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All main topics / Finance & Investment / Derivatives

Derivatives (239 Cards)

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1.A one-year forward contract is an agreement where
A.One side has the right to buy an asset for a certain price in one year’s time.
B.One side has the obligation to buy an asset for a certain price in one year’s time.
C.One side has the obligation to buy an asset for a certain price at some time during the next year.
D.One side has the obligation to buy an asset for the market price in one year’s time.
Answer: B
A one-year forward contract is an obligation to buy or sell in one year’s time for a predetermined price. By contrast, an option is the right to buy or sell.
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2.Which of the following is NOT true 
A.When a CBOE call option on IBM is exercised, IBM issues more stock
B.An American option can be exercised at any time during its life
C.An call option will always be exercised at maturity if the underlying asset price is greater than the strike price
D.A put option will always be exercised at maturity if the strike price is greater than the underlying asset price.
Answer: A
When an IBM call option is exercised the option seller must buy shares in the market to sell to the option buyer. IBM is not involved in any way. Answers B, C, and D are true.
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A one-year call option on a stock with a strike price of $30 costs $3; a one-year put option on the stock with a strike price of $30 costs $4. Suppose that a trader buys two call options and one put option. The breakeven stock price above which the trader makes a profit is
A.$35
B.$40
C.$30
D.$36

Answer: A
When the stock price is $35, the two call options provide a payoff of 2×(35−30) or $10. The put option provides no payoff. The total cost of the options is 2×3+ 4 or $10.  The stock price in A, $35, is therefore the breakeven stock price above which the position is profitable because it is the price for which the cost of the options equals the payoff.


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4.A one-year call option on a stock with a strike price of $30 costs $3; a one-year put option on the stock with a strike price of $30 costs $4. Suppose that a trader buys two call options and one put option. The breakeven stock price below which the trader makes a profit is
A.$25
B.$28
C.$26
D.$20
Answer: D
When the stock price is $20 the two call options provide no payoff. The put option provides a payoff of 30−20 or $10. The total cost of the options is 2×3+ 4 or $10.  The stock price in D, $20, is therefore the breakeven stock price below which the position is profitable because it is the price for which the cost of the options equals the payoff.
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Which of the following is approximately true when size is measured in terms of the underlying principal amounts or value of the underlying assets
A.The exchange-traded market is twice as big as the over-the-counter market.
B.The over-the-counter market is twice as big as the exchange-traded market.
C.The exchange-traded market is ten times as big as the over-the-counter market.
D.The over-the-counter market is ten times as big as the exchange-traded market.
Answer: D
The OTC market is about $600 trillion whereas the exchange-traded market is about $60 trillion.
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Which of the following best describes the term “spot price”
A.The price for immediate delivery
B.The price for delivery at a future time
C.The price of an asset that has been damaged
D.The price of renting an asset
Answer: A
The spot price is the price for immediate delivery. The futures or forward price is the price for delivery in the future
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Which of the following is true about a long forward contract
A.The contract becomes more valuable as the price of the asset declines
B.The contract becomes more valuable as the price of the asset rises
C.The contract is worth zero if the price of the asset declines after the contract has been entered into
D.The contract is worth zero if the price of the asset rises after the contract has been entered into
Answer:  B
A long forward contract is an agreement to buy the asset at a predetermined price. The contract becomes more attractive as the market price of the asset rises. The contract is only worth zero when the predetermined price in the forward contract equals the current forward price (as it usually does at the beginning of the contract).

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An investor sells a futures contract an asset when the futures price is $1,500. Each contract is on 100 units of the asset. The contract is closed out when the futures price is $1,540. Which of the following is true
A.The investor has made a gain of $4,000
B.The investor has made a loss of $4,000
C.The investor has made a gain of $2,000
D.The investor has made a loss of $2,000
Answer: B
An investor who buys (has a long position) has a gain when a futures price increases. An investor who sells (has a short position) has a loss when a futures price increases.

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Which of the following describes European options?
A.Sold in Europe
B.Priced in Euros
C.Exercisable only at maturity
D.Calls (there are no European puts)
Answer: C
European options can be exercised only at maturity. This is in contrast to American options which can be exercised at any time. The term “European” has nothing to do with geographical location, currencies, or whether the option is a call or a put.

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Which of the following is NOT true
A.A call option gives the holder the right to buy an asset by a certain date for a certain price
B.A put option gives the holder the right to sell an asset by a certain date for a certain price
C.The holder of a call or put option must exercise the right to sell or buy an asset
D.The holder of a forward contract is obligated to buy or sell an asset
Answer: C
The holder of a call or put option has the right to exercise the option but is not required to do so. A, B, and C are correct
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Which of the following is NOT true about call and put options:
A.An American option can be exercised at any time during its life
B. A European option can only be exercised only on the maturity date
C.Investors must pay an upfront price (the option premium) for an option contract
D.The price of a call option increases as the strike price increases
Answer: D
A call option is the option to buy for the strike price. As the strike price increases this option becomes less attractive and is therefore less valuable. A, B, and C are true.
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The price of a stock on July 1 is $57. A trader buys 100 call options on the stock with a strike price of $60 when the option price is $2. The options are exercised when the stock price is $65. The trader’s net profit is
A. $700
B.$500
C. $300
D.$600
Answer: C
The payoff from the options is 100×(65-60) or $500. The cost of the options is 2×100 or $200. The net profit is therefore 500−200 or $300.

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The price of a stock on February 1 is $124. A trader sells 200 put options on the stock with a strike price of $120 when the option price is $5. The options are exercised when the stock price is $110. The trader’s net profit or loss is
A. Gain of $1,000
B.Loss of $2,000
C.Loss of $2,800
D.Loss of $1,000
Answer: D
The payoff that must be made on the options is 200×(120−110) or $2000. The amount received for the options is 5×200 or $1000. The net loss is therefore 2000−1000 or $1000.

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The price of a stock on February 1 is $84. A trader buys 200 put options on the stock with a strike price of $90 when the option price is $10. The options are exercised when the stock price is $85. The trader’s net profit or loss is
A.Loss of $1,000
B.Loss of $2,000
C.Gain of $200
D.Gain of $1000
Answer: A
The payoff is 90−85 or $5 per option. For 200 options the payoff is therefore 5×200 or $1000. However the options cost 10×200 or $2000. There is therefore a net loss of $1000.
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The price of a stock on February 1 is $48. A trader sells 200 put options on the stock with a strike price of $40 when the option price is $2. The options are exercised when the stock price is $39. The trader’s net profit or loss is
A. Loss of $800
B.Loss of $200
C. Gain of $200
D.Loss of $900
Answer: C
The payoff is 40−39 or $1 per option. For 200 options the payoff is therefore 1×200 or $200. However the premium received by the trader is 2×200 or $400. The trader therefore has a net gain of $200.
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16.A speculator can choose between buying 100 shares of a stock for $40 per share and buying 1000 European call options on the stock with a strike price of $45 for $4 per option. For second alternative to give a better outcome at the option maturity, the stock price must be above
A.$45
B.$46
C.$55
D.$50
Answer: D
When the stock price is $50 the first alternative leads to a position in the stock worth 100×50 or $5000. The second alternative leads to a payoff from the options of 1000×(50−45) or $5000. Both alternatives cost $4000. It follows that the alternatives are equally profitable when the stock price is $50. For stock prices above $50 the option alternative is more profitable.
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A company knows it will have to pay a certain amount of a foreign currency to one of its suppliers in the future. Which of the following is true
A.A forward contract can be used to lock in the exchange rate
B.A forward contract will always give a better outcome than an option
C.An option will always give a better outcome than a forward contract
D.An option can be used to lock in the exchange rate
Answer: A
A forward contract ensures that the effective exchange rate will equal the current  forward exchange rate. An option provides insurance that the exchange rate will not be worse than a certain level, but requires an upfront premium. Options sometimes give a better outcome and sometimes give a worse outcome than forwards.
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A short forward contract on an asset plus a long position in a European call option on the asset with a strike price equal to the forward price is equivalent to
A. A short position in a call option
B.A short position in a put option
C. A long position in a put option
D.None of the above
Answer: C
Suppose that ST is the final asset price and K is the strike price/forward price. A short forward contract leads to a payoff of K−ST. A long position in a European call option leads to a payoff of max(ST−K, 0). When added together we see that the total position leads to a payoff of max(0, K−ST), which is the payoff from a long position in a put option. C can also be seen to be true by plotting the payoffs as a function of the final stock price. 
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A trader has a portfolio worth $5 million that mirrors the performance of a stock index. The stock index is currently 1,250. Futures contracts trade on the index with one contract being on 250 times the index. To remove market risk from the portfolio the trader should
A. Buy 16 contracts
B.Sell 16 contracts
C.Buy 20 contracts
D.Sell 20 contracts
Answer: B
One futures contract protects a portfolio worth 1250×250. The number of contract required is therefore 5,000,000/(1250×250)=16.  To remove market risk we need to gain on the contracts when the market declines. A short futures position is therefore required.

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Which of the following best describes a central clearing party
A.It is a trader that works for an exchange
B.It stands between two parties in the over-the-counter market
C.It is a trader that works for a bank
D.It helps facilitate futures trades
Answer: B
A central clearing party (CCP) is a clearing house that stands between two parties in the over-the-counter market. It serves the same purpose as an exchange clearing house.
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Which of the following is true
A.Both forward and futures contracts are traded on exchanges.
B. Forward contracts are traded on exchanges, but futures contracts are not.
C.Futures contracts are traded on exchanges, but forward contracts are not.
D.Neither futures contracts nor forward contracts are traded on exchanges.
Answer: C

Futures contracts trade only on exchanges. Forward contracts trade only in the over-the-counter market.
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Which of the following is NOT true
A.Futures contracts nearly always last longer than forward contracts
B.Futures contracts are standardized; forward contracts are not.
C.Delivery or final cash settlement usually takes place with forward contracts; the same is not true of futures contracts.
D.Forward contracts usually have one specified delivery date; futures contract often have a range of delivery dates.
Answer: A

Forward contracts often last longer than futures contracts. B, C, and D are true
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In the corn futures contract a number of different types of corn can be delivered (with price adjustments specified by the exchange) and there are a number of different delivery locations. Which of the following is true
A. This flexibility tends increase the futures price.
B.This flexibility tends decrease the futures price.
C.This flexibility may increase and may decrease the futures price.
D.This flexibility has no effect on the futures price
Answer: B

The party with the short position chooses between the alternatives. The alternatives therefore make the futures contract more attractive to the party with the short position. The lower the futures price the less attractive it is to the party with the short position. The benefit of the alternatives available to the party with the short position is therefore compensated for by the futures price being lower than it would otherwise be.
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A company enters into a short futures contract to sell 50,000 units of a commodity for 70 cents per unit. The initial margin is $4,000 and the maintenance margin is $3,000. What is the futures price per unit above which there will be a margin call?
A.78 cents
B.76 cents
C.74 cents
D.72 cents 

Answer: D

There will be a margin call when more than $1000 has been lost from the margin account so that the balance in the account is below the maintenance margin level.  Because the company is short, each one cent rise in the price leads to a loss or 0.01×50,000 or $500. A greater than 2 cent rise in the futures price will therefore lead to a margin call. The future price is currently 70 cents. When the price rises above 72 cents there will be a margin call.
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A company enters into a long futures contract to buy 1,000 units of a commodity for $60 per unit. The initial margin is $6,000 and the maintenance margin is $4,000. What futures price will allow $2,000 to be withdrawn from the margin account?  
A.    $58
B.    $62
C.    $64
D. $66
Answer: B

Amounts in the margin account in excess of the initial margin can be withdrawn. Each $1 increase in the futures price leads to a gain of $1000. When the futures price increases by $2 the gain will be $2000 and this can be withdrawn. The futures price is currently $60. The answer is therefore $62.
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One futures contract is traded where both the long and short parties are closing out existing positions. What is the resultant change in the open interest?
A. No change
B.Decrease by one
C.Decrease by two
D.Increase by one
Answer: B

The open interest goes down by one. There is one less long position and one less short position.
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Who initiates delivery in a corn futures contract
A.The party with the long position
B.The party with the short position
C.Either party 
D.The exchange

Answer: B

The party with the short position initiates delivery by sending a “Notice of Intention to Deliver” to the exchange. The exchange has a procedure for choosing a party with a long position to take delivery.

