MCQ

13.13. 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

10.1. 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.

10.5. 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.

5.12. 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

11.9. Which of the following is NOT true? (Assume that present values are calculated from the end of the life of the option to today.) 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 an American put option is exercised today is pays off the strike price minus the stock price. If the stock price is very low A is not true. The other three are all true.

10.9. 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.

11.15. 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 (10.11).

5.3. 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.

5.6. 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.

11.10. 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.

15.9. What was the original Black-Scholes-Merton model designed to value? A. A European option on a stock providing no dividends B. A European or American option on a stock providing no dividends C. A European option on any stock D. A European or American option on any stock

Answer: A The original Black-Scholes-Merton model was designed to value a European option on a stock paying no dividends .

3.1. 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.

10.7. 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.

3.2. 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).

11.3. 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.

10.11. 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

15.15. When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 5%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European put option on the stock A. 19.7N(-0.1)-20N(-0.2) B. 20N(-0.1)-20N(-0.2) C. 19.7N(-0.2)-20N(-0.1) D. 20N(-0.2)-20N(-0.1)

Answer: A In this case S0 = K =20, r = 0.06, =0.2, and T=0.25 so that Ke^-rT=20e^-0.06×0.25=19.7. Also d1 = [ln(1)+(0.06+0.04/2)×0.25]/(0.2×0.5) =0.2 and d2=0.2 - 0.2×0.5=0.1. A is therefore the correct answer.

3.4. 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.

13.6. 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 when the option is in the money.

15.12. Which of the following is NOT true? A. Risk-neutral valuation provides prices that are only correct in a world where investors are risk-neutral B. Options can be valued based on the assumption that investors are risk neutral C. In risk-neutral valuation the expected return on all investment assets is set equal to the risk-free rate D. In risk-neutral valuation the risk-free rate is used to discount expected cash flows

Answer: A Risk-neutral valuation produces a valuation that is correct in all situations not just those where investors are risk-neutral. The expected return on all investments is assumed to be the risk-free rate and the risk-free rate is used to discount expected payoffs.

15. 3. Which of the following is a definition of volatility A. The standard deviation of the return, measured with continuous compounding, in one year B. The variance of the return, measured with continuous compounding, in one year C. The standard deviation of the stock price in one year D. The variance of the stock price in one year

Answer: A Volatility when multiplied by square root of t is the standard deviation of the return in a short period of time of length t. It is also the standard deviation of the continuously compounded return in one year.

13.3. 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 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.)

13.12. 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

13.16. 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

Answer: B

13.17. 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

5.9. 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.

11.11. 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.

5.13. 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.

3.7. 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.)

3.6. 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

13.14. 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.

15.14. When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European call option on the stock A. 20N(0.1)-19.7N(0.2) B. 20N(0.2)-19.7N(0.1) C. 19.7N(0.2)-20N(0.1) D. 19.7N(0.1)-20N(0.2)

Answer: B In this case S0 =K =20, r=0.06, =0.2, and T=0.25 so that Ke^-rT=20e^-0.06×0.25=19.7. Also d1 = [ln(1)+(0.06+0.04/2)×0.25]/(0.2×0.5) =0.2 and d2=0.2 - 0.2×0.5=0.1. B is therefore the correct answer.

3.11. 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.

5.2. 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.

3.10. 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.

5.4. 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.

13.1. 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.

5.14. 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.

11.16. 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.

11.7. 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.

15.6. What is the number of trading days in a year usually assumed for equities? A. 365 B. 252 C. 262 D. 272

Answer: B Analysts usually assume that there are 252 trading days in a year for equities.

15.1. Which of the following is assumed by the Black-Scholes-Merton model? A. The return from the stock in a short period of time is lognormal B. The stock price at a future time is lognormal C. The stock price at a future time is normal D. None of the above

Answer: B Black-Scholes-Merton assumes that the return from a stock in a short period of time is normally distributed. This means that the stock price at a future time is lognormally distributed.

15. 4 A stock price is $100. Volatility is estimated to be 20% per year. What is an estimate of the standard deviation of the change in the stock price in one week? A. $0.38 B. $2.77 C. $3.02 D. $0.76

Answer: B The estimate is 100x0.2x root 1/52 =2.77$

15. 5. What does N(x) denote? A. The area under a normal distribution from zero to x B. The area under a normal distribution up to x C. The area under a normal distribution beyond x D. The area under the normal distribution between -x and x

Answer: B The normal distribution runs from minus infinity to plus infinity. N(x) is the area under the distribution between minus infinity and x.

15.10. A stock provides an expected return of 10% per year and has a volatility of 20% per year. What is the expected value of the continuously compounded return in one year? A. 6% B. 8% C. 10% D. 12%

Answer: B The expected value of the continuously compounded return per year is In this case it is 0.1 - 0.2^2/2 = 0.08 or 8%.

