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FRM II
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Questions in this quiz (87)
1209. 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 .
- Ao e times the square root of dt
- Bo e times dt
- Co dt times the square root of e
- Do The square root of e times the square root of dt
2210. Which of the following is true when the stock price follows geometric Brownian motion
- Ao The future stock price has a normal distribution
- Bo The future stock price has a lognormal distribution
- Co The future stock price has geometric distribution
- Do The future stock price has a truncated normal distribution
3211. Which of the following is NOT a property of a Wiener process?
- Ao The change during a short period of time dt has a variance dt
- Bo The changes in two different short periods of time are independent
- Co The mean change in any time period is zero
- Do The standard deviation over two consecutive time periods is the sum of the standard deviations over each of the periods
4212. 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?
- Ao 1
- Bo 2
- Co 3
- Do 4
5213. 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?
- Ao 4
- Bo 8
- Co 12
- Do 16
6214. 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
- Ao 16
- Bo 20
- Co 24
- Do 30
7215. If a stock price follows a Markov process which of the following could be true
- Ao Whenever the stock price has gone up for four successive days it has a 70% chance of going up on the fifth day.
- Bo Whenever the stock price has gone up for four successive days there is almost certain to be a correction on the fifth day.
- Co The way the stock price moves on a day is unaffected by how it moved on the previous four days.
- Do Bad years for stock price returns are usually followed by good years.
8216. 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.)
- Ao $0.20
- Bo $0.23
- Co $0.26
- Do $0.29
9217. Which of the following gives a random sample from a standard normal distribution in Excel?
- Ao =NORMSINV()
- Bo =NORMSINV(RAND())
- Co =RND(NORMSINV())
- Do =RAND()
10218. 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.
- Ao sX
- Bo sX^2
- Co 2sX^2
- Do msX
11219. The variance of a Wiener process in time t is
- Ao t
- Bo t squared
- Co the square root of t
- Do t to the power of 4
12220. Which of the following defines an Ito process?
- Ao A process where the drift is non-constant and can be stochastic
- Bo A process where the coefficient of dz is non-constant and can be stochastic
- Co A process where either the drift or the coefficient of dz or both are non-constant and can be stochastic
- Do A process where proportional changes follow a generalized Wiener process
13221. 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?
- Ao 12
- Bo 14
- Co 16
- Do 18
14222. 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.
- Ao dX = m dt + s dz
- Bo dX = (m−r) dt + s dz
- Co dX = (m − s^2) dt + s dz
- Do dX = (m − s^2 /2) dt + s dz
15223. 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.)
- Ao $20.41
- Bo $20.51
- Co $20.61
- Do $20.71
16224. 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.
- Ao 2 m X^2 + s^2 X^2
- Bo 2 m X^2
- Co m X^2 + 2 s^2 X^2
- Do m X^2 + s^2 X^2
17225. 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).
- Ao dy = b y dz
- Bo dy = 0.5 b^2 y dt +b y dz
- Co dy = (y + 0.5 b^2 y) dt +b y dz
- Do dy = 0.5 b^2 y dt +b dz
18226. 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
- Ao 6.2
- Bo 6.7
- Co 7.2
- Do 7.7
19227. If the risk-free rate is r and price of a non dividend 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.
- Ao m
- Bo m − s^2 / 2
- Co m−r
- Do r− s^2 /2
20228. Which of the following is a definition of volatility
- Ao The standard deviation of the return, measured with continuous compounding, in one year
- Bo The variance of the return, measured with continuous compounding, in one year
- Co The standard deviation of the stock price in one year
- Do The variance of the stock price in one year
21229. What was the original Black-Scholes-Merton model designed to value?
- Ao A European option on a stock providing no dividends
- Bo A European or American option on a stock providing no dividends
- Co A European option on any stock
- Do A European or American option on any stock
22230. What does N(x) denote?
- Ao The area under a normal distribution from zero to x
- Bo The area under a normal distribution up to x
- Co The area under a normal distribution beyond x
- Do The area under the normal distribution between -x and x
23231. Which of the following is NOT true?
- Ao Risk-neutral valuation provides prices that are only correct in a world where investors are risk-neutral
- Bo Options can be valued based on the assumption that investors are risk neutral
- Co In risk-neutralvaluation the expected return on all investment assets is set equal to the risk-free rate
- Do In risk-neutral valuation the risk-free rate is used to discount expected cash flows
24232. A stock price is $100. Volatility is estimated to be 20% per year. What is an estimate of the standard deviation of the absolute change in the stock price in one week?