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You sell one December futures contracts when the futures price is $1,010 per unit. Each contract is on 100 units and the initial margin per contract that you provide is $2,000. The maintenance margin per contract is $1,500. During the next day the futures price rises to $1,012 per unit. What is the balance of your margin account at the end of the day?  
A.$1,800
B.$3,300
C.$2,200
D.$3,700
Answer: B

The price has increased by $2. Because you have a short position you lose 2×100 or $200.  The balance in the margin account therefore goes down from $3,500 to $3,300.
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A hedger takes a long position in a futures contract on a commodity on November 1, 2012 to hedge an exposure on March 1, 2013. The initial futures price is $60. On December 31, 2012 the futures price is $61. On March 1, 2013 it is $64. The contract is closed out on March 1, 2013. What gain is recognized in the accounting year January 1 to December 31, 2013? Each contract is on 1000 units of the commodity.
A. $0
B.$1,000
C.$3,000
D.$4,000
Answer: D

Hedge accounting is used. The whole of the gain or loss on the futures is therefore recognized in 2013. None is recognized in 2012. In this case the gain is $4 per unit or $4,000 in total.
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A speculator takes a long position in a futures contract on a commodity on November 1, 2012 to hedge an exposure on March 1, 2013. The initial futures price is $60. On December 31, 2012 the futures price is $61. On March 1, 2013 it is $64. The contract is closed out on March 1, 2013. What gain is recognized in the accounting year January 1 to December 31, 2013? Each contract is on 1000 units of the commodity.

A. $0
B. $1,000
C.$3,000
D.$4,000
Answer: C

In this case there is no hedge accounting. Gains or losses are accounted for as they are accrued. The price per unit increases by $3 in 2013. The total gain in 2013 is therefore $3,000.
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The frequency with which futures margin accounts are adjusted for gains and losses is
A.Daily
B.Weekly
C.Monthly
D.Quarterly

Answer: A

In futures contracts margin accounts are adjusted for gains or losses daily.
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Margin accounts have the effect of
A.Reducing the risk of one party regretting the deal and backing out
B. Ensuring funds are available to pay traders when they make a profit
C.Reducing systemic risk due to collapse of futures markets
D.All of the above
Answer: D

Initial margin requirements dramatically reduce the risk that a party will walk away from a futures contract. As a result they reduce the risk that the exchange clearing house will not have enough funds to pays profits to traders. Furthermore, if traders are less likely to suffer losses because of counterparty defaults there is less systemic risk.
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Which entity in the United States takes primary responsibility for regulating futures market?
A.Federal Reserve Board
B.Commodities Futures Trading Commission (CFTC)
C.Security and Exchange Commission (SEC)
D.US Treasury
Answer: B

The CFTC has primary responsibility for regulating futures markets
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For a futures contract trading in April 2012, the open interest for a June 2012 contract, when compared to the open interest for Sept 2012 contracts, is usually
A.Higher
B.Lower
C.The same
D.Equally likely to be higher or lower
Answer: A

The contracts which are close to maturity tend to have the highest open interest. However, during the  maturity month itself the open interest declines.
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15.Clearing houses are
A.Never used in futures markets and sometimes used in OTC markets
B.Used in OTC markets, but not in futures markets
C.Always used in futures markets and sometimes used in OTC markets
D.Always used in both futures markets and OTC markets
Answer: C

Clearing houses are always used by exchanges trading futures. Increasingly, OTC products are cleared through CCPs, which are a type of clearing house.
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A haircut of 20% means that
A.A bond with a market value of $100 is considered to be worth $80 when used to satisfy a collateral request
B. A bond with a face value of $100 is considered to be worth $80 when used to satisfy a collateral request
C.A bond with a market value of $100 is considered to be worth $83.3 when used to satisfy a collateral request
D.A bond with a face value of $100 is considered to be worth $83.3 when used to satisfy a collateral request
Answer: A

A haircut is the amount the market price of asset is reduced by for the purposes of determining its value for collateral purposes. A is therefore correct.
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With bilateral clearing, the number of agreements between four dealers, who trade with each other,  is
A.12
B.1
C.6
D.2
Answer: C
Suppose the dealers are W, X, Y , and Z. The agreements are between W and X, W and Y, W and Z, X and Y, X and Z, and Y and Z. There are therefore a total of 6 agreements.
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Which of the following best describes central clearing parties
A.Help market participants to value derivative transactions
B.Must be used for all OTC derivative transactions
C.Are used for futures transactions
D.Perform a similar function to exchange clearing houses

Answer: D

CCPs do for the OTC market what exchange clearing houses do for the exchange-traded market. The correct answer is therefore D. CCPs must be used for most standard OTC derivatives transactions, but not for all derivatives transactions.
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Which of the following are cash settled
A.All futures contracts
B.All option contracts
C.Futures on commodities
D.Futures on stock indices
Answer: D

Futures on stock indices are usually cash settled. The rest are settled by delivery of the underlying assets
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A limit order
A.Is an order to trade up to a certain number of futures contracts at a certain price
B. Is an order that can be executed at a specified price or one more favorable to the investor
C.Is an order that must be executed within a specified period of time
D.None of the above
Answer: B

In a limit order a trader specifies the worst price (from the trader’s perspective) at which the trade can be carried out.
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The compounding frequency for an interest rate defines
A.The frequency with which interest is paid
B. A unit of measurement for the interest rate
C.The relationship between the annual interest rate and the monthly interest rate
D.None of the above
Answer: B

The compounding frequency is a unit of measurement. The frequency with which interest is paid may be different from the compounding frequency used for quoting the rate.
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An interest rate is 6% per annum with annual compounding. What is the equivalent rate with continuous compounding?
A.5.79%
B.6.21%
C.5.83%
D.6.18%
Answer: C

The equivalent rate with continuous compounding is ln(1.06) = 0.0583 or 5.83%.

Rc= mln(1+ Rm/m)
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An interest rate is 5% per annum with continuous compounding. What is the equivalent rate with semiannual compounding?
A.5.06%
B.5.03%
C.4.97%
D.4.94%
Answer: A

The equivalent rate with semiannual compounding is 2×(e0.05/2−1) = 0.0506 or 5.06%.

m(eRc/m -1) =Rm
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An interest rate is 12% per annum with semiannual compounding. What is the equivalent rate with quarterly compounding?
A. 11.83%
B.11.66%
C.11.77%
D.11.92%
Answer: A

The equivalent rate per quarter is 1.06^0.5 -1=2.956% . The annualized rate with quarterly compounding is four times this or 11.83%.
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The two-year zero rate is 6% and the three year zero rate is 6.5%. What is the forward rate for the third year? All rates are continuously compounded.
A.6.75%
B.7.0%
C.7.25%
D.7.5%
Answer: D

The forward rate for the third year is (3×0.065−2×0.06)/(3−2) = 0.075 or 7.5%
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The six-month zero rate is 8% per annum with semiannual compounding. The price of a one-year bond that provides a coupon of 6% per annum semiannually is 97. What is the one-year continuously compounded zero rate?
A.8.02%
B.8.52%
C.9.02%
D.9.52%
Answer: C

If the rate is R we must have

97= 3/1.04+103e^-R*1

or

e^-R= (97-3/1.04)/103 =0.9137

so that R = ln(1/0.9137) = 0.0902 or 9.02%.

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The yield curve is flat at 6% per annum. What is the value of an FRA where the holder receives interest at the rate of 8% per annum for a six-month period on a principal of $1,000 starting in two years?  All rates are compounded semiannually.
A.$9.12
B.$9.02
C.$8.88
D.$8.63
Answer: D

The value of the FRA is the value of receiving an extra 0.5×(0.08−0.06)×1000 = $10 in 2.5 years. This is 10/(1.035) = $8.63
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Under liquidity preference theory, which of the following is always true? 
A.The forward rate is higher than the spot rate when both have the same maturity.
B.Forward rates are unbiased predictors of expected future spot rates.
C.The spot rate for a certain maturity is higher than the par yield for that maturity.
D.Forward rates are higher than expected future spot rates.

Answer: D

Liquidity preference theory argues that individuals like their borrowings to have a long maturity and their deposits to have a short maturity. To induce people to lend for long periods forward rates are raised relative to what expected future short rates would predict.
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The zero curve is upward sloping. Define X as the 1-year par yield, Y as the 1-year zero rate and Z as the forward rate for the period between 1 and 1.5 year. Which of the following is true?
A.X is less than Y which is less than Z
B.Y is less than X which is less than Z
C.X is less than Z which is less than Y 
D.Z is less than Y which is less than X
E.X is less than Y which is less than Z
F.Y is less than X which is less than Z
G.X is less than Z which is less than Y 
H.Z is less than Y which is less than X
Answer: A

When the zero curve is upward sloping, the one-year zero rate is higher than the one-year par yield and the forward rate corresponding to the period between 1.0 and 1.5 years is higher than the one-year zero rate. The correct answer is therefore A.
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10.Which of the following is true of the fed funds rate
A.It is the same as the Treasury rate
B.It is an overnight interbank rate
C.It is a rate for which collateral is posted
D.It is a type of repo rate

Answer: B

At the end of each day some banks have surplus reserves on deposit with the Federal Reserve others have deficits. They use overnight borrowing and lending at what is termed the fed funds rate to rectify this.

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The modified duration of a bond portfolio worth $1 million is 5 years. By approximately how much does the value of the portfolio change if all yields increase by 5 basis points?
A.Increase of  $2,500
B.Decrease of $2,500
C.Increase of  $25,000
D.Decrease of $25,000
Answer: B

When yields increase bond prices decrease. The proportional decrease is the modified duration times the yield increase. In this case, it is 5×0.0005=0.0025. The decrease is therefore 0.0025×1,000,000 or $2,500.
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A company invests $1,000 in a five-year zero-coupon bond and $4,000 in a ten-year zero-coupon bond. What is the duration of the portfolio?
A. 6 years
B. 7 years
C. 8 years
D. 9 years
Answer: D

The duration of the first bond is 5 years and the duration of the second bond is 10 years. The duration of the portfolio is a weighted average with weights corresponding to the amounts invested in the bonds. It is 0.2×5+0.8×10=9 years.
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Which of the following is true of LIBOR
A.The LIBOR rate is free of credit risk
B.A LIBOR rate is lower than the Treasury rate when the two have the same maturity
C.It is a rate used when borrowing and lending takes place between banks
D.It is subject to favorable tax treatment in the U.S.


Answer:  C

LIBOR is a rate used for interbank transactions.
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Which of following describes forward rates?
A.Interest rates implied by current zero rates for future periods of time
B.Interest rate earned on an investment that starts today and last for n-years in the future without coupons
C.The coupon rate that causes a bond price to equal its par (or principal) value
D.A single discount rate that gives the value of a bond equal to its market price when applied to all cash flows
Answer: A

The forward rate is the interest rate implied by the current term structure for future periods of time. For example, earning the zero rate for one year and the forward rate for the period between one and two years gives the same result as earning the zero rate for two years.

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Which of the following is NOT a theory of the term structure
A.Expectations theory
B.Market segmentation theory
C.Liquidity preference theory
D.Maturity preference theory
Answer: C

Maturity preference theory is not a theory of the term structure. The other three are.
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A repo rate is
A.An uncollateralized rate
B.A rate where the credit risk is relative high
C.The rate implicit in a transaction where securities are sold and bought back later at a higher price
D.None of the above
Answer:  C

A repo transaction is one where a company agrees to sell securities today and buy them back at a future time. It is a form of collateralized borrowing. The credit risk is very low.
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Bootstrapping involves
A.Calculating the yield on a bond
B.Working from short maturity instruments to longer maturity instruments determining zero rates at each step
C.Working from long maturity instruments to shorter maturity instruments determining zero rates at each step
D.The calculation of par yields
Answer: B


Bootstrapping is a way of constructing the zero coupon yield curve from coupon-bearing bonds. It involves working from the shortest maturity bond to progressively longer maturity bonds making sure that the calculated zero coupon yield curve is consistent with the market prices of the instruments.
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The zero curve is downward sloping. Define X as the 1-year par yield, Y as the 1-year zero rate and Z as the forward rate for the period between 1 and 1.5 year. Which of the following is true?
A.X is less than Y which is less than Z
B.Y is less than X which is less than Z
C.X is less than Z which is less than Y 
D.Z is less than Y which is less than X

Answer: D

The forward rate accentuates trends in the zero curve. The par yield shows the same trends but in a less pronounced way.
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Which of the following is true?
A.When interest rates in the economy increase, all bond prices increase
B. As its coupon increases, a bond’s price decreases
C.Longer maturity bonds are always worth more that shorter maturity bonds when the coupon rates are the same
D.None of the above

Answer: D

When interest rates increase the impact of discounting is to make future cash flows worth less. Bond prices therefore decline. A is therefore wrong. As coupons increase a bond becomes more valuable because higher cash flows will be received. B is therefore wrong. When the coupon is higher than prevailing interest rates, longer maturity bonds are worth more than shorter maturity bonds. When it is less than prevailing interest rates, longer maturity bonds are worth less than shorter maturity bonds. C is therefore not true. The correct answer is therefore D.
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The six month and one-year rates are 3% and 4% per annum with semiannual compounding. Which of the following is closest to the one-year par yield expressed with semiannual compounding?
A.3.99%
B.3.98%
C.3.97%
D.3.96%
Answer: A

The six month rate is 1.5% per six months. The one year rate is 2% per six months. The one year par yield is the coupon that leads to a bond being worth par. A is the correct answer because  (3.99/2)/1.015+(100+3.99/2)/1.022 = 100. The formula in the text can also be used to give the par yield as [(100-100/1.022)×2]/(1/1.015+1.022)=3.99.

