Exams Fin dera

Question 1

When interest rates are zero, a delta and gamma neutral option portfolio will in the Black-Scholes-Merton model have a theta that is approximately equal to zero.

Answer 1

True

Question 2

In a symmetric implied volatility smile, the BSM model is underpricing at-the-money options and overpricing deep in- and out-of-the-money options.

Answer 2

False

Question 3

A stock with price 100 kr follows a one step Binomial model and will either increase in value to 110 kr or decrease to 90 kr in one year’s time. The risk-free interest rate is zero. An investor considers the increase and decrease in this model to be equally likely to happen. Is it true or false that the investor is risk-averse?

Answer 3

False

Question 4

Assuming the Black-Scholes model with the standard assumptions, the function f(t, St) = 3t + S³t cannot be the price of any derivative.

Answer 4

True

Question 5

Two different portfolios that have the same value at risk must also have the same expected shortfall.

Answer 5

False

Question 6

When using Monte Carlo simulations in the Black-Scholes-Merton model to price options, then the change in the stock’s return under the risk-neutral measure over the interval 0.02 is given by: d lnS = rdt + σε√0.02 where ε is drawn from a standard normal distribution.

Answer 6

False

Question 7

To simulate the change in a standard Wiener process (also called Brownian Motion) over a time interval of length 0.03 you first ask the computer to provide a random number , drawn from a standard normal distribution with mean 0 and standard deviation 1. Then you multiply with the square root of 0.03.

Answer 7

True

Question 7

Consider a forward contract with delivery price K and a European call option with strike K. Both contracts are currently deep in the money and both contracts mature in twelve months. If interest rates are positive, the theta of the option and the theta of the forward will be approximately the same.

Answer 7

True

Question 8

A European at-the-money call option and a European at-the-money put option with the same time to maturity will always provide the same implied volatility in an arbitrage free market without transaction costs and nonnegative interest rates.

Answer 8

True

Question 9

If the one day 99% Expected shortfall is 10 million kr, then the expected portfolio loss tomorrow will be 10 million kr.

Answer 9

False

Question 10

Assuming the Black-Scholes model with the standard assumptions, the so-called forward price is a function that cannot be the value of any derivative.

Answer 10

True

Question 11

(a) Consider a forward contract with delivery price K and a European call option with strike K. Both contracts are currently deep in the money and both contracts mature in twelve months. If interest rates are positive, the theta of the option and the theta of the forward will be approximately the same.

Answer 11

True

Question 12

The Black-Scholes-Merton pricing formula for a call option is correct only for investors that are risk-neutral.

Answer 12

False

Question 13

(b) When valuing an American put option using a two step Binomial tree model, early exercise will be optimal in at least one of the nodes.

Answer 13

False

Question 14

(d) Assuming the normal distribution for a given asset’s return, the 10-day Expected Shortfall at 95% confidence level will always be larger than the 10-day Value-at-Risk at 95% confidence.

Answer 14

True

Question 15

(c) To simulate the change in a standard Wiener process (also called Brownian Motion) over a time interval of length 0.02, you first ask the computer to provide a random number X, drawn from a uniform distribution (0,1). Then you multiply X with the square root of 0.02.

Answer 15

False

Question 16

(e) Consider a call option and a put option that have the same strike price and mature in six months. Assume that the market is free of arbitrage, has no transaction costs and that interest rates are positive. Then if the put option is out of the money, the implied volatility of the put option must be higher than the implied volatility of the call option.

Answer 16

False

Question 17

(b) It is always optimal to early exercise an American call option written on a dividend paying stock.

Answer 17

False

Question 17

(a) A European put option that is deep in the money has a delta close to zero.

Answer 17

False

Question 18

(e) An option’s implied volatility is a measure of the variation in the option price.

Answer 18

False

Question 18

(c) To simulate the change in a Generalized Wiener process over a time interval of length 0.02, you first ask the computer to provide a random number X, drawn from a standard normal distribution with mean 0 and variance 1. Then you multiply X with the square root of 0.02.

Answer 18

False

Question 18

(d) Two portfolios AAA and BBB with the same 10-day Value-at-Risk at 95% confidence level have the same maximum loss.