- Ao $0.38
- Bo $2.77
- Co $3.02
- Do $0.76
25233. The volatility of a stock is 18% per year. Which is closest to the volatility per month?
- Ao 1.5%
- Bo 3.0%
- Co 5.2%
- Do 6.3%
26234. 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 put option on the stock
- Ao 19.7 N(-0.1) -20 N(-0.2)
- Bo 20 N(-0.1) – 20 N(-0.2)
- Co 19.7 N(-0.2) - 20 N(-0.1)
- Do 20 N(-0.2) – 20 N(-0.1)
27235. 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.
- Ao 1.33%
- Bo 1.23%
- Co 1.13%
- Do 0.93%
28236. Which of the following is assumed by the Black-Scholes-Merton model?
- Ao The return from the stock in a short period of time is lognormal
- Bo The stock price at a future time is lognormal
- Co The stock price at a future time is normal
- Do None of the above
29237. 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 put option on the stock
- Ao 20 N(-0.1) –19.7 N(-0.2)
- Bo 20 N(-0.2) – 19.7N(-0.1)
- Co 19.7 N(-0.2) - 20 N(-0.1)
- Do 19.7 N(-0.1) -20 N(-0.2)
30238. 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?
- Ao The binomial tree price converges to a price slightly above the Black-Scholes-Merton price as the number of time steps is increased
- Bo The binomial tree price converges to a price slightly below the Black-Scholes-Merton price as the number of time steps is increased
- Co The binomial tree price converges to the Black-Scholes-Merton price as the number of time steps is increased
31239. 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?
- Ao Assume that the expected growth rate for the stock price is 17% and discount the expected payoff at 12%
- Bo Assuming that the expected growth rate for the stock price is 5% and discounting the expected payoff at 12%
- Co Assuming that the expected growth rate for the stock price is 5% and discounting the expected payoff at 5%
- Do Assuming that the expected growth rate for the stock price is 12% and discounting the expected payoff at 5%
32240. 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
- Ao The value of a European option maturing just before the first dividend
- Bo The value of a European option maturing just before the second (final) dividend
- Co The greater of the values in A and B
- Do The greater of the value in B and the value assuming no early exercise
33241. 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?
- Ao 50%
- Bo 60%
- Co 70%
- Do 80%
34242. Which of the following is true for a one-year call option on a stock that pays dividends every three months?
- Ao It is never optimal to exercise the option early
- Bo It can be optimal to exercise the option at any time
- Co It is only ever optimal to exercise the option immediately after an ex-dividend date
- Do None of the above
35243. Which of the following is a way of extending the Black-Scholes-Merton formula to value a European call option on a stock paying a single dividend?
- Ao Reduce the maturity of the option so that it equals the time of the dividend
- Bo Subtract the dividend from the stock price
- Co Add the dividend to the stock price
- Do Subtract the present value of the dividend from the stock price
36244. 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?
- Ao 6%
- Bo 8%
- Co 10%
- Do 12%
37245. Which of the following is measured by the VIX index
- Ao Implied volatilities for stock options trading on the CBOE
- Bo Historical volatilities for stock options trading on CBOE
- Co Historical volatilities for options trading on the S&P 500 index
- Do Implied volatilities for options trading on the S&P 500 index
- Eo VIX index measures the implied volatilities of one-month options trading on the S&P 500 index.
38246. What is the number of trading days in a year usually assumed for equities?
- Ao 365
- Bo 252
- Co 262
- Do 272
39Which of the following best describes the term “spot price”
- AThe price for delivery at a future time
- BThe price of renting an asset
- CThe price of an asset that has been damaged
- DThe price for immediate delivery
40Which of the following is approximately true when size is measured in terms of the underlying principal amounts or value of the underlying assets
- AThe exchange-traded market is twice as big as the over-the-counter market.
- BThe over-the-counter market is twice as big as the exchange-traded market.
- CThe over-the-counter market is ten times as big as the exchange-traded market
- DThe exchange-traded market is ten times as big as the over-the-counter market.
41A 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
- AA forward contract can be used to lock in the exchange rate
- BA forward contract will always give a better outcome than an option
- CAn option will always give a better outcome than a forward contract
- DAn option can be used to lock in the exchange rate
42 Which of the following best describes a central clearing party
- AIt helps facilitate futures trades
- BIt is a trader that works for a bank
- CIt stands between two parties in the over-the-counter market
- DIt is a trader that works for an exchange
43A limit order
- ANone of the above
- BIs an order that must be executed within a specified period of time
- CIs an order that can be executed at a specified price or one more favorable to the investor
- DIs an order to trade up to a certain number of futures contracts at a certain price
4450. Which of the following is an example of an option series?
- Ao All calls on a certain stock
- Bo All calls with a particular time to maturity and strike price on a certain stock
- Co All calls with a particular strike price on a certain stock
- Do All calls with a particular time to maturity on a certain stock
4541. 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?