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Which of the following is a consumption asset?
A.The S&P 500 index
B.The Canadian dollar
C.Copper
D.IBM stock
Answer: C

A, B, and D are investment assets (held by at least some investors purely for investment purposes). C is a consumption asset.
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An investor shorts 100 shares when the share price is $50 and closes out the position six months later when the share price is $43. The shares pay a dividend of $3 per share during the six months. How much does the investor gain? 
A.  $1,000
B.  $400
C.  $700
D.  $300
Answer: B

The investor gains $7 per share because he or she sells at $50 and buys at $43. However, the investor has to pay the $3 per share dividend. The net profit is therefore 7−3 or $4 per share. 100 shares are involved. The total gain is therefore $400.
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The spot price of an investment asset that provides no income is $30 and the risk-free rate for all maturities (with continuous compounding) is 10%. What is the three-year forward price? 
A.  $40.50
B.  $22.22
C.  $33.00
D.$33.16
Answer:  A

The 3-year forward price is the spot price grossed up for 3 years at the risk-free rate. It is 30e0.1×3 =$40.50.
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The spot price of an investment asset is $30 and the risk-free rate for all maturities is 10% with continuous compounding. The asset provides an income of $2 at the end of the first year and at the end of the second year. What is the three-year forward price?
A.  $19.67
B.  $35.84
C.  $45.15
D.  $40.50
Answer: B

The present value of the income is 2e-0.1×1+2e-0.1×2= $3.447. The three year forward price is obtained by subtracting the present value of the income from the current stock price and then grossing up the result for three years at the risk-free rate. It is (30−3.447)e0.1×3 = $35.84.

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An exchange rate is 0.7000 and the six-month domestic and foreign risk-free interest rates are 5% and 7% (both expressed with continuous compounding). What is the six-month forward rate?
A. 0.7070
B.0.7177
C.0.7249
D.0.6930
Answer: D

The six-month forward rate is 0.7000e−(0.05−0.07)×0.5=0.6930.
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Which of the following is true? 
A.The convenience yield is always positive or zero.
B.The convenience yield is always positive for an investment asset.
C.The convenience yield is always negative for a consumption asset.
D.The convenience yield measures the average return earned by holding futures contracts.
Answer: A

The convenience yield measures the benefit of owning an asset rather than having a forward/futures contract on an asset. For an investment asset it is always zero. For a consumption asset it is greater than or equal to zero.
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A short forward contract that was negotiated some time ago will expire in three months and has a delivery price of $40. The current forward price for three-month forward contract is $42. The three month risk-free interest rate (with continuous compounding) is 8%. What is the value of the short forward contract?
A.  +$2.00
B.  −$2.00
C.  +$1.96
D. −$1.96
Answer: D

The contract gives one the obligation to sell for $40 when a forward price negotiated today would give one the obligation to sell for $42. The value of the contract is the present value of −$2 or   −2e-0.08×0.25 = −$1.96.
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The spot price of an asset is positively correlated with the market. Which of the following would you expect to be true?
A.The forward price equals the expected future spot price.
B.The forward price is greater than the expected future spot price.
C.The forward price is less than the expected future spot price.
D.The forward price is sometimes greater and sometimes less than the expected future spot price.

Answer:  C

When the spot price is positively correlated with the market the forward price is less than the expected future spot price. This is because the spot price is expected to provide a return greater than the risk-free rate and the forward price is the spot price grossed up at the risk-free rate.
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Which of the following describes the way the futures price of a foreign currency is quoted by the CME group?
A.The  number of U.S. dollars per unit of the foreign currency
B.The number of the foreign currency per U.S. dollar
C.Some futures prices are always quoted as the number of U.S. dollars per unit of the foreign currency and some are always quoted the other way round
D.There are no quotation conventions for futures prices
Answer:  A

The futures price is quoted as the number of US dollars per unit of the foreign currency. Spot exchange rates and forward exchange rates are sometimes quoted this way and sometimes quoted the other way round.
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Which of the following describes the way the forward price of a foreign currency is quoted?
A.The  number of U.S. dollars per unit of the foreign currency
B.The number of the foreign currency per U.S. dollar
C.Some forward prices are quoted as the number of U.S. dollars per unit of the foreign currency and some are quoted the other way round
D.There are no quotation conventions for forward prices
Answer:  C
The futures price is quoted as the number of US dollars per unit of the foreign currency. Spot exchange rates and forward exchange rates are sometimes quoted this way and sometimes quoted the other way round.
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Which of the following is NOT a reason why a short position in a stock is closed out?
A.The investor with the short position chooses to close out the position
B.The lender of the shares issues instructions to close out the position
C.The broker is no longer able to borrow shares from other clients
D.The investor does not maintain margins required on his/her margin account
Answer: B

A, C, and D are all reasons why the short position might be closed out. B is not. The lender of shares cannot issue instructions to close out the short position.

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Which of the following is NOT true?
A.Gold and silver are investment assets
B.Investment assets are held by significant numbers of investors for investment purposes
C.Investment assets are never held for consumption
D.The forward price of an investment asset can be obtained from the spot price, interest rates, and the income paid on the asset
Answer: C

Investment assets are sometimes held for consumption. Silver is an example. To be an investment asset, an asset has to be held for investment by at least some traders

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What should a trader do when the one-year forward price of an asset is too low? Assume that the asset provides no income.
A.The trader should borrow the price of the asset, buy one unit of the asset and enter into a short forward contract to sell the asset in one year.
B.The trader should borrow the price of the asset, buy one unit of the asset and enter into a long forward contract to buy the asset in one year.
C.The trader should short the asset, invest the proceeds of the short sale at the risk-free rate, enter into a short forward contract to sell the asset in one year 
D.The trader should short the asset, invest the proceeds of the short sale at the risk-free rate, enter into a long forward contract to buy the asset in one year 
Answer: D

If the forward price is too low relative to the spot price the trader should short the asset in the spot market and buy it in the forward market.
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Which of the following is NOT true about forward and futures contracts?
A.Forward contracts are more liquid than futures contracts
B.The futures contracts are traded on exchanges while forward contracts are traded in the over-the-counter market
C.In theory forward prices and futures prices are equal when there is no uncertainty about future interest rates
D.Taxes and transaction costs can lead to forward and futures prices being different
Answer: A
Futures contracts are more liquid than forward contracts. To unwind a futures position it is simply necessary to take an offsetting position. The statements in B, C, and D are correct
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As the convenience yield increases, which of the following is true?
A.The one-year futures price as a percentage of the spot price increases
B.The one-year futures price as a percentage of the spot price decreases
C.The one-year futures price as a percentage of the spot price stays the same
D.Any of the above can happen
Answer: B
As the convenience yield increases, the futures price declines relative to the spot price. This is because the convenience of owning the asset (as opposed to having a futures contract) becomes more important.
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As inventories of a commodity decline, which of the following is true?
A.The one-year futures price as a percentage of the spot price increases
B.The one-year futures price as a percentage of the spot price decreases
C.The one-year futures price as a percentage of the spot price stays the same
D.Any of the above can happen
Answer: B

When inventories decline, the convenience yield increases and the futures price as a percentage of the spot price declines.

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Which of the following describes a known dividend yield on a stock?
A.The size of the dividend payments each year is known
B.Dividends per year as a percentage of today’s stock price are known
C.Dividends per year as a percentage of the stock price at the time when dividends are paid are known 
D.Dividends will yield a certain return to a person buying the stock today

Answer: C

The dividend yield is the dividend per year as a percent of the stock price at the time when the dividend is paid.
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Which of the following is an argument used by Keynes and Hicks?
A.If hedgers hold long positions and speculators holds short positions, the futures price will tend to be higher than the expected future spot price
B.If hedgers hold long positions and speculators holds short positions, the futures price will tend to be lower than the expected future spot price
C.If hedgers hold long positions and speculators holds short positions, the futures price will tend to be lower than today’s spot price
D.If hedgers hold long positions and speculators holds short positions, the futures price will tend to be higher than today’s spot price
Answer: A

Keynes and Hicks argued that hedgers will be prepared to accept negative returns on average because of the benefits of hedging whereas speculators require positive returns on average. This leads to A.
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19.Which of the following describes contango?
A.The futures price is below the expected future spot price
B.The futures price is below today’s spot price
C.The futures price is a declining function of  the time to maturity
D.The futures price is above the expected future spot price

Answer: D

Contango is defined as the futures price being above the expected future spot price. It is also sometimes used to describe the situation where the futures price is above the spot price.
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Which of the following is true for a consumption commodity?
A.There is no limit to how high or low the futures price can be, except that the futures price cannot be negative
B.There is a lower limit to the futures price but no upper limit
C.There is an upper limit to the futures price but no lower limit, except that the futures price cannot be negative
D.The futures price can be determined with reasonable accuracy from the spot price and interest rates
Answer: C

If the futures price of a consumption commodity becomes too high an arbitrageur will buy the commodity and sell futures to lock in a profit. An arbitrageur cannot follow the opposite strategy of buying futures and selling or shorting the asset when the futures price is low.  This is because consumption assets cannot be shorted . Furthermore, people who hold the asset in general do so because they need the asset for their business. They are not prepared to swap their position in the asset for a similar position in a futures. Consequently, there is an upper limit but no lower limit to the futures price.


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A company can invest funds for five years at LIBOR minus 30 basis points. The five-year swap rate is 3%. What fixed rate of interest can the company earn by using the swap? 
A.2.4%
B.2.7%
C.3.0%
D.3.3%
Answer: B

When the company invests at LIBOR minus 0.3% and then enters into a swap where it pays LIBOR and receives 3% it earns 2.7% per annum. Note that it is the bid rate that will apply to the swap.
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Which of the following is true?
A.Principals are not usually exchanged in a currency swap
B.The principal amounts usually flow in the opposite direction to interest payments at the beginning of a currency swap and in the same direction as interest payments at the end of the swap.
C.The principal amounts usually flow in the same direction as interest payments at the beginning of a currency swap and in the opposite direction to interest payments at the end of the swap.
D.Principals are not usually specified in a currency swap
Answer: B

The correct answer is B. There are two principals in a currency swap, one for each currency. They flow in the opposite direction to the corresponding interest payments at the beginning of the life of the swap and in the same direction as the corresponding interest payments at the end of the life of the swap.
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Company X and Company Y have been offered the following rates
        
              Fixed Rate  Floating Rate
Company X3.5%  3-month LIBOR plus 10bp
Company Y4.5%  3-month LIBOR plus 30 bp

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?
A.3-month LIBOR−30bp
B.3.1%
C.3-month LIBOR−10bp
D.3.3%
Answer: A
The interest rate differential between the fixed rates is 100 basis points. The interest rate differential between the floating rates is 20 basis points. The difference between the interest rates differentials is 100 – 20 = 80 basis points. This is the total apparent gain from the swap to the two sides. Since the benefits are shared equally company X should be able to borrow at 40 bp less than it is currently offered in the floating rate market, i.e., at LIBOR minus 30 bp.
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Which of the following describes the five-year swap rate?
A.The fixed rate of interest which a swap market maker is prepared to pay in exchange for LIBOR on a 5-year swap
B.The fixed rate of interest which a swap market maker is prepared to receive in exchange for LIBOR on a 5-year swap
C.The average of A and B
D.The higher of A and B
Answer:  C

The swap rate is the average of the bid swap rate (i.e. A) and the offer swap rate (i.e. B)
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Which of the following is a use of a currency swap?
A.To exchange an investment in one currency for an investment in another currency
B.To exchange borrowing in one currency for borrowings in another currency
C.To take advantage situations where the tax rates in two countries are different
D.All of the above
Answer: D

A currency swap can be used for any of A, B, and C.

