is $20 and the strike price of a 1 year European

call option on the stock is $20. The risk-free rate is 4% (continuously compounded). Which of the

following is a lower bound for the option such that there are arbitrage opportunities if the price is

below the lower bound and no arbitrage opportunities if it is above the lower bound?

Select one:

a) $0.00

b) $0.40

c) $0.78

d) $0.59

16. Use the spreadsheet output below to answer the following questions. You must use the values in

this table to calculate your answers.

Black Scholes Merton option price calculation

stock price 20.00

strike price 20.00

interest rate 0.040

volatility 0.300

time to maturity (years) 0.500

d1 0.200

N(d1) 0.579

d2 -0.012

N(d2) 0.495

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BSM call option 1.8781

N(-d1) 0.4206

N(-d2) 0.5047

BSM put option 1.4821

Delta (Call) 0.5794

Delta (Put) -0.4206

gamma 0.0922

a) When the share price decreases by $0.10 the put option price will ? . Using the delta, the new

put option price after the share price has decreased by $0.10 is estimated to be $ ? Write your

answer to two decimal places or your answer will be incorrect.

b) Assume the share price is $20 as in the table. When the share price decreases to $19.90 the

delta of the put option changes. The new delta for the put is estimated using the ? . Using this

approach, the new estimated delta is ? . You must give your answer correct to three decimal

places or your answer will be incorrect.

17. A portfolio contains only long (bought) put options. The gamma of the portfolio is negative.

Select one:

a) True

b) False

18. You are given the following information on shares and options on Sunless Ltd. Sunless shares are

currently trading at $10.30. You construct a portfolio of 200 long call options at an exercise price of

$9.50 and 100 long call options at an exercise price of $10.50 over Sunless shares. Each call option is

over one share.

Black Scholes Merton option price calculation

stock price 10.30 10.30 10.30

strike price 9.50 10.00 10.50

interest rate 0.050 0.050 0.050

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volatility 0.300 0.300 0.300

time to maturity (years) 0.500 0.500 0.500

d1 0.605 0.363 0.133

N(d1) 0.727 0.642 0.553

d2 0.393 0.151 -0.079

N(d2) 0.653 0.560 0.469

BSM call option 1.4439 1.1481 0.8975

N(-d1) 0.2726 0.3582 0.4470

N(-d2) 0.3472 0.4399 0.5314

BSM put option 0.4094 0.6012 0.8382

delta(Call) 0.7274 0.6418 0.5530

delta(Put) -0.2726 -0.3582 -0.4470

Answer the following questions.

a) The cost to set the portfolio up is $ ? . Give your answer correct to 2 decimal places or your

answer will be incorrect.

b) The delta of your portfolio of long call options is ?. Give your answer correct to 2 decimal

places or your answer will be incorrect.

c) To construct a delta neutral portfolio you add shares to the portfolio of long call options. You

must add a ? position in ? shares (round to the nearest whole number) to make your portfolio

(close to) delta neutral. Do NOT include the dollar sign ($) in any answer.

19. What is the value of a European put futures option where the futures price is 50, the strike price is

50, the risk-free rate is 5%, the volatility is 20% and the time to maturity is three months?

a) 50N(0.1)-49.38N(-0.1)

b) 49.38N(0.05)-49.38N(-0.05)

c) 49.38N(0.1)-49.38N(-0.1)

d) 50N(0.05)-50N(-0.05)

Using the Black model spreadsheet:

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Black’s model for futures options

futures (forward) price 50.00

strike price 50.00

interest rate 0.050

volatility 0.200

time to maturity (years) 0.250

d1 0.050

N(d1) 0.520

d2 -0.050

N(d2) 0.480

Black call 1.9691

N(-d1) 0.4801

N(-d2) 0.5199

Black put 1.9691

20.You operate a business in Australia that frequently exports goods to partners in China. As a result,

you often need to receive payments denominated in Renminbi (RMB). This exposes you to foreign

exchange risk, therefore you make use of option contracts to manage this risk. The spot exchange rate

between AUD and RMB is RMB1.0000 = AUD0.2200. The risk free rates of interest (continuously

compounded) in Australia and China are 2% and 4% respectively. The volatility of the RMB/AUD

exchange rate is 10% pa. In this particular case you will be sending a shipment of goods to China

with payment in three (3) months’ time. You will receive RMB1,000,000 for the goods. You decide

to use an at-the-money spot (Strike price is RMB1=AUD0.2200) 3-month maturity option to hedge

your exposure.

a) In this particular case your business is exposed to ? of the RMB.

b) One strategy you could use to hedge your exposure to currency risk is to purchase a ? option

on RMB1,000,000.