- Ao Loss of $500
- Bo Loss of $1,000
- Co Gain of $1,500
- Do Loss of $1,500
4687. The spot price of an asset is positively correlated with the market. Which of the following would you expect to be true?
- Ao The forward price equals the expected future spot price.
- Bo The forward price is greater than the expected future spot price.
- Co The forward price is sometimes greater and sometimes less than the expected future spot price.
- Do The forward price is less than the expected future spot price.
4777. An interest rate is 5% per annum with continuous compounding. What is the equivalent rate with semiannual compounding? (%)
- Ao 5.06
- Bo 5.03
- Co 4.97
- Do 4.94
4879. 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. (%)
- Ao 7.5
- Bo 7.25
- Co 7
- Do 6.75
4971. An interest rate is 12% per annum with semiannual compounding. What is the equivalent rate with quarterly compounding? (%)
- Ao 11.92
- Bo 11.77
- Co 11.66
- Do 11.83
5080. 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?
- Ao $40.50
- Bo $45.15
- Co $35.84
- Do $19.67
51134. 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, s = 0.011. What does the model in the text estimate as the difference between the futures and the forward interest rate?
- A1. 0.105%
- B2. 0.103%
- C3. 0.098%
- D4. 0.093%
52153. 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.
- Ao 2.66
- Bo 2.06
- Co 1.06
- Do 0.00
5390. 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?
- Ao 0.7249
- Bo 0.693
- Co 0.707
- Do 0.7177
54202. 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.
- Ao $3.93
- Bo $2.93
- Co $1.93
- Do $0.93
55157. 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?
- Ao The value of the swap to the company increases
- Bo The value of the swap to the company decreases
- Co The value of the swap does not change providing the swap rate remains the same
- Do The value of the swap can either increase or decrease
5698. As inventories of a commodity decline, which of the following is true?
- Ao The one-year futures price as a percentage of the spot price increases
- Bo The one-year futures price as a percentage of the spot price decreases
- Co The one-year futures price as a percentage of the spot price stays the same
- Do Any of the above can happen
5799. As the convenience yield increases, which of the following is true?
- Ao The one-year futures price as a percentage of the spot price increases
- Bo The one-year futures price as a percentage of the spot price decreases
- Co The one-year futures price as a percentage of the spot price stays the same
- Do Any of the above can happen
58A call option on a stock has a delta of 0.3. A trader has sold 1,000 options. What position should the trader take to hedge the position?
- AA) Sell 300 shares
- BB) Buy 300 shares
- CC) Sell 700 shares
- D) Buy 700 shares
59What does theta measure?
- AA) The rate of change of delta with the asset price
- BC) The sensitivity of a portfolio value to interest rate changes
- CD) None of the above
- DB) The rate of change of the portfolio value with the passage of time
60What does gamma measure?
- AA) The rate of change of delta with the asset price
- B) The rate of change of the portfolio value with the passage of time
- CC) The sensitivity of a portfolio value to interest rate changes
- DD) None of the above
61What does vega measure
- A?A) The rate of change of delta with the asset price
- BB) The rate of change of the portfolio value with the passage of time
- CC) The sensitivity of a portfolio value to interest rate changes
- DD) None of the above
62What does rho measure?
- AA) The rate of change of delta with the asset price
- BC) The sensitivity of a portfolio value to interest rate changes
- CD) None of the above
- DB) The rate of change of the portfolio value with the passage of time
63Which of the following is true?
- AA) The delta of a European put equals minus the delta of a European call
- BB) The delta of a European put equals the delta of a European call
- CC) The gamma of a European put equals minus the gamma of a European call
- DD) The gamma of a European put equals the gamma of a European call
64A portfolio of derivatives on a stock has a delta of 2400 and a gamma of -10. An option on the stock with a delta of 0.5 and a gamma of 0.04 can be traded. What position in the option is necessary to make the portfolio gamma neutral?
- ALong position in 250 options
- BShort position in 250 options
- CLong position in 20 options
- DShort position in 20 options
65 A trader uses a stop-loss strategy to hedge a short position in a three-month call option with astrike price of 0.7000 on an exchange rate. The current exchange rate is 0.6950 and value of theoption is 0.1. The trader covers the option when the exchange rate reaches 0.7005 and uncovers(i.e., assumes a naked position) if the exchange rate falls to 0.6995. Which of the following isNOT true?
- AA) The exchange rate trading might cost nothing so that the trader gains 0.1 for each option sold
- BThe present value of the gain or loss from the exchange rate trading should be about 0.1 on average for each option sold
- CThe hedge works reasonably well
- DThe exchange rate trading might cost considerably more than 0.1 for each option sold so that the trader loses money
66Maintaining a delta-neutral portfolio is an example of which of the following?