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The reference entity in a credit default swap is
A.The buyer of protection
B.The seller of protection
C.The company or country whose default is being insured against
D.None of the above
Answer: C
In a credit default swap the buyer of protection pays a CDS spread to the seller of protection and the protection seller has to make a payoff if there is a default by the reference entity.
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Which of the following describes an interest rate swap?
A.The exchange of a fixed rate bond for a floating rate bond
B.A portfolio of forward rate agreements
C.An agreement to exchange interest at a fixed rate for interest at a floating rate
D.All of the above
Answer: D

The answer is D because all of A, B, and C are true for an interest rate swap.
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Which of the following is true for an interest rate swap?
A.A swap is usually worth close to zero when it is first negotiated
B. Each forward rate agreement underlying a swap is worth close to zero when the swap is first entered into
C.Comparative advantage is a valid reason for entering into the swap
D.None of the above
Answer: A

A swap is worth close to zero at the beginning of its life. (It may not be worth exactly zero because of the impact of the market maker’s bid-offer spread.) It is not true that each of the forward contracts underlying the swap are worth zero. (The sum of the value of the forward contracts is zero, but this does not mean that each one is worth zero.) The remaining floating payments on a swap are worth the notional principal immediately after a swap payment date, but this is not necessarily true for the remaining fixed payments.
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Which of the following is true for the party paying fixed in a newly negotiated interest rate swap when the yield curve is upward sloping?
A.The early forward contracts underlying the swap have a positive value and the later ones have a negative value
B.The early forward contracts underlying the swap have a negative value and the later ones have a positive value
C.The swap is designed so that all forward rates have zero value
D.Sometimes A is true and sometimes B is true
Answer: B

The forward contracts are contracts where fixed is paid and floating is received. They can be  valued assuming that forward rates are realized. Forward rates increase with maturity. This means that the value of the forward contracts increase with maturity.  The total value of the forward contracts is zero. This means that the value of the early contracts is negative and the value of the later contracts is positive.
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A bank enters into a 3-year swap with company X where it pays LIBOR and receives 3.00%. It enters into an offsetting swap with company Y where is receives LIBOR and pays 2.95%. Which of the following is true:
A.If company X defaults, the swap with company Y is null and void
B.If company X defaults, the bank will be able to replace company X at no cost
C.If company X defaults, the swap with company Y continues
D.The bank’s bid-offer spread is 0.5 basis points
Answer: C

The bank`s bid-offer spread is 5 basis points not 0.5 basis points. The bank has quite separate transactions with X and Y. If one defaults, it still has to honor the swap with the other.
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When LIBOR is used as the discount rate:
A.The value of a swap is worth zero immediately after a payment date
B. The value of a swap is worth zero immediately before a payment date
C.The value of the floating rate bond underlying a swap is worth par immediately after a payment date
D.The value of the floating rate bond underlying a swap is worth par immediately before a payment date
Answer:  C

The value of the floating rate bond underlying an interest rate swap is worth par immediately after a swap payment date. This result is used when the swap is valued as the difference between two bonds. 
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A company enters into an interest rate swap where it is paying fixed and receiving LIBOR. When interest rates increase, which of the following is true?
A.The value of  the swap to the company increases
B.The value of the swap to the company decreases
C.The value of the swap can either increase or decrease
D.The value of the swap does not change providing the swap rate remains the same

Answer:  A

It is receiving the floating rate. When interest rates increase the floating rate can be expected to be higher and so the swap becomes more valuable. The answer is therefore A.
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A floating for floating currency swap is equivalent to
A.Two interest rate swaps, one in each currency
B.A fixed-for-fixed currency swap and one interest rate swap
C.A fixed-for-fixed currency swap and two interest rate swaps, one in each currency
D.None of the above
Answer: C

A floating-for-floating currency swap where the currency paid is  X and the currency received is Y is equivalent to (a) a fixed-for-fixed currency swap where, say, 5% in currency X is paid and say, say, 4% in currency Y is received, (b) a regular interest rate swap where 5% in currency X is received and floating in currency X is paid and (c) a regular interest rate swap where 4% in currency Y is paid and floating in currency Y is received.

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A floating-for-fixed currency swap is equivalent to
A.Two interest rate swaps, one in each currency
B.A fixed-for-fixed currency swap and one interest rate swap
C.A fixed-for-fixed currency swap and two interest rate swaps, one in each currency
D.None of the above

Answer: B

A floating-for-fixed currency swap where the floating rate is paid in currency X and the fixed rate is received in currency Y is equivalent to (a) a fixed-for-fixed currency swap where, say, 5% in currency X is paid and the fixed rate in currency Y is received, (b) a regular interest rate swap where 5% in currency X is received and floating in currency X is paid.
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An interest rate swap has three years of remaining life. Payments are exchanged annually. Interest at 3% is paid and 12-month LIBOR is received. A exchange of payments has just taken place. The one-year, two-year and three-year LIBOR/swap zero rates are 2%, 3% and 4%. All rates an annually compounded. What is the value of the swap as a percentage of the principal when LIBOR discounting is used.
A. 0.00
B.2.66
C. 2.06
D.1.06
Answer: B

Suppose the principal 100. The value of the floating rate bond underlying the swap is 100. The value of the fixed rate bond is 3/1.02+3/(1.03)2+103/(1.04)3=97.34. The value of the swap is therefore 100−97.34 = 2.66 or 2.66% of the principal
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A semi-annual pay interest rate swap where the fixed rate is 5.00% (with semi-annual compounding) has a remaining life of nine months.  The six-month LIBOR rate observed three months ago was 4.85% with semi-annual compounding. Today’s three and nine month LIBOR rates are 5.3% and 5.8% (continuously compounded) respectively. From this it can be calculated that the forward LIBOR rate for the period between three- and nine-months is 6.14% with semi-annual compounding.  If the swap has a principal value of $15,000,000, what is the value of the swap to the party receiving a fixed rate of interest?
A.$74,250
B.−$70,760
C.−$11,250
D.$103,790
Answer:  B

The forward rates for the floating payment at time 9 months is 6.14%. The swap can be valued assuming that the fixed payments are 2.5% of principal at 3 months and 9 months and that the floating payments are 2.425% and 3.07% of the principal at 3 months and 9 months. The value of the swap to the party receiving fixed is therefore
1,000,000(0.025-0.02425)e-0.053×0.25+1,000,000(0.025-0.0307)e-0.058×0.75  = –$70,760
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Which of the following describes the way a LIBOR-in-arrears swap differs from a plain vanilla interest rate swap?
A.Interest is paid at the beginning of the accrual period in a LIBOR-in-arrears swap
B.Interest is paid at the end of the accrual period in a LIBOR-in-arrears swap
C.No floating interest is paid until the end of the life of the swap in a LIBOR-in-arrears swap, but fixed payments are made throughout the life of the swap
D.Neither floating nor fixed payments are made until the end of the life of the swap
Answer:  A

In a LIBOR-in-arrears swap interest is observed for an accrual period and paid at the beginning of that accrual period (not at the end of the accrual period which is normal)

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In a fixed-for-fixed currency swap, 3% on a US dollar principal of $150 million is received and 4% on a British pound principal of 100 million pounds is paid. The current exchange rate is 1.55 dollar per pound. Interest rates in both countries for all maturities are currently 5% (continuously compounded). Payments are exchanged every year. The swap has 2.5 years left in its life. What is the value of the swap?
A. −$7.15
B.−$8.15
C.−$9.15
D.−$10.15
Answer: C

The value of the British pound bond underlying the swap is in millions of pounds
4e-0.05×0.5+4e-0.05×1.5+104e-0.05×2.5 = 99.39
The value of the U.S. dollar bond is in millions of dollars
4.5e-0.05×0.5+4.5e-0.05×1.5+154.5e-0.05×2.5  = 144.91
The value of the swap is 144.91 – 99.39×1.55 = –9.15


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19.Which of the following is a typical bid-offer spread on the swap rate for a plain vanilla interest rate swap?
A.3 basis points
B.8 basis points
C.13 basis points
D.18 basis points
Answer: A

3 basis points is a typical spread between the bid and the offer on a plain vanilla interest rate swap.

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Which of the following describes the five-year swap rate?
A.The rate on a five-year loan to a AA-rated company
B.The rate on a five-year loan to an A-rated company
C.The rate that can be earned over five years from a series of short-term loans to AA-rated companies
D.The rate that can be earned over five years from a series of short-term loans to A-rated companies 
Answer:  C

By considering the effect of making a series of LIBOR loans to AA-rated companies and entering into a swap we see that the swap rate corresponds to the risk in a series of short-term loans.


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The basis is defined as spot minus futures. A trader is hedging the sale of an asset with a short futures position.  The basis increases unexpectedly. Which of the following is true?
A. The hedger’s position improves.
B.The hedger’s position worsens.
C.The hedger’s position sometimes worsens and sometimes improves.
D.The hedger’s position stays the same.
Answer: A

The price received by the trader is the futures price plus the basis. It follows that the trader’s position improves when the basis increases.
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Futures contracts trade with every month as a delivery month. A company is hedging the purchase of the underlying asset on June 15. Which futures contract should it use?
A.The June contract
B.The July contract
C.The May contract
D.The August contract
Answer: B

As a general rule the futures maturity month should be as close as possible to but after the month when the asset will be purchased. In this case the asset will be purchased in June and so the best contract is the July contract.
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On March 1 a commodity’s spot price is $60 and its August futures price is $59. On July 1 the spot price is $64 and the August futures price is $63.50. A company entered into futures contracts on March 1 to hedge its purchase of the commodity on July 1. It closed out its position on July 1. What is the effective price (after taking account of hedging) paid by the company?
A.$59.50
B.$60.50
C.$61.50
D.$63.50
Answer: A

The user of the commodity takes a long futures position. The gain on the futures is 63.50−59 or $4.50. The effective paid realized is therefore 64−4.50 or $59.50. This can also be calculated as the March 1 futures price (=59) plus the basis on July 1 (=0.50).
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On March 1 the price of a commodity is $1,000 and the December futures price is $1,015. On November 1 the price is $980 and the December futures price is $981. A producer of the commodity entered into a December futures contracts on March 1 to hedge the sale of the commodity on November 1. It closed out its position on November 1. What is the effective price (after taking account of hedging) received by the company for the commodity? 
A.   $1,016
B.   $1,001
C.$981
D.$1,014
Answer: D

The producer of the commodity takes a short futures position. The gain on the futures is 1015−981 or $34. The effective price realized is therefore 980+34 or $1014. This can also be calculated as the March 1 futures price (=1015) plus the November 1 basis (=−1).
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5.Suppose that the standard deviation of monthly changes in the price of commodity A is $2. The standard deviation of monthly changes in a futures price for a contract on commodity B (which is similar to commodity A) is $3. The correlation between the futures price and the commodity price is 0.9. What hedge ratio should be used when hedging a one month exposure to the price of commodity A?  
A. 0.60
B. 0.67
C. 1.45
D.0.90
Answer: A

The optimal hedge ratio is 0.9×(2/3) or 0.6.
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A company has a $36 million portfolio with a beta of 1.2. The futures price for a contract on an index is 900. Futures contracts on $250 times the index can be traded. What trade is necessary to reduce beta to 0.9?
A.Long 192 contracts
B.Short 192 contracts
C.Long 48 contracts
D.Short 48 contracts
               Answer: D

To reduce the beta by 0.3 we need to short 0.3×36,000,000/(900×250) or 48 contracts.
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A company has a $36 million portfolio with a beta of 1.2. The futures price for a contract on an index is 900. Futures contracts on $250 times the index can be traded. What trade is necessary to increase beta to 1.8?
A.Long 192 contracts
B.Short 192 contracts
C.Long 96 contracts
D.Short 96 contracts
Answer: C

To increase beta by 0.6 we need to go long  0.6×36,000,000/(900×250) or 96 contracts
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Which of the following is true? 
A.The optimal hedge ratio is the slope of the best fit line when the spot price (on the y-axis) is regressed against the futures price (on the x-axis).
B.The optimal hedge ratio is the slope of the best fit line when the futures price (on the y-axis) is regressed against the spot price (on the x-axis).
C.The optimal hedge ratio is the slope of the best fit line when the change in the spot price (on the y-axis) is regressed against the change in the futures price (on the x-axis).
D.The optimal hedge ratio is the slope of the best fit line when the change in the futures price (on the y-axis) is regressed against the change in the spot price (on the x-axis).
Answer: C

The optimal hedge ratio reflects the ratio of movements in the spot price to movements in the futures price.
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Which of the following describes tailing the hedge?
A.A strategy where the hedge position is increased at the end of the life of the hedge
B.A strategy where the hedge position is increased at the end of the life of the futures contract
C.A more exact calculation of the hedge ratio when forward contracts are used for hedging
D.None of the above 

Answer: D

Tailing the hedge is a calculation appropriate when futures are used for hedging. It corrects for daily settlement

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A company due to pay a certain amount of a foreign currency in the future decides to hedge with futures contracts.  Which of the following best describes the advantage of hedging?
A.It leads to a better exchange rate being paid
B.It leads to a more predictable exchange rate being paid
C.It caps the exchange rate that will be paid
D.It provides a floor for the exchange rate that will be paid
Answer: B

Hedging is designed to reduce risk not increase expected profit. Options can be used to create a cap or floor on the price. Futures attempt to lock in the price

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11.Which of the following best describes the capital asset pricing model?
A.Determines the amount of capital that is needed in particular situations
B.Is used to determine the price of futures contracts
C.Relates the return on an asset to the return on a stock index
D.Is used to determine the volatility of a stock index
Answer: C

CAPM  relates the return on an asset to its beta. The parameter beta measures the sensitivity of the return on the asset to the return on the market. The latter is usually assumed to be the return on a stock index such as the S&P 500.

