Question 1

Consider five put options at strikes 40,45,50,55,and 60.The price of the 40-strike option is $4,the price of the 50-strike put is $5,and the price of the 60-strike option is $8.Which of the following statements is most accurate? (Assume all options have the same maturity. )

A)The maximum difference between the prices of the 55-strike put and the 45-strike put is $0.
B)The difference between the maximum prices of the 55-strike put and the 45-strike put is $0.50.
C)The minimum difference between the prices of the 55-strike put and the 45-strike put is $1.00.
D)The difference between the maximum prices of the 55-strike put and the 45-strike put is $2.00.

D
By convexity,the maximum price of the 45-strike put is $4.5,and that of the 55-strike put is $6.5.Hence,the difference in maximum prices is $2.The other options are not true.

Accepted Answer

By convexity,the maximum price of the 45-strike put is $4.5,and that of the 55-strike put is $6.5.Hence,the difference in maximum prices is $2.The other options are not true.

Question 2

Consider three put options at strikes 40,50,and 60.The price of the 40-strike option is $4 and the price of the 60-strike option is $8.Which of the following statements is most accurate? (Assume all options have the same maturity. )

A)The price of the 50-strike put is less than $10.
B)The price of the 50-strike put is less than $6.
C)The price of the 50-strike put is greater than $6.
D)The price of the 50-strike put is greater than $7.

B
By convexity,the price of the 50-strike put must be less than the average of the prices of the options at strikes 40 and 60.

Accepted Answer

By convexity,the price of the 50-strike put must be less than the average of the prices of the options at strikes 40 and 60.

Question 3

A &#34;no-arbitrage restriction&#34; on option prices is the statement that&#10;A)Options on possibly different stocks that trade at the same price must have the same payoffs.&#10;B)An option written on a specific stock will be perfectly correlated with the stock.&#10;C)The price of an option is such that no strategy can be constructed using the option and the underlying that generates arbitrage profits.&#10;D)Arbitrage trading in options is prohibited by the SEC.

Accepted Answer

The "no-arbitrage restriction" on option prices means that the price of an option is such that no strategy can be constructed using the option and the underlying that generates arbitrage profits. This is because if there were a possibility for arbitrage, then traders could take advantage of the mispricing and make profits without taking on any risk. Therefore, option prices are set to prevent such opportunities for arbitrage. Choices A and B are not accurate because the payoffs of options on different stocks can differ, and options are not perfectly correlated with their underlying assets. Choice D is also not accurate as the SEC does not prohibit arbitrage trading in options, but instead regulates the market to ensure fairness and transparency.

Question 4

Consider two American call options,with maturities three months and six months on the same stock and same strike price.The stock pays dividends in two months time and every quarter thereafter.Which of the following statements is most accurate?

A)The three-month option must be worth more than the six-month option.
B)The three-month option must be worth less than the six-month option.
C)The three-month option must be worth less than the six-month option if the dividend payments remain constant.
D)The three-month option may be worth more than the six-month option.

Accepted Answer

These are American options,and therefore,the six-month option can always be exercised at the same time as the three-month option,and will always be worth as much if not more than the shorter-maturity option.

Question 5

The current price of a non-dividend paying stock is $40.A European call option with three months maturity and strike $39 is priced at $2.The risk free rate of interest for three months is 2%.Which of the following statements is correct?

A)The price of the call obeys no-arbitrage restrictions.
B)It is possible to construct a risk-less arbitrage strategy to yield a gain of at least $0.81.
C)It is possible to onstruct a risk-less arbitrage strategy to yield a gain of at least $1.00.
D)It is possible to construct a risk-less arbitrage strategy to yield a gain of at least $1.19.