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c) Using the Garman-Kohlhagen model, the cost of this option is AUD ? . Give your answer as a

whole number to the nearest dollar.

d) Suppose in three months time the exchange rate is RMB1=AUD0.2000. Then the total receipt

for the goods in AUD (you must account for the cost of the option-ignore the time differential

between payment of the option premium and receipt of payment for the goods) is AUD ? .

Give your answer as a whole number to the nearest dollar.

DO NOT put dollar signs or commas in any of your numerical answers.

GARMAN KOHLHAGEN MODEL

spot rate 0.22

strike rate 0.22

interest rate base

currency (“foreign”) 0.04

interest rate terms

currency (“domestic”) 0.02

volatility 0.1

time to maturity 0.25

d1 -0.075

N(d1) 0.470107356

d2 -0.125

N(d2) 0.450261775

Gk call price 0.003830997

N(-d1) 0.529892644

N(-d2) 0.549738225

Gk put price 0.004922779

21.One-year European call and put options on an asset are worth $4 and $3 respectively when the

strike price is $20 and the one-year risk-free rate is 5%. What is the one-year futures price of the

asset if there are no arbitrage opportunities? (Use put-call parity.)

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Select one:

a) $19.05

b) $21.05

c) $20.95

d) $18.95

22. The yield curve is flat at 5% per annum. What is the value of an FRA where the holder receives

interest at the rate of 7% per annum for a six-month period on a principal of $1,000 starting in two

years? All rates are compounded semi annually.

Select one:

a) $9.52

b) $8.88

c) $8.85

d) $8.84

23. The zero coupon curve is given as follows:

Time (years) 1 2 3 4

Zero Rate 0.03 0.035 0.04 0.046

The three year forward one year rate f(3,4) is equal to

Select one:

a) 4.8%

b) 6.42%

c) 6.00%

d) 5.00%

24. Company X and Company Y have been offered the following rates

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Suppose that Company X borrows fixed and company Y borrows floating. If they enter into a swap

with each other where the apparent benefits are shared equally, what is company X’s effective

borrowing rate?

Select one:

a) 3.3%

b) 3-month LIBOR minus 30bp

c) 3-month LIBOR minus 60 bp

d) 3.0%

25. An interest rate is 6% per annum with semi-annual compounding. What is the equivalent rate

with continuous compounding?

Select one:

a) 6.21%

b) 5.83%

c) 5.91%

d) 5.79%

26. The sale of a covered call is equivalent to the portfolio consisting of a short put and a long bond.

Select one:

a) True

b) False

27. An investor uses a long covered call strategy. The current stock price is $20, strike price for the

option is $22, and the option premium is $1. The investor will start making a profit on this covered

call strategy as long as the stock price at maturity is

Select one:

a) greater than $22

b) greater than $19

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c) the investor will always make a profit from this strategy

d) greater than $20

28.A trader creates a spread by buying a 6-month call option with a $15.00 strike price for $3.45 and

selling a 6-month call option with a $18.00 strike price for $2.20.

The initial cost to set up the strategy is $ ?

Give your answer correct to two decimal places, or your answer will be incorrect.

The breakeven share price for the strategy is $ ?

Give your answer correct to two decimal places, or your answer will be incorrect.

This strategy is called a ? -spread.

29. An investor uses a long protective put strategy. The current stock price is $12.00, the strike price

for the option is $10.00, and the option premium is $1.00. The investor will start making a profit on

this protective put strategy as long as the stock price at maturity is

Select one:

a) Greater than $12.00

b) Greater than $13.00

c) The investor will always make a profit from this strategy

d) Greater than $10.00

30. When we apply the binomial option pricing model to calculate an option value, one approach is to

use risk neutral valuation, where we calculate the risk neutral probability.

We have constructed a 3 period binomial tree to price a 6 month put option. The tree parameters are u

=1.177, d = 0.849, r = 0.05.

The risk neutral probability correct to three decimal places is:

Select one:

a) 0.486

b) 0.538 You have used 0.5 (6/12) as the time step

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c) 0.527

d) 0.617

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