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Which of the following best describes “stack and roll”?
A.Creates long-term hedges from short term futures contracts
B.Can avoid losses on futures contracts by entering into further futures contracts
C.Involves buying a futures contract with one maturity and selling a futures contract with a different maturity
D.Involves two different exposures simultaneously
Answer: A
Stack and roll is a procedure where short maturity futures contracts are entered into. When they are close to maturity they are replaced by more short maturity futures contracts and so on. The result is the creation of a long term hedge from short-term futures contracts.
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Which of the following increases basis risk?
A.A large difference between the futures prices when the hedge is put in place and when it is closed out
B.Dissimilarity between the underlying asset of the futures contract and the hedger’s exposure
C.A reduction in the time between the date when the futures contract is closed and its delivery month
D.None of the above
Answer: B
Basis is the difference between futures and spot at the time the hedge is closed out.  This increases as the time between the date when the futures contract is put in place and the delivery month increases. (C is not therefore correct). It also increases as the asset underlying the futures contract becomes more different from the asset being hedged.  (B is therefore correct.)
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Which of the following is a reason for hedging a portfolio with an index futures?
A.The investor believes the stocks in the portfolio will perform better than the market but is uncertain about the future performance of the market
B.The investor believes the stocks in the portfolio will perform better than the market and the market is expected to do well
C.The portfolio is not well diversified and so its return is uncertain
D.All of the above

Answer: A

Index futures can be used to remove the impact of the performance of the overall market on the portfolio. If the market is expected to do well hedging against the performance of the market is not appropriate. Hedging cannot correct for a poorly diversified portfolio.
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Which of the following does NOT describe beta?
A.A measure of the sensitivity of the return on an asset to the return on an index
B.The slope of the best fit line when the return on an asset is regressed against the return on the market
C.The hedge ratio necessary to remove market risk from a portfolio
D.Measures correlation between futures prices and spot prices for a commodity
Answer: D

A, B, and C all describe beta but beta has nothing to do with the correlation between futures and spot prices for a commodity

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Which of the following is true?
A.Hedging can always be done more easily by a company’s shareholders than by the company itself
B.If all companies in an industry hedge, a company in the industry can sometimes reduce its risk by choosing not to hedge  
C.If all companies in an industry do not hedge, a company in the industry can reduce its risk by hedging
D.If all companies in an industry do not hedge, a company is liable increase its risk by hedging
Answer: D

If all companies in a industry hedge, the prices of the end product tends to reflect movements in relevant market variables. Attempting to hedge those movements can therefore increase risk.

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Which of the following is necessary for tailing a hedge?
A.Comparing the size in units of the position being hedged with the size in units of the futures contract
B.Comparing the value of the position being hedged with the value of one futures contract
C.Comparing the futures price of the asset being hedged to its forward price
D.None of the above

Answer: B

When tailing a hedge the optimal hedge ratio is applied to the ratio of the value of the position being hedged to the value of one futures contract.


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Which of the following is true?
A.Gold producers should always hedge the price they will receive for their production of gold over the next three years
B.Gold producers should always hedge the price they will receive for their production of gold over the next one year
C.The hedging strategies of a gold producer should depend on whether it shareholders want exposure to the price of gold
D.Gold producers can hedge by buying gold in the forward market
Answer: C

Some shareholders buy gold stocks to gain exposure to the price of gold. They do not want the company they invest in to hedge. In practice gold mining companies make their hedging strategies clear to shareholders.


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A silver mining company has used futures markets to hedge the price it will receive for everything it will produce over the next 5 years. Which of the following is true?
A.It is liable to experience liquidity problems if the price of silver falls dramatically
B.It is liable to experience liquidity problems if the price of silver rises dramatically
C.It is liable to experience liquidity problems if the price of silver rises dramatically or falls dramatically
D.The operation of futures markets protects it from liquidity problems

Answer: B

The mining company shorts futures. It gains on the futures when the price decreases and loses when the price increases.  It may get margin calls which lead to liquidity problems when the price rises even though the silver in the ground is worth more.
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A company will buy 1000 units of a certain commodity in one year. It decides to hedge 80% of its exposure using futures contracts. The spot price and the futures price are currently $100 and $90, respectively. If the spot price and the futures price in one year turn out to be $112 and $110, respectively. What is the average price paid for the commodity?
A.$92
B.$96
C.$102
D.$106

Answer: B

On the 80% (hedged) part of the commodity purchase the price paid will 112−(110−90) or $92. On the other 20% the price paid will be the spot price of $112. The weighted average of the two prices is 0.8×92+0.2×112 or $96. 
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Which of following is applicable to corporate bonds in the United States?
A.Actual/360
B.Actual/Actual
C.30/360
D.Actual/365

Answer: C

Corporate bonds in the U.S are usually quoted with a 30/360 day count. This means that there are assumed to be 30 days per month and 360 days per year when the length of an accrual period is calculated.
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It is May 1. The quoted price of a bond with an Actual/Actual (in period) day count and 12% per annum coupon (paid semiannually) in the United States is 105. It has a face value of 100 and pays coupons on April 1 and October 1. What is the cash price?
A.106.00
B. 106.02
C.105.98
D.  106.04
Answer: C

The cash price is the quoted price plus accrued interest. There are 30 actual days between April 1 and May 1 and 183 actual days between April 1 and October 1.  In this case the quoted price is 105 and the accrued interest is 0.06×100×30/183=0.98. The answer is therefore 105.98.
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It is May 1. The quoted price of a bond with a 30/360 day count and 12% per annum coupon in the United States is 105. It has a face value of 100 and pays coupons on April 1 and October 1. What is the cash price?
A.  106.00
B. 106.02
C.105.98
D.  106.04
Answer: A

The cash price is the quoted price plus accrued interest. There are 30 assumed days between April 1 and May 1 and 180 assumed days between April 1 and October 1. In this case the quoted price is 105 and the accrued interest is 0.06×100×30/180 = 1.00. The answer is therefore 106.00.
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The most recent settlement bond futures price is 103.5. Which of the following four bonds is cheapest to deliver?
A.Quoted bond price = 110; conversion factor = 1.0400.
B.Quoted bond price = 160; conversion factor = 1.5200.
C.Quoted bond price = 131; conversion factor = 1.2500.
D.Quoted bond price = 143; conversion factor = 1.3500.

Answer:  C

The cost of delivering a bond is the quoted bond price minus the most recent settlement price times the conversion factor. This is 2.36, 2.68, 1.625, and 3.275 for bonds in A, B, C, and D, respectively. The bond in C is therefore cheapest to deliver.
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Which of the following is NOT an option open to the party with a short position in the Treasury bond futures contract?
A.The ability to deliver any of a number of different bonds
B.The wild card play
C.The fact that delivery can be made any time during the delivery month
D.The interest rate used in the calculation of the conversion factor
Answer: D

A, B, and C describe options that the party with the short position has. D does not
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A trader enters into a long position in one Eurodollar futures contract. How much does the trader gain when the futures price quote increases by 6 basis points?
A.   $6
B.  $150
C.$60
D.   $600                                                                                               

Answer: B

The trader gains $25 for each basis point. The gain is therefore 25×6 or $150.
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The bonds that can be delivered in a Treasury bond futures contract are
A.Assets that provide no income
B.Assets that provide a known cash income
C.Assets that provide a known yield
D.None of the above

       Answer: B
A bond is an asset that provides a known cash income (the coupons)
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An ultra T-bond futures contract is one where
A.Bonds with maturities less than 3 years can be delivered
B.Bonds with maturities less than 10 years can be delivered
C.Bonds with maturities greater than 15 years can be delivered
D.Bonds with maturities greater than 25 year can be delivered

Answer: D

In the ultra T-bond futures contract bonds with maturities over 25 years can be delivered.
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A portfolio is worth $24,000,000. The futures price for a Treasury note futures contract is 110 and each contract is for the delivery of bonds with a face value of $100,000. On the delivery date the duration of the bond that is expected to be cheapest to deliver is 6 years and the duration of the portfolio will be 5.5 years. How many contracts are necessary for hedging the portfolio?  
A.100
B.200
C.300
D.400
Answer: B

The contract price is 110,000. The number of contracts is (24,000,000×5.5)/(110,000×6.0)=200

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Which of the following is true?
A.The futures rates calculated from a Eurodollar futures quote are always less than the corresponding forward rate
B.The futures rates calculated from a Eurodollar futures quote are always greater than the corresponding forward rate
C.The futures rates calculated from a Eurodollar futures quote should equal the corresponding forward rate
D.The futures rates calculated from a Eurodollar futures quote are sometimes greater than and sometimes less than the corresponding forward rate

Answer: B

The futures rate must be reduced by what is known as a convexity adjustment to get the forward rate.
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How much is a basis point?
A.1.0%
B.0.1%
C.0.01%
D.0.001%
Answer: C

A basis point is 0.01%.
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Which of the following day count conventions applies to a US Treasury bond?
A.Actual/360
B.Actual/Actual (in period)
C.30/360
D.Actual/365
Answer: B

Actual/Actual (in period) is used for US Treasury bonds. This means that the interest earned during a period that lies between two coupon payment dates is calculated by dividing the actual number of days in the period by the number of days between the coupon payments and multiplying the result by the next coupon payment
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What is the quoted discount rate on a money market instrument?
A.The interest rate earned as a percentage of the final face value of a bond
B.The interest rate earned as a percentage of the initial price of a bond
C.The interest rate earned as a percentage of the average price of a bond
D.The risk-free rate used to calculate the present value of future cash flows from a bond

Answer: A
The quoted discount rate is the interest earned as a percentage of the final face value
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Which of the following is closest to the duration of a 2-year bond that pays a coupon of 8% per annum semiannually? The yield on the bond is 10% per annum with continuous compounding.
A.1.82
B.1.85
C.1.88
D.1.92
Answer: C
The duration of the bond is the weighted average of the times when cash flows are received with weights proportional to the present values of the cash flows.

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Which of the following is NOT true about duration?
A.It equals the years-to-maturity for a zero coupon bond
B.It equals the weighted average of payment times for a bond, where weights are proportional to the present value of payments
C.Equals the weighted average of individual bond durations for a portfolio, where weights are proportional to the present value of bond prices
D.The prices of two bonds with the same duration change by the same percentage amount when interest rate moves up by 100 basis points
Answer: D

D is only approximately true. A, B, and C are exactly true.
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16.The conversion factor for a bond is approximately
The price it would have if all cash flows were discounted at 6% per annum
B.The price it would have if it paid coupons at 6% per annum
C.The price it would have if all cash flows were discounted at 8% per annum
D.The price it would have if it paid coupons at 8% per annum

Answer: A

The calculation of the conversion factor involves discounting the cash flows on the bond at 6%.
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The time-to-maturity of a Eurodollars futures contract is 4 years and the time-to-maturity of the rate underlying the futures contract is 4.25 years. The standard deviation of the change in the short term interest rate,  = 0.011. What does the model in the text estimate as the difference between the futures and the forward interest rate?
A.   0.105%
B.   0.103%
C.   0.098%
D.   0.093%
Answer: B

With the notation in the text, the futures rate exceeds the forward rate by 0.52T1T2. In this case =0.011, T1=4 and T2=4.25 so the difference between the forward and futures price is 0.5×0.011×4×4.25=0.00103.


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A trader uses 3-month Eurodollar futures to lock in a rate on $5 million for six months. How many contracts are required?
A.5
B.10
C.15
D.20

Answer: B

Each contract locks in the rate on $1 million dollars for three months. A six month instrument is approximately twice as sensitive to rate movements as a three month instrument because it has twice the duration. 2×5 = 10 contracts are therefore required
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In the U.S. what is the longest maturity for 3-month Eurodollar futures contracts?
A:  2 years
B:  5 years
C:  10 years
D:  20 years

Answer: C

Eurodollar futures have maturities out to 10 years.

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Duration matching immunizes a portfolio against
A.Any parallel shift in the yield curve
B.All shifts in the yield curve
C.Changes in the steepness of the yield curve
D.Small parallel shifts in the yield curve
Answer: D
Duration matching only protects against small parallel shifts. It does not provide protection against large parallel shifts and non-parallel shifts.

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Which of the following describes a call option?
A.The right to buy an asset for a certain price
B.The obligation to buy an asset for a certain price
C.The right to sell an asset for a certain price
D.The obligation to sell an asset for a certain price
Answer: A

A call option is the right, but not the obligation to buy.