Accepted Answer

BTo check if the price of the call option obeys no-arbitrage restrictions, we can use the put-call parity for European options, which simplifies to $C + Ke^{-rt} = P + S$ for non-dividend paying stocks, where $C$ is the call price, $K$ is the strike price, $e^{-rt}$ is the discount factor, $P$ is the put price (not directly relevant here as we're not given or asked about it), and $S$ is the current stock price. However, without the price of a put option, we directly look at the possibility of arbitrage given the information.Given:- Current stock price ($S$) = $40- Strike price ($K$) = $39- Call option price ($C$) = $2- Risk-free rate ($r$) = 2% for three monthsIf you buy the call option for $2, your total investment to potentially control the stock is $2. If the stock price is above the strike price at expiration, you can exercise the option, buy the stock at $39, and sell it at the market price. The break-even point at expiration is a stock price of $41 ($39 strike + $2 option premium).To check for arbitrage, consider creating a synthetic position by buying the call and lending/borrowing the present value of the strike price ($Ke^{-rt}$).The present value of the strike price is $39e^{-0.02 \times 0.25} \approx 38.81$. If you buy the call for $2 and borrow $38.81 (the present value of the strike price), your total outlay is $2 + $38.81 = $40.81.If at expiration, the stock is above $39, you exercise the call, buy the stock at $39, repay the borrowed amount (which has grown to $39 due to interest), and sell the stock at its market price. The minimum profit occurs if the stock is exactly at $39, in which case, after repaying the borrowed amount, you break even, but any price above $39 yields a profit.However, since the initial outlay ($40.81) is more than the current stock price ($40), this suggests there's no immediate risk-less arbitrage opportunity based on these prices alone. The mistake in my explanation is assuming an arbitrage exists without correctly calculating the arbitrage profit, which would require a specific strategy showing how the profit is locked in regardless of the stock's future price movement. The correct approach to identify arbitrage involves comparing the cost of a replicating portfolio to the cost of the option, but with the given information, we primarily assess the call's pricing in the context of its intrinsic and time value, which seems reasonable given the stock's current price, the strike price, and the cost of the call. Thus, indicating the call price might obey no-arbitrage restrictions without a clear arbitrage strategy presented. The possibility of constructing a risk-less arbitrage strategy (B) is incorrectly justified without a detailed strategy that exploits price discrepancies. The correct analysis should involve comparing the option's price to the cost of a replicating portfolio or using other arbitrage relationships, which was not accurately done.

Question 6

The 50-strike call on a stock is trading at $13 and a 60-strike call on the same stock with the same maturity is trading at $4.The minimum price of the 100-strike call is&#10;A)$0&#10;B)$1&#10;C)$2&#10;D)$3

Accepted Answer

The minimum price of a call option is $0, as an option cannot have a negative value.

Question 7

The current price of a stock is $100.The stock pays a dividend of $2 in three months.The risk free rate of interest for all maturities in annualized and continuously-compounded terms is 2%.What is the minimum price of an at-the-money American call option on the stock with six months maturity?

A)

- \$ 1

B)$0
C)$1
D)$2

Accepted Answer

The minimum price of an at-the-money American call option cannot be negative, as the option gives the holder the right but not the obligation to buy the stock. Therefore, the worst-case scenario is that the option expires worthless, making its minimum value $0.

Question 8

Consider two European call options,with maturities three months and six months on the same stock and same strike price.The stock pays dividends in two months time and every quarter thereafter.Which of the following statements is most accurate?

A)The three-month option must be worth more than the six-month option.
B)The three-month option must be worth less than the six-month option.
C)The three-month option must be worth less than the six-month option if the dividend payments remain constant.
D)The three-month option may be worth more than the six-month option.

Accepted Answer

Without dividends the six-month option will always have a lower bound that is greater than that of the three-month option (see equation 9.27 in Chapter 9).However,there is a dividend in five months that lowers the lower bound on the six-month option but not on the three-month option.This may result in the three-month option being worth more than the six-month option,especially if the dividend in five months is expected to be large.

Question 9

Consider a pair of at-the-money European call and a put options written on the same non-dividend-paying stock with the same maturity.Which of the following statements is most accurate?&#10;A)The minimum price of the call is at least as much as that of the put.&#10;B)The minimum price of the call is equal to that of the put.&#10;C)The minimum price of the call is less than that of the put.&#10;D)There is not enough information to decide.

Accepted Answer

For at-the-money European options on a non-dividend-paying stock, the call option's price is at least as much as the put option's price due to the cost of carry and the fact that calls have unlimited upside potential.

Question 10

Consider two six-month American puts at strikes 90 and 100.The risk free rate is 2%.The difference between the two put prices at any time before maturity will always be&#10;A)Less than $10.&#10;B)Equal to $10.&#10;C)Greater than $10.&#10;D)One cannot be sure of which of the preceding three choices is valid and more information may be required.