- AStop-loss strategy
- BDynamic hedging
- CStatic hedging
- DHedge and forget strategy
67Which of the following could NOT be a delta-neutral portfolio?
- AA long position in call options plus a short position in the underlying stock
- BA short position in call options plus a short position in the underlying stock
- CA long position in put options and a long position in the underlying stock
- DA long position in a put option and a long position in a call option
68Which of the following is NOT true about gamma?
- AA long position in either a call or a put has a positive gamma
- BA big positive value for gamma indicates that a big movement in the asset price in eitherdirection will lead to a loss
- CThe magnitude of gamma is a measure of the curvature of the portfolio value as a function of the underlying asset price
- DA highly positive or highly negative value of gamma indicates that a portfolio needs frequent rebalancing to stay delta neutral
69 Gamma tends to be high for which of the following?
- AAt-the money options
- BOut-of-the money options
- CIn-the-money options
- DOptions with a long time to maturity
70Which of the following is true for a call option on a non-dividend-paying stock?
- AIf the strike price equals the forward price of the stock, it must have a delta of 0.5
- BIf the option has a delta of 0.5, it must be out of the money
- CIf the option has a delta of 0.5, it must be in of the money
- DIf the option is at the money (stock price equals strike price) it must have a delta of 0.5
71The risk-free rate is 5% and the dividend yield on an index is 2%. Which of the following isthe delta with respect to the index of a one-year futures on the index?
- A1.02
- B1.03
- C1.05
- D0.98
72The gamma of a delta-neutral portfolio is 500. What is the impact of a jump of $3 in theprice of the underlying asset?
- AA gain of $2,250
- BA loss of $2,250
- CA gain of $750
- DA loss of $750
73Vega tends to be high for which of the following?
- AAt-the money options
- BOut-of-the money options
- CIn-the-money options
- DOptions with a short time to maturity
74 The delta of a call option on a non-dividend-paying stock is 0.4. What is the delta of thecorresponding put option?
- A-0.4
- B-0.6
- C0.4
- D0.6
75A call option on a non-dividend-paying stock has a strike price of $30 and a time to maturityof six months. The risk-free rate is 4% and the volatility is 25%. The stock price is $28. What isthe delta of the option?
- AN(-0.1342)
- BN(-0.1888)
- CN(-0.2034)
- DN(-0.2241)
76Which of the following is NOT a letter in the Greek alphabet?
- Adelta
- Bgamma
- Cvega
- Drho
77 When the interest rate is zero which of the following is true for a delta-neutral portfolio witha positive gamma?
- AAs gamma increases theta becomes more negative
- BTheta is zero
- CAs gamma decreases theta declines
- DAs gamma increases theta becomes more positive
78119. 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.
- Ao 1.82
- Bo 1.85
- Co 1.88
- Do 1.92
79126. 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?
- Ao 106.04
- Bo 105.98
- Co 106.02
- Do 106
80123. What is the quoted discount rate on a money market instrument?
- Ao The interest rate earned as a percentage of the final face value of a bond
- Bo The risk-free rate used to calculate the present value of future cash flows from a bond
- Co The interest rate earned as a percentage of the initial price of a bond
- Do The interest rate earned as a percentage of the average price of a bond
81159. Which of the following describes a covered call?
- Ao A long call option on a stock plus a long position in the stock
- Bo A long call option on a stock plus a short put option on the stock
- Co A short call option on a stock plus a long position in the stock
- Do A short call option on a stock plus a short position in the stock
82162. 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?
- Ao $200
- Bo $400
- Co $300
- Do $100
83184. 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?
- Ao $5.00
- Bo $4.98
- Co $3.98
- Do $5.98
84192. 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?
- Ao $0.86
- Bo $2.86
- Co $3.86
- Do $4.00
85133. The conversion factor for a bond is approximately
- Ao The price it would have if it paid coupons at 8% per annum
- Bo The price it would have if all cash flows were discounted at 6% per annum
- Co The price it would have if all cash flows were discounted at 8% per annum
- Do The price it would have if it paid coupons at 6% per annum
8674. 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.
- Ao 8.88
- Bo 8.63
- Co 9.02
- Do 9.12
87207. Which of the following describes delta?
- Ao The ratio of the option price to the stock price
- Bo The ratio of the stock price to the option price
- Co The ratio of a change in the stock price to the corresponding change in the option price
- Do The ratio of a change in the option price to the corresponding change in the stock price
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- Quiz on FRM II, created by carolina salcedo on 17/08/2021.
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