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2.Which of the following is true?
A.A long call is the same as a short put
B.A short call is the same as a long put
C.A call on a stock plus a stock the same as a put
D.None of the above
Answer: D

None of the statements are true. Long calls, short calls, long puts, and short puts all have different payoffs as indicated by Figure 10.5. A put on a stock plus the stock provides a payoff that is similar to a call, as explained in Chapters 11 and 12. But a call on a stock plus a stock does not provide a similar payoff to a put.
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3.An investor has exchange-traded put options to sell 100 shares for $20. There is a 2 for 1 stock split. Which of the following is the position of the investor after the stock split?
A.Put options to sell 100 shares for $20
B.Put options to sell 100 shares for $10
C.Put options to sell 200 shares for $10
D.Put options to sell 200 shares for $20
Answer: C

When there is a stock split the number of shares increases and the strike price decreases. In this case, because it is a 2 for 1 stock split, the number of shares doubles and the strike price halves.
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4.An investor has exchange-traded put options to sell 100 shares for $20. There is 25% stock dividend. Which of the following is the position of the investor after the stock dividend?
A.Put options to sell 100 shares for $20
B.Put options to sell 75 shares for $25
C.Put options to sell 125 shares for $15
D.Put options to sell 125 shares for $16
Answer: D

The stock dividend is equivalent to a 5 for 4 stock split. The number of shares goes up by 25% and the strike price is reduced to 4/5 of its previous value.
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5.An investor has exchange-traded put options to sell 100 shares for $20. There is a $1 cash dividend. Which of the following is then the position of the investor?
A.The investor has put options to sell 100 shares for $20
B.The investor has put options to sell 100 shares for $19
C.The investor has put options to sell 105 shares for $19
D.The investor has put options to sell 105 shares for $19.05
Answer: A

       Cash dividends unless they are unusually large have no effect on the terms of an option.
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6.Which of the following describes a short position in an option?
A.A position in an option lasting less than one month
B.A position in an option lasting less than three months
C.A position in an option lasting less than six months
D.A position where an option has been sold

Answer: D

A short position is a position where the option has been sold (the opposite to a long     position).
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7.Which of the following describes a difference between a warrant and an exchange-traded stock option?
A.In a warrant issue, someone has guaranteed the performance of the option seller in the event that the option is exercised
B.The number of warrants is fixed whereas the number of exchange-traded options in existence depends on trading
C.Exchange-traded stock options have a strike price
D.Warrants cannot be traded after they have been purchased

Answer: B

A warrant is a fixed number of options issued by a company. They often trade on an exchange after they have been issued.
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1.Which of the following describes LEAPS?
A.Options which are partly American and partly European
B.Options where the strike price changes through time
C.Exchange-traded stock options with longer lives than regular exchange-traded stock options
D.Options on the average stock price during a period of time
Answer:  C

LEAPS are long-term equity anticipation securities. They are exchange-traded options with relatively long maturities.
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8.Which of the following is an example of an option class?
A.All calls on a certain stock
B.All calls with a particular strike price on a certain stock
C.All calls with a particular time to maturity on a certain stock
D.All calls with a particular time to maturity and strike price on a certain stock
Answer: A

An option class is all calls on a certain stock or all puts on a certain stock.
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9.Which of the following is an example of an option series?
A.All calls on a certain stock
B.All calls with a particular strike price on a certain stock
C.All calls with a particular time to maturity on a certain stock
D.All calls with a particular time to maturity and strike price on a certain stock

Answer: D

All options on a certain stock of a certain type (calls or put) with a certain strike price and time to maturity are referred to as an option series
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10.Which of the following must post margin?
A.The seller of an option
B.The buyer of an option
C.The seller and the buyer of an option
D.Neither the seller nor the buyer of an option

Answer: A

The seller of the option must post margin as a guarantee that the payoff on the option (if there is one) will be made. The buyer of the option usually pays for the option upfront and so no margin is required.

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11.Which of the following describes a long position in an option?
A.A position where there is more than one year to maturity
B.A position where there is more than five years to maturity
C.A position where an option has been purchased
D.A position that has been held for a long time
Answer: C

A long position is a position where an option has been purchased. It can be contrasted with a short position which is a position where an option has been sold.

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12.Which of the following is NOT traded by the CBOE?
A.Weeklys
B.Monthlys
C.Binary options
D.DOOM options

Answer: B

Monthlys are not a CBOE product. The others are.
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13.When a six-month option is purchased
A.The price must be paid in full
B.Up to 25% of the option price can be borrowed using a margin account
C.Up to 50% of the option price can be borrowed using a margin account
D.Up to 75% of the option price can be borrowed using a margin account
Answer: A

       Only options lasting more than 9 months can be bought on margin.
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14.Which of the following are true for CBOE stock options?
A.There are no margin requirements
B.The initial margin and maintenance margin are determined by formulas and are equal
C.The initial margin and maintenance margin are determined by formulas and are different
D.The maintenance margin is usually about 75% of the initial margin
Answer: B

Margin accounts for options must be brought up to the initial/maintenance margin level every day.
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15.The price of a stock is $67. A trader sells 5 put option contracts on the stock with a strike price of $70 when the option price is $4. The options are exercised when the stock price is $69. What is the trader’s net profit or loss?
A.Loss of $1,500
B.Loss of $500
C.Gain of $1,500
D.Loss of $1,000
Answer: C

The option payoff is 70−69 = $1. The amount received for the option is $4. The gain is $3 per option. In total 5×100 = 500 options are sold. The total gain is therefore $3 × 500 = $1,500.
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16.A trader buys a call and sells a put with the same strike price and maturity date. What is the position equivalent to?
A.A long forward
B.A short forward
C.Buying the asset
D.None of the above
Answer: A

From adding up the two payoffs we see that A is true: max(ST−K,0)−max(K−ST,0)= ST−K
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17.The price of a stock is $64. A trader buys 1 put option contract on the stock with a strike price of $60 when the option price is $10. When does the trader make a profit?
A.When the stock price is below $60
B.When the stock price is below $64
C.When the stock price is below $54
D.When the stock price is below $50
Answer: D
The payoff must be more than the $10 paid for the option. The stock price must therefore be below $50.
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18.Consider a put option and a call option with the same strike price and time to maturity. Which of the following is true?
A.It is possible for both options to be in the money
B.It is possible for both options to be out of the money
C.One of the options must be in the money
D.One of the options must be either in the money or at the money

Answer: D

If the stock price is greater than the strike price the call is in the money and the put is out of the money. If the stock price is less than the strike price the call is out of the money and the put is in the money. If the stock price is equal to the strike price both options are at the money.
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19.In which of the following cases is an asset NOT considered constructively sold?
A.The owner shorts the asset
B.The owner buys an in-the-money put option on the asset 
C.The owner shorts a forward contract on the asset
D.The owner shorts a futures contract on the stock

Answer: B

Profits on the asset have to be recognized in A, C, and D. The holder of the asset cannot defer recognition of profits with the trades indicated. In the case of B the asset is not considered constructively sold. Buying a deep-in-the-money put option is a way of almost certainly locking in a profit on an asset without triggering an immediate tax liability.
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1.When the stock price increases with all else remaining the same, which of the following is true?
A.Both calls and puts increase in value
B.Both calls and puts decrease in value
C.Calls increase in value while puts decrease in value
D.Puts increase in value while calls decrease in value
Answer: C

Stock price increases cause the values of calls to increase and the values of puts to decline.
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2.When the strike price increases with all else remaining the same, which of the following is true? 
A.Both calls and puts increase in value
B.Both calls and puts decrease in value
C.Calls increase in value while puts decrease in value
D.Puts increase in value while calls decrease in value
Answer: D

Strike price increases cause the values of puts to increase and the values of calls to decline.
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3.When volatility increases with all else remaining the same, which of the following is true? 
A.Both calls and puts increase in value
B.Both calls and puts decrease in value
C.Calls increase in value while puts decrease in value
D.Puts increase in value while calls decrease in value

Answer: A

Volatility increases the likelihood of a high payoff from either a call or a put option. The payoff can never be negative. It follows that as volatility increases the value of all options increase.
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4.When dividends increase with all else remaining the same, which of the following is true? 
A.Both calls and puts increase in value
B.Both calls and puts decrease in value
C.Calls increase in value while puts decrease in value
D.Puts increase in value while calls decrease in value
Answer: D

Dividends during the life of an option reduce the final stock price. As a result dividend increases cause puts to increase in value and calls to decrease in value.
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5.When interest rates increase with all else remaining the same, which of the following is true? 
A.Both calls and puts increase in value
B.Both calls and puts decrease in value
C.Calls increase in value while puts decrease in value
D.Puts increase in value while calls decrease in value
Answer: C

Calls increase and puts decrease in value. As explained in the text an increase in interest rates causes the growth rate of the stock price to increase and the discount rate to increase. An increase in interest rates therefore reduces the value of puts because puts are hurt by both a discount rate increase and a growth rate increase. For calls it turns out that the growth rate increase is more important than the discount rate increase so that their values increase when interest rates increase. (Note that we are assuming all else equal and so the asset price does not change.)

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6.When the time to maturity increases with all else remaining the same, which of the following is true? 
A.European options always increase in value
B.The value of European options either stays the same or increases
C.There is no effect on European option values
D.European options are liable to increase or decrease in value
Answer: D

When the time to maturity increases from X to Y, European options usually increase in value. But they can decrease in value if a big dividend expected between X and Y.

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7.The price of a stock, which pays no dividends, is $30  and the strike price of a one year European call option on the stock is $25. 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?
A.$5.00
B.$5.98
C.$4.98
D.$3.98

Answer: B

The lower bound in S0 − Ke-rT. In this case it is 30 – 25e-0.04×1  = $5.98.
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8.A stock price (which pays no dividends) is $50 and the strike price of a two year European put option is $54. The risk-free rate is 3% (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?
A.$4.00
B.$3.86
C.$2.86
D.$0.86

Answer: D

The lower bound in Ke-rT −S0  In this case it is 54e−0.03×2   – 50= $0.86.

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9.Which of the following is NOT true? (Present values are calculated from the end of the life of the option to the beginning.)
A.An American put option is always worth less than the present value of the strike price
B.A European put option is always worth less than the present value of the strike price
C.A European call option is always worth less than the stock price
D.An American call option is always worth less than the stock price
Answer: A

If it is optimal to exercise an American option today and the stock price is very low the option will be worth more than the present value of the strike price
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10.Which of the following best describes the intrinsic value of an option?
A.The value it would have if the owner had to exercise it immediately or not at all
B.The Black-Scholes-Merton price of the option
C.The lower bound for the option’s price
D.The amount paid for the option

Answer: A

The intrinsic value of an option is the value it would have if it were about the expire which is the same as the value in A.
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11.Which of the following describes a situation where an American put option on a stock becomes more likely to be exercised early?
A.Expected dividends increase
B.Interest rates decrease
C.The stock price volatility decreases
D.All of the above
Answer:  C

As the volatility of the option decreases the time value declines and the option becomes more likely to be exercised early. In the case of A and B, time value increases and the option is less likely to be exercised early.
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12.Which of the following is true?
A.An American call option on a stock should never be exercised early
B.An American call option on a stock should never be exercised early when no dividends are expected
C.There is always some chance that an American call option on a stock will be exercised early
D.There is always some chance that an American call option on a stock will be exercised early when no dividends are expected

Answer: B

An American call option should never be exercised early when the underlying stock does not pay dividends. There are two reasons. First, it is best to delay paying the strike price. Second the insurance provided by the option (that the stock price will fall below the strike price) is lost.
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13.Which of the following is the put-call parity result for a non-dividend-paying stock?
A.The European put price plus the European call price must equal the stock price plus the present value of the strike price
B.The European put price plus the present value of the strike price must equal the European call price plus the stock price
C.The European put price plus the stock price must equal the European call price plus the strike price
D.The European put price plus the stock price must equal the European call price plus the present value of the strike price

Answer: D

The put-call parity result is c+Ke-rT=p+S0.
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14.Which of the following is true when dividends are expected?
A.Put-call parity does not hold
B.The basic put-call parity formula can be adjusted by subtracting the present value of expected dividends from the stock price
C.The basic put-call parity formula can be adjusted by adding the present value of expected dividends to the stock price
D.The basic put-call parity formula can be adjusted by subtracting the dividend yield from the interest rate

Answer:  B

Put call parity still holds for European options providing the present value of the dividends is subtracted from the stock price.

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15.The price of a European call option on a non-dividend-paying stock with a strike price of $50 is $6. The stock price is $51, the continuously compounded risk-free rate (all maturities) is 6% and the time to maturity is one year. What is the price of a one-year European put option on the stock with a strike price of $50? 
A.$9.91
B.$7.00
C.$6.00
D.$2.09
Answer: D

Put-call parity is c+Ke-rT=p+S0. In this case K=50, S0=51, r=0.06, T=1, and c=6. It follows that
p=6+50e-0.06×1−51 = 2.09.