Accepted Answer

The value of an American put option depends on several factors including the current price of the underlying asset, volatility, time to expiration, and the risk-free rate. The difference between the prices of two American puts with different strike prices cannot be determined to be less than, equal to, or greater than $10 without additional information about the underlying asset's price and its volatility.

Question 11

There are three- and six-month American calls on $A B C$ stock.Suppose the three-month option costs $5 and the six-month option costs $3.Which of the following statements is most accurate given this information?

A)There is an arbitrage strategy which involves buying the three-month call and selling the six-month call.
B)There is an arbitrage strategy which involves buying the six-month call and selling the three-month call.
C)There is no arbitrage available in this setting.
D)If there is a sufficiently large dividend payment between three and six months,there is no arbitrage in this situation.

Accepted Answer

The answer of There are three- and six-month American calls...

Question 12

Non-dividend paying stock XYZ is trading at $20.The risk free rate is 2%.The minimum price of a four-month American put option at strike $22 is&#10;A)$0&#10;B)$1.27&#10;C)$1.89&#10;D)$2.00

Accepted Answer

The answer of Non-dividend paying stock XYZ is trading at...

Question 13

All else the same,when the interest rate rises,the lower bound on a call option&#10;A)Falls or stays the same.&#10;B)Stays the same.&#10;C)Increases or stays the same.&#10;D)May become negative.

Accepted Answer

The answer of All else the same,when the interest rate...

Question 14

There are three- and six-month European calls on $A B C$ stock.Suppose the three-month option costs $5 and the six-month option costs $3.Then,there is an arbitrage strategy that involves,among other things,

A)Buying the three-month call and selling the six-month call.
B)Buying the six-month call and selling the three-month call.
C)Buying the three-month and six-month calls,shorting the stock,and investing.
D)There is not enough information given to answer this question.

Accepted Answer

The answer of There are three- and six-month European calls...

Question 15

Consider two six-month European calls at strikes 90 and 100.The risk free rate is 2%.Which of the following alternatives best describes the condition that must be met by the difference in prices $C ( 90 ) - C ( 100 )$ ?

A)It must be strictly less than $10.
B)It must be less than or equal to $10.
C)It must be strictly greater than $10.
D)There is insufficient information to answer this question.

Accepted Answer

The answer of Consider two six-month European calls at strikes...

Question 16

An arbitrage opportunity is any situation in which&#10;A)Your estimated probability of a stock appreciating in price is three times that of the stock falling in price,and you hold a long position in the stock.&#10;B)An insider trading on privileged information is able to make supernormal profits.&#10;C)You are able to generate superior risk-adjusted returns to the market.&#10;D)You can construct a strategy whose cash flows are non-negative at all times with at least one positive cash flow at one or more points in time.

Accepted Answer

The answer of An arbitrage opportunity is any situation in...

Question 17

Consider two European put options,with maturities three months and six months on the same stock and same strike price.The stock pays no dividends.Which of the following statements is most accurate?&#10;A)The three-month option must be worth more than the six-month option.&#10;B)The three-month option must be worth less than the six-month option.&#10;C)The three-month option must be worth less than the six-month option if the option is deep in-the-money.&#10;D)The three-month option may be worth more than the six-month option.

Accepted Answer

The answer of Consider two European put options,with maturities three...

Question 18

All else the same,when the interest rate rises,the lower bound on a put option&#10;A)Falls or stays the same.&#10;B)Stays the same.&#10;C)Increases or stays the same.&#10;D)May become negative.

Accepted Answer

The answer of All else the same,when the interest rate...

Question 19

A stock is trading at $70.A one-month at-the-money call option on the stock is priced at $0.10.The risk free rate of interest is 2%.Which of the following statements is most accurate?&#10;A)There is no arbitrage opportunity here.&#10;B)A risk-less arbitrage strategy is to buy the call,short the stock and lend the present value of the strike for one month.&#10;C)A risk-less arbitrage strategy is to short the call,buy the stock and borrow the present value of the strike for one month.&#10;D)There is not enough information to decide whether or not there is an arbitrage.

Accepted Answer

The answer of A stock is trading at $70.A one-month...

Deck 9: No-Arbitrage Restrictions