13.18. 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

13.20. 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.

10.8. 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.

5.1. 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.

13.2. 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 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.

5.10. 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

10.4. 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.

3.9. 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.

11.1. 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.

5.15. 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.

13.8. 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.

13.7. 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.

3.5. 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 the change in the spot price to the change in the futures price.

10.10. 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.

13.4. 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.

15.17. The volatility of a stock is 18% per year. Which of the following is closest to the volatility per month? A. 1.5% B. 3.0% C. 5.2% D. 6.3%

Answer: C The volatility per month is the volatility per year multiplied by the square root of 1/12 . The square root of 1/12 is 0.2887 and 18% multiplied by this is 5.2%.

11.5. 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 that no other variables change. In particular the stock price remains the same.)

15.7. The risk-free rate is 5% and the expected return on a non-dividend-paying stock is 12%. Which of the following is a way of valuing a derivative? A. Assume that the expected growth rate for the stock price is 17% and discount the expected payoff at 12% B. Assuming that the expected growth rate for the stock price is 5% and discounting the expected payoff at 12% C. Assuming that the expected growth rate for the stock price is 5% and discounting the expected payoff at 5% D. Assuming that the expected growth rate for the stock price is 12% and discounting the expected payoff at 5%

Answer: C Risk-neutral valuation shows that a derivative can be correctly valued by assuming that the stock grows at the risk-free rate and discounting the expected payoff at the risk-free rate. It follows that C is the correct answer.

13.11. 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.

5.8. 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.

13.15. 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

13.19. 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

5.7. 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 (40-42)e-0.08×0.25 = −$1.96.

10.3. 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).

10.6. 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.

11.4. 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.

15. 8. When there are two dividends on a stock, Black's approximation sets the value of an American call option equal to which of the following A. The value of a European option maturing just before the first dividend B. The value of a European option maturing just before the second (final) dividend C. The greater of the values in A and B D. The greater of the value in B and the value assuming no early exercise

Answer: D For Black's approximation we calculate a) the value of the option assuming no early exercise and b) the value of the option assuming that the exercise decision is made immediately before the final ex-dividend date. The value of the option is set equal to the greater of these two values.

3.8. 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.

5.11. 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.

10.13. 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.

11.14. 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.

13.5. 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 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

10.2. 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. Acallonastockplusastockthesameasaput 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 9.5. A put on a stock plus the stock provides a payoff that is similar to a call, as explained in Chapters 10 and 11. But a call on a stock plus a stock does not provide a similar payoff to a put.

15.11. An investor has earned 2%, 12% and -10% on equity investments in successive years (annually compounded). This is equivalent to earning which of the following annually compounded rates for the three year period. A. 1.33% B. 1.23% C. 1.13% D. 0.93%

Answer: D Over the three year period $100 grows to 100×1.02×1.12×0.9 = $102.816. This corresponds to an annually compounded return per year of 3rd root 1.02816 or 0.93%. One plus the return is the geometric average of 1.02, 1.12, and 0.90.

11.13. 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

11.2. 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.

13.9. 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.

13.10. 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.

10.12. 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.

15.16. A stock price is 20, 22, 19, 21, 24, and 24 on six successive Fridays. Which of the following is closest to the volatility per annum estimated from this data? A. 50% B. 60% C. 70% D. 80%

Answer: D The price relative for the first week is 22/20 or 1.1. The natural log of the price relative is ln(1.1) or 0.09531. Similarly the ln of the price relatives for the other weeks are -0.1466, 0.1001, 0.1335, and 0. The standard deviation of 0.09531, -0.1466, 0.1001, 0.1335, and 0 is 0.1138. The volatility per week is therefore 11.38%. This corresponds to a volatility per year of 0.1138 multiplied by the square root of 52 or about 82%. The answer is therefore D.

3.3. 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).

5.5. 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.

11.6. 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.

11.8. 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.

15.13. When the Black-Scholes-Merton and binomial tree models are used to value an option on a non-dividend-paying stock, which of the following is true? A. The binomial tree price converges to a price slightly above the Black-Scholes-Merton price as the number of time steps is increased B. The binomial tree price converges to a price slightly below the Black-Scholes-Merton price as the number of time steps is increased C. Either A or B can be true D. The binomial tree price converges to the Black-Scholes-Merton price as the number of time steps is increased

Answer: D The binomial tree valuation method and the Black-Scholes formula are based on the same set of assumptions. As the number of time steps is increased the answer given by the binomial tree approach converges to the answer given by the Black-Scholes-Merton formula.

15.2. The original Black-Scholes and Merton papers on stock option pricing were published in which year? A. 1983 B. 1984 C. 1974 D. 1973

Answer: D The correct answer is 1973. By coincidence this is also the year that organized trading in call options started. Put option trading started a few years later.

11.12. 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.