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16.The price of a European call option on a stock with a strike price of $50 is $6. The stock price is $51, the continuously compounded risk-free rate (all maturities) is 6% and the time to maturity is one year. A dividend of $1 is expected in six months. What is the price of a one-year European put option on the stock with a strike price of $50? 
A.$8.97
B.$6.97
C.$3.06
D.$1.12
Answer: C

Put-call parity is c+Ke-rT=p+S0. In this case K=50, S0=51, r=0.06, T=1, and c=6. The present value of the dividend is 1×e−0.06×0.5 = 0.97.   It follows that
p=6+50e-0.06×1−(51-0.97) = 3.06
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17. A European call and a European put on a stock have the same strike price and time to maturity. At 10:00am on a certain day, the price of the call is $3 and the price of the put is $4. At 10:01am news reaches the market that has no effect on the stock price or interest rates, but increases volatilities. As a result the price of the call changes to $4.50. Which of the following is correct?
A.The put price increases to $6.00
B.The put price decreases to $2.00
C.The put price increases to $5.50
D.It is possible that there is no effect on the put price

Answer: C

The price of the call has increased by $1.50. From put-call parity the price of the put must increase by the same amount. Hence the put price will become 4.00 +1.50 = $5.50.
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18.Interest rates are zero. A European call with a strike price of $50 and a maturity of one year is worth $6. A European put with a strike price of $50 and a maturity of one year is worth $7. The current stock price is $49. Which of the following is true?
A.The call price is high relative to the put price
B.The put price is high relative to the call price
C.Both the call and put must be mispriced
D.None of the above
Answer: D

In this case because interest rates are zero c+K=p+S0. The left side of this equation is 50+6=56. The right side is 49+7=56. There is no mispricing.
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19.Which of the following is true for American options?
A.Put-call parity provides an upper and lower bound for the difference between call and put prices
B.Put call parity provides an upper bound but no lower bound for the difference between call and put prices
C.Put call parity provides an lower bound but no upper bound for the difference between call and put prices
D.There are no put-call parity results 
Answer: A

Put call parity provides both an upper and lower bound for the difference between call and put prices. See equation (11.11).
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20.Which of the following can be used to create a long position in a European put option on a stock?
A.Buy a call option on the stock and buy the stock
B.Buy a call on the stock and short the stock
C.Sell a call option on the stock and buy the stock
D.Sell a call option on the stock and sell the stock

Answer: B

As payoff diagrams show a call on a stock combined with a short position in the stock gives a payoff similar to a put option. Alternatively we can use put-call parity, which shows that a call minus the stock equals the put minus the present value of the strike price.

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Which of the following creates a bull spread?
A.Buy a low strike price call and sell a high strike price call
B.Buy a high strike price call and sell a low strike price call
C.Buy a low strike price call and sell a high strike price put
D.Buy a low strike price put and sell a high strike price call

Answer: A

A bull spread is created by buying a low strike call and selling a high strike call. Alternatively, it can be created by buying a low strike put and selling a high strike put.
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2.Which of the following creates a bear spread?
A.Buy a low strike price call and sell a high strike price call
B.Buy a high strike price call and sell a low strike price call
C.Buy a low strike price call and sell a high strike price put
D.Buy a low strike price put and sell a high strike price call
Answer: B

A bear spread is created by buying a high strike call and selling a low strike call. Alternatively, it can be created by buying a high strike put and selling a low strike put
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3.Which of the following creates a bull spread?
A.Buy a low strike price put and sell a high strike price put
B.Buy a high strike price put and sell a low strike price put
C.Buy a high strike price call and sell a low strike price put
D.Buy a high strike price put and sell a low strike price call
Answer: A

A bull spread is created by buying a low strike call and selling a high strike call. Alternatively, it can be created by buying a low strike put and selling a high strike put.
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4. Which of the following creates a bear spread?
A.Buy a low strike price put and sell a high strike price put
B.Buy a high strike price put and sell a low strike price put
C.Buy a high strike price call and sell a low strike price put
D.Buy a high strike price put and sell a low strike price call
Answer: B

A bear spread is created by buying a high strike call and selling a low strike call. Alternatively, it can be created by buying a high strike put and selling a low strike put.
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5.What is the number of different option series used in creating a butterfly spread?
A.1
B.2
C.3
D.4
Answer: C

Three different options all with the same maturity are involved in creating a butterfly spread. The strike prices are usually equally spaced. The creator buys the low strike option, buys the high strike option, and sells two of the intermediate strike option
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1.A stock price is currently $23. A reverse (i.e short)  butterfly spread is created from options with strike prices of $20, $25, and $30. Which of the following is true?
A.The gain when the stock price is greater that $30 is less than the gain when the stock price is less than $20
B.The gain when the stock price is greater that $30 is greater than the gain when the stock price is less than $20
C.The gain when the stock price is greater that $30 is the same as the gain when the stock price is less than $20
D.It is incorrect to assume that there is always a gain when the stock price is greater than $30 or less than $20 
Answer: C

The gain from a very high stock price or a very low stock price is the same. Suppose calls are used. In the case of a very low stock price none are exercised and the gain is c1+c3−2c2 from the option premium. In the case of a very high stock price all options are exercised. The net payoff is zero and the gain is the same.

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7.Which of the following is correct?
A.A calendar spread can be created by buying a call and selling a put when the strike prices are the same and the times to maturity are different
B.A calendar spread can be created by buying a put and selling a call when the strike prices are the same and the times to maturity are different
C.A calendar spread can be created by buying a call and selling a call when the strike prices are different and the times to maturity are different
D.A calendar spread can be created by buying a call and selling a call when the strike prices are the same and the times to maturity are different
Answer: D

A calendar spread is created by buying an option with one maturity and selling an option with another maturity when the strike prices are the same and the option types (calls or puts) are the same.
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8.What is a description of the trading strategy where an investor sells a 3-month call option and buys a one-year call option, where both options have a strike price of $100 and the underlying stock price is $75?
A.Neutral Calendar Spread
B.Bullish Calendar Spread
C.Bearish Calendar Spread
D.None of the above
Answer: B

This is a bullish calendar spread because a big increase in the stock price between three months and one year is necessary for the trading strategy to be profitable.

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9.Which of the following is correct?
A.A diagonal spread can be created by buying a call and selling a put when the strike prices are the same and the times to maturity are different
B.A diagonal spread can be created by buying a put and selling a call when the strike prices are the same and the times to maturity are different
C.A diagonal spread can be created by buying a call and selling a call when the strike prices are different and the times to maturity are different
D.A diagonal spread can be created by buying a call and selling a call when the strike prices are the same and the times to maturity are different
Answer: C

Both the strike prices and times to maturity are different in a diagonal spread.
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10.Which of the following is true of a box spread?
A.It is a package consisting of a bull spread and a bear spread
B.It involves two call options and two put options
C.It has a known value at maturity 
D.All of the above
Answer: D

    A, B, and C are all true.
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11.How can a straddle be created?
A.Buy one call and one put with the same strike price and same expiration date
B.Buy one call and one put with different strike prices and same expiration date
C.Buy one call and two puts with the same strike price and expiration date
D.Buy two calls and one put with the same strike price and expiration date
Answer: A

A straddle consists of one call and one put where the strike price and time to maturity are the same. It has a V-shaped payoff.
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12.How can a strip trading strategy be created?
A.Buy one call and one put with the same strike price and same expiration date
B.Buy one call and one put with different strike prices and same expiration date
C.Buy one call and two puts with the same strike price and expiration date
D.Buy two calls and one put with the same strike price and expiration date

Answer: C

A strip consists of one call and two puts with the same strike price and time to maturity.
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13.How can a strap trading strategy be created?
A.Buy one call and one put with the same strike price and same expiration date
B.Buy one call and one put with different strike prices and same expiration date
C.Buy one call and two puts with the same strike price and expiration date
D.Buy two calls and one put with the same strike price and expiration date
Answer: D

A strap consists of two calls and one put with the same strike price and time to maturity.

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14.How can a strangle trading strategy be created?
A.Buy one call and one put with the same strike price and same expiration date
B.Buy one call and one put with different strike prices and same expiration date
C.Buy one call and two puts with the same strike price and expiration date
D.Buy two calls and one put with the same strike price and expiration date
Answer: B
A straddle consists of one call and one put where the times to maturity are the same but the call strike price is greater than the put strike price.
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15.Which of the following describes a protective put?
A.A long  put option on a stock plus a long position in the stock
B.A long put option on a stock plus a short position in the stock
C.A short put option on a stock plus a short call option on the stock
D.A short put option on a stock plus a long position in the stock
Answer: A

A protective put consists of a long put plus the stock. The holder of the put owns the stock that might become deliverable.

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16.Which of the following describes a covered call?
A.A long  call option on a stock plus a long position in the stock
B.A long call option on a stock plus a short put option on the stock
C.A short call option on a stock plus a short position in the stock
D.A short call option on a stock plus a long position in the stock

Answer: D

A covered call consists of a short call plus a long position in the stock. The if the call is exercised the owner of the position has the stock ready to deliver if the other side exercises the call.
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17.When the interest rate is 5% per annum with continuous compounding, which of the following creates a principal protected note worth $1000?
A.A one-year zero-coupon bond plus a one-year call option worth about $59
B.A one-year zero-coupon bond plus a one-year call option worth about $49
C.A one-year zero-coupon bond plus a one-year call option worth about $39
D.A one-year zero-coupon bond plus a one-year call option worth about $29

Answer: B

A one-year zero-coupon bond is worth 1000e-0.05×1 or about $951. This leaves 1000−951 = $49 for buying the option.
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18.A trader creates a long butterfly spread from options with strike prices $60, $65, and $70 by trading a total of 400 options. The options are worth $11, $14, and $18. What is the maximum net gain (after the cost of the options is taken into account)?
A.$100
B.$200
C.$300
D.$400
Answer: D

The butterfly spread involves buying 100 options with strike prices $60 and $70 and selling 200 options with strike price $65. The maximum gain is when the stock price equals the middle strike price, $65. The payoffs from the options are then, $500, 0, and 0, respectively. The total payoff is $500. The cost of setting up the butterfly spread is 11×100+18×100−14×200 = $100. The gain is 500−100 or $400.
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19.A trader creates a long butterfly spread from options with strike prices $60, $65, and $70 by trading a total of 400 options. The options are worth $11, $14, and $18. What is the maximum net loss (after the cost of the options is taken into account)?
A.$100
B.$200
C.$300
D.$400

Answer:  A

The butterfly spread involves buying 100 options with strike prices $60 and $70 and selling 200 options with strike price $65. The maximum loss is when the stock price is less than $60 or greater than $70. The total payoff is then zero. The cost of setting up the butterfly spread is 11×100+18×100−14×200 = $100. The loss is therefore $100.
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20.Six-month call options with strike prices of $35 and $40 cost $6 and $4, respectively. What is the maximum gain when a bull spread is created by trading a total of 200 options?
A.$100
B.$200
C.$300
D.$400

Answer: C

The bull spread involves buying 100 calls with strike $35 and selling 100 calls with strike price $40. The cost is 6×100−4×100=$200. The maximum payoff (when the stock price is greater than or equal to $40 is $500. The maximum gain is therefore 500 −200 = $300.

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1.The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. An investor sells call options with a strike price of $32. Which of the following hedges the position?
A.Buy 0.6 shares for each call option sold
B.Buy 0.4 shares for each call option sold
C.Short 0.6 shares for each call option sold
D.Short 0.6 shares for each call option sold
Answer: B

The value of the option will be either $4 or zero.  If  is the position in the stock we require 36−4=26
so that =0.4. it follows that 0.4 shares should be purchased for each option sold.
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2.The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. What is the risk-neutral probability of that the stock price will be $36?
A.0.6
B.0.5
C.0.4
D.0.3

Answer: C

The formula for the risk-neutral probability of an up movement is

In this case u=36/30 or 1.2 and d=26/30 =0.8667.  Also r=0 and T=0.5. The formula gives
p=(1-0.8667/(1.2-0.8667) =0.4.

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3.The current price of a non-dividend-paying stock is $30. Over the next six months it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. An investor sells call options with a strike price of $32. What is the value of each call option?
A.$1.6
B.$2.0
C.$2.4
D.$3.0
Answer: A

The formula for the risk-neutral probability of an up movement is

In this case u=36/30 or 1.2 and d=26/30 =0.8667.  Also r=0 and T=0.5. The formula gives
p=(1-0.8667/(1.2-0.8667) =0.4.
The payoff from the call option is $4 if there is an up movement and $0 if there is a down movement. The value of the option is therefore 0.4×4 +0.6×0 = $1.6. (We do not do any discounting because the interest rate is zero.)
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4.The current price of a non-dividend-paying stock is $40. Over the next year it is expected to rise to $42 or fall to $37. An investor buys put options with a strike price of $41. Which of the following is necessary to hedge the position?
A.Buy 0.2 shares for each option purchased
B.Sell 0.2 shares for each option purchased
C.Buy 0.8 shares for each option purchased
D.Sell 0.8 shares for each option purchased
Answer: C

The payoff from the put option is zero if there is an up movement and 4 if there is a down movement. Suppose that the investor buys one put option and buys  shares. If there is an up movement the value of the portfolio is ×42. If there is a down movement it is worth ×37+4. These are equal when 37+4=42 or =0.8. The investor should therefore buy 0.8 shares for each option purchased. 
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5.The current price of a non-dividend-paying stock is $40. Over the next year it is expected to rise to $42 or fall to $37. An investor buys put options with a strike price of $41. What is the value of each option? The risk-free interest rate is 2% per annum with continuous compounding.
A.$3.93
B.$2.93
C.$1.93
D.$0.93
Answer:  D

The formula for the risk-neutral probability of an up movement is


In this case r=0.02, T= 1, u=42/40=1.05 and d=37/40=0.925 so that p=0.76 and the value of the option is (0.76×0+0.24×4)e-0.02×1=0.93
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6.Which of the following describes how American options can be valued using a binomial tree?
A.Check whether early exercise is optimal at all nodes where the option is in-the-money
B.Check whether early exercise is optimal at the final nodes
C.Check whether early exercise is optimal at the penultimate nodes and the final nodes
D.None of the above

Answer: A

For an American option we must check whether exercising is better than not exercising at each node where the option is in the money.  (It is clearly not worth exercising when the option is out of the money)

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7.In a binomial tree created to value an option on a stock, the expected return on stock is
A.Zero
B.The return required by the market
C.The risk-free rate
D.It is impossible to know without more information

Answer: C

The expected return on the stock on the tree is the risk-free rate. This is an application of risk-neutral valuation.

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8.In a binomial tree created to value an option on a stock, what is the expected return on the option?
A.Zero
B.The return required by the market
C.The risk-free rate
D.It is impossible to know without more information
Answer: C

The expected return on the option on the tree is the risk-free rate. This is an application of risk-neutral valuation. The expected return on all assets in a risk-neutral world is the risk-free rate.
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9.A stock is expected to return 10% when the risk-free rate is 4%. What is the correct discount rate to use for the expected payoff on an option in the real world?
A.4%
B.10%
C.More than 10%
D.It could be more or less than 10%
Answer: D

The correct answer is D. There is no easy way of determining the correct discount rate for an option’s expected payoff in the real world. For a call option the correct discount rate in the real world is often quite high and for a put option it is often quite low (even negative). The example in the text illustrates this
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10.Which of the following is true for a call option on a stock worth $50
A.As a stock’s expected return increases the price of the option increases
B.As a stock’s expected return increases the price of the option decreases
C.As a stock’s expected return increases the price of the option might increase or decrease
D.As a stock’s expected return increases the price of the option on the stock stays the same
Answer:  D

The option price when expressed in terms of the underlying stock price is independent of the return on the stock. To put this another way, everything relevant about the expected return is incorporated in the stock price.
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11.Which of the following are NOT true
A.Risk-neutral valuation and no-arbitrage arguments give the same option prices
B.Risk-neutral valuation involves assuming that the expected return is the risk-free rate and then discounting expected payoffs at the risk-free rate
C.A hedge set up to value an option does not need to be changed
D.All of the above
Answer: C 

The hedge set up to value an option needs to be changed as time passes. A and B are true.

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12.The current price of a non-dividend paying stock is $30. Use a two-step tree to value a European call option on the stock with a strike price of $32 that expires in 6 months. Each step is 3 months, the risk free rate is 8% per annum with continuous compounding. What is the option price when u = 1.1 and d = 0.9.
A.$1.29
B.$1.49
C.$1.69
D.$1.89
Answer: B

The probability of an up movement is

The tree is



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13.The current price of a non-dividend paying stock is $30. Use a two-step tree to value a European put option on the stock with a strike price of $32 that expires in 6 months with u = 1.1 and d = 0.9. Each step is 3 months, the risk free rate is 8%.
A.$2.24
B.$2.44
C.$2.64
D.$2.84
Answer: A

The probability of an up movement is

The tree is
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14.Which of the following is NOT true in a risk-neutral world?
A.The expected return on a call option is independent of its strike price
B.Investors expect higher returns to compensate for higher risk
C.The expected return on a stock is the risk-free rate
D.The discount rate used for the expected payoff on an option is the risk-free rate
Answer: B

In a risk-neutral world investors require an expected return equal to the risk-free rate and the discount rate that should be used for all expected payoffs is the risk-free rate.
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15.If the volatility of a non-dividend paying stock is 20% per annum and a risk-free rate is 5% per annum, which of the following is closest to the Cox, Ross, Rubinstein parameter u for a tree with a three-month time step?
A.1.05
B.1.07
C.1.09
D.1.11
Answer: D

The formula for u is

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16.If the volatility of a non-dividend-paying stock is 20% per annum and a risk-free rate is 5% per annum, which of the following is closest to the Cox, Ross, Rubinstein parameter p for a tree with a three-month time step?
A.0.50
B.0.54
C.0.58
D.0.62

The formula for p is

=0.538
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17.The current price of a non-dividend paying stock is $50. Use a two-step tree to value an American put option on the stock with a strike price of $48 that expires in 12 months. Each step is 6 months, the risk free rate is 5% per annum, and the volatility is 20%.  Which of the following is the option price?
A.$1.95
B.$2.00
C.$2.05
D.$2.10
Answer: B


In this case

The tree is
     
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18.Which of the following describes delta?
A.The ratio of the option price to the stock price
B.The ratio of the stock price to the option price
C.The ratio of a change in the option price to the corresponding change in the stock price
D.The ratio of a change in the stock price to the corresponding change in the option price

Answer: C

Delta is f/S where S is a small change in the stock price (with nothing else changing) and f is the corresponding change in the option price.


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When moving from valuing an option on a non-dividend paying stock to an option on a currency which of the following is true?
A.The risk-free rate is replaced by the excess of the domestic risk-free rate over the foreign risk-free rate in all calculations
B.The formula for u changes
C.The risk-free rate is replaced by the excess of the domestic risk-free rate over the foreign risk-free rate for discounting
D.The risk-free rate is replaced by the excess of the domestic risk-free rate over the foreign risk-free rate when p is calculated 
Answer: D 

The formula for u does not change. The discount rate does not change. The formula for p becomes

showing that D is correct.


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20.A tree is constructed to value an option on an index which is currently worth 100 and has a volatility of 25%. The index provides a dividend yield of 2%. Another tree is constructed to value an option on a non-dividend-paying stock which is currently worth 100 and has a volatility of 25%. Which of the following are true?
A.The parameters p and u are the same for both trees
B.The parameter p is the same for both trees but u is not
C.The parameter u is the same for both trees but p is not
D.None of the above
Answer: C


The formula for u is the same in the two cases so that the values of the index on its tree are the same as the values of the stock on its tree. However, in the formula for p,  r is replaced by r−q.
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1.A variable x starts at 10 and follows the generalized Wiener process
dx = a dt + b dz
where time is measured in years. If a = 2 and b =3 what is the expected value after 3 years?
A.12
B.14
C.16
D.18
Answer: C
The drift is 2 per year and so the expected increase over three years is 2×3 = 6 and the expected value at the end of 3 years is 10+6 = 16.
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2.A variable x starts at 10 and follows the generalized Wiener process
dx = a dt + b dz
where time is measured in years. If a = 3 and b =4 what is the standard deviation of the value in 4 years?
A.4
B.8
C.12
D.16
Answer: B
The variance per year is 42  or 16. The variance over four years is 16×4 = 64. The standard deviation is .
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3.A variable x starts at 10 and follows the generalized Wiener process
dx = a dt + b dz
If a = 3 and b =4 what is the standard deviation of the value in three months?
A.1
B.2
C.3
D.4
Answer: B

The variance per year is 42  or 16. The variance over three months is 16×0.25 = 4. The standard deviation is .
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4.The variance of a Wiener process in time t is
A. t
B. t squared
C. the square root of t
D. t to the power of 4
Answer: A

The variance of a Wiener process is 1 per unit time or t in time t.

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5.The process followed by a variable X is
dX = mX dt+sX dz
What is the coefficient of dz in the process for the square of X.
A.sX
B.sX2
C.2sX2
D.msX
Answer: C

From Ito’s lemma, the coefficient of dz is   where f = X2. Because   , the coefficient of dz is 2sX2.
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6.The process followed by a variable X is
dX = mX dt+sX dz
What is the coefficient of dt  in the process for the square of X.
A.2mX2+s2X2
B.2mX2
C.mX2+2s2X2
D.mX2+s2X2
Answer: A

From Ito’s lemma, the coefficient of dt is

where  f = X2. Because   and  the coefficient of dt is 2mX2+s2X2

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7.Which of the following is true when the stock price follows geometric Brownian motion
A.The future stock price has a normal distribution
B.The future stock price has a lognormal distribution
C.The future stock price has  geometric distribution
D.The future stock price has a truncated normal distribution
Answer: B

Ito’s lemma show that the log of the stock price follows a generalized Wiener process. This means that the log of the stock price is normally distributed so that the stock price is lognormally distributed.

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8.If a stock price follows a Markov process which of the following could be true
A.Whenever the stock price has gone up for four successive days it has a 70% chance of going up on the fifth day.
B.Whenever the stock price has gone up for four successive days there is almost certain to be a correction on the fifth day.
C.The way the stock price moves on a day is unaffected by how it moved on the previous four days.
D.Bad years for stock price returns are usually followed by good years.

Answer: C

A Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. Stock prices are usually assumed to follow Markov processes. This corresponds to a weak form market efficiency assumption.
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9.A variable x starts at zero and follows the generalized Wiener process
dx = a dt + b dz
where time is measured in years. During the first two years a=3 and b=4. During the following three years a=6 and b=3. What is the expected value of the variable at the end of 5 years
A.16
B.20
C.24
D.30
Answer: C
During the first two years, the drift per year is 3 and so the total drift is 3×2 or 6. During the next three years, the drift per year is 6 and the total drift is 6×3 = 18. The total drift over the five years is 6+18 =24. Given that the variable starts at zero, its expected value at the end of the five years is therefore 24.

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10.A variable x starts at zero and follows the generalized Wiener process
dx = a dt + b dz
where time is measured in years. During the first two years a=3 and b=4. During the following three years a=6 and b=3. What the standard deviation  of the value of the variable at the end of 5 years
A.6.2
B.6.7
C.7.2
D.7.7
Answer: D
The variance per year for the first two years is 42 or 16. The variance per year for the next three years is 32 or 9.  The total variance of the change over five years is 2×16+3×9= 59. The standard deviation of the value of the variable at the end of the five years is therefore 
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11.If a variable x follows the process dx = b dz where dz is a Wiener process, which of the following is the process followed by y = exp(x).
A.dy = by dz
B.dy = 0.5b2y dt+by dz
C.dy = (y+0.5b2y) dt+by dz
D.dy = 0.5b2y dt+b dz
Answer: B

Ito’s lemma shows that the process followed by y is dy = 0.5b2exp(x) dt +bexp(x) dz.  Substituting y = exp(x) we get the answer in B.

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12.If the risk-free rate is r and price of a nondividend paying stock grows at rate mwith volatility s, at what rate does a forward price of the stock grow for a forward contract maturing at a future time T.
A.m
B.m−s2/2
C.m−r
D.r−s2/2

Answer: C

This is the application of Ito’s lemma in Section 14.6.





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13.When a stock price, S,  follows geometric Brownian motion with mean return m and volatility s what is the process follows by X where X = ln S.
A.dX = m dt + s dz
B.dX = (m−r) dt + s dz
C.dX = (m−s2) dt + s dz
D.dX = (m − s2/2) dt + s dz
Answer: D

This is the example in Section 14.7
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15.Which of the following defines an Ito process?
A.A process where the drift is non-constant and can be stochastic
B.A process where the coefficient of dz is non-constant and can be stochastic
C.A process where either the drift or the coefficient of dz or both are non-constant and can be stochastic
D.A process  where proportional changes follow a generalized Wiener process
Answer: C

In a generalized Wiener process the drift and coefficient of dz are both constant.  In an Ito process they are not both constant .

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16.A stock price is $20. It has an expected return of 12% and a volatility of 25%. What is the standard deviation of the change in the price  in one day. (For this question assume that there are 365 days in the year.)
A.$0.20
B.$0.23
C.$0.26
D.$0.29
Answer: C
The standard deviation of the change in one day  is 
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17. A stock price is $20. It has an expected return of 12% and a volatility of 25%. What is the stock price that has a 2.5% chance of being exceeded in one day? (For this question assume that there are 365 days in the year.)
A.$20.41
B.$20.51
C.$20.61
D.$20.71
Answer: B

From the previous question the standard deviation of the change in one day is $0.26. There is a 2.5% chance that the stock price will increase by more than 1.96 standard deviations. The answer is therefore 20+1.96×0.26 = $20.51. The expected return in one day is small and can be ignored.
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18.Which of the following is NOT a property of a Wiener process?
A.The change during a short period of time dt has a variance dt
B.The changes in two different short periods of time are independent
C.The mean change in any time period is zero
D.The standard deviation over two consecutive time periods is the sum of the standard deviations  over each of the periods
Answer D

Variances of Wiener processes are additive but standard deviations are not.
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19. If e is a random sample from a standard normal distribution, which of the following is the change in a Wiener process in time dt .
A.e times the square root of dt
B.e times dt
C.dt times the square root of e
D.The square root of e times the square root of dt
Answer: A

The change is   . This result is used when the process is simulated.
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20. For what value of the correlation between two Wiener processes is the sum of the processes also a Wiener process?
A.0.5
B.−0.5
C.0
D.1

Answer: B

The variance of each process is 1 per unit time. The variance of the sum is 1+1+2where is the correlation. This is 1 when =−0.5.
Flashcard set info:
Author: CoboCards-User
Main topic: Finance & Investment
Topic: Derivatives
Published: 27.10.2015
 
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