Answer 18

False

Question 19

Which of the following is equivalent to a short position in a European put option?
A. A short position in an asset-or-nothing put option plus a long position in a cash-or-nothing put option
B. A long position in an asset-or-nothing put option plus a long position in a cash-or-nothing put option
C. A long position in an asset-or-nothing call option plus a long position in a cash-or-nothing call option
D. A long position in an asset-or-nothing call option plus a short position in a cash-or-nothing call option
E. None of the alternatives

Answer 19

A

Question 20

A futures price is currently 40 cents. It is expected to move up to 44 cents or down to 34 cents in the next six months. The risk-free interest rate is 6%. What is the value of a six-month put option on the futures with a strike price of 37 cents?
A. 3.00 cents
B. 2.91 cents
C. 1.16 cents
D. 1.20 cents
E. None of the alternatives

Answer 20

C

Question 21

A European at-the-money call option on a currency has four years until maturity. The exchange rate volatility is 10%, the domestic risk-free rate is 2% and the foreign risk-free rate is 5%. The current exchange rate is 1.2000. What is the value of the option? A. 0.98N(0.25)-1.11(0.05) B. 0.98N(-0.3)-1.11N(-0.5) C. 0.98N(-0.5)-1.11N(-0.7) D. 0.98N(0.10)-1.11N(0.06) E. None of the alternatives

Answer 21

C

Question 22

A volatility surface is a table showing the relationship between which of the following? A. Implied volatility, time to maturity, and strike price B. Implied volatility, historical volatility, and time to maturity C. Historical volatility, strike price, and time to maturity D. None of the above

Answer 22

A

Question 23

In a Binomial model, the values of a stock price at the end of the second time step are $80, $100, $120. The corresponding values of an option are $0, $5, and $20 respectively. What is an estimate of gamma? A. 0.013 B. 0.014 C. 0.015 D. 0.025

Answer 23

E

Question 24

What should a trader do to grab an arbitrage, 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 E. None of the alternatives

Answer 24

D

Question 25

Which of the following is acquired (in addition to a cash payoff) when the holder of a put futures exercises the option? A. A long position in a futures contract B. A short position in a futures contract C. A long position in the underlying asset D. A short position in the underlying asset E. None of the alternatives

Answer 25

B

Question 26

Which of the following is equivalent to a long position in a European call option? A. A short position in a cash-or-nothing put option plus a long position in an asset-or-nothing put option B. A long position in an asset-or-nothing put option plus a long position in a cash-or-nothing put option C. A long position in an asset-or-nothing call option plus a long position in a cash-or-nothing call option D. A long position in an asset-or-nothing call option plus a short position in a cash-or-nothing call option E. None of the alternatives

Answer 26

D

Question 27

A futures price is currently 40 cents. It is expected to move up to 44 cents or down to 34 cents in the next six months. The risk-free interest rate is 6%. What is the value of a six-month call option with a strike price of 39 cents? A. 5.00 cents B. 2.91 cents C. 3.00 cents D. 4.21 cents E. None of the alternatives

Answer 27

B

Question 28

How many different paths are there through a (Cox-Ross-Rubinstein) Binomial tree with four steps? A. 5 B. 9 C. 12 D. 16 E. None of the alternatives

Answer 28

D

Question 29

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 E. None of the alternatives

Answer 29

B

Question 30

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, the price of a European call option on the stock is: 20N(0.2)−19.7N(0.1) What does N(0.2)denote? A. The area under a normal distribution from zero to 0.2 B. The area under a normal distribution up to 0.2 C. The area under a normal distribution beyond 0.2 D. The area under the normal distribution between -0.2 and 0.2 E. None of the alternatives

Answer 30

B

Question 31

For a European call option on a currency, the exchange rate is 1.0000, the strike price is 0.9100, the time to maturity is one year, the domestic risk-free rate is 5% per annum, and the foreign risk-free rate is 3% per annum. How low can the option price be without there being an arbitrage opportunity? A. 0.0900 B. 0.1048 C. 0.1344 D. 0.1211 E. None of the alternatives

Answer 31

B

Question 32

Which one of the following strategies could NOT be a delta-neutral portfolio? A. A long position in call options plus a short position in the underlying stock B. A short position in call options plus a short position in the underlying stock C. A long position in put options and a long position in the underlying stock D. A long position in a put option and a long position in a call option E. None of the alternatives

Answer 32

B

Question 33

A binomial tree prices an American option at $3.12 and the corresponding European option at $3.04. The Black-Scholes-Merton price of the European option is $2.98. What is the control variate price of the American option? A. $3.06 B. $3.18 C. $2.90 D. $3.08 E. None of the alternatives

Answer 33

A

Question 34

Today’s price of a non-dividend-paying stock is $30. Over the next six months the stock price is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. An investor sells six-month call options with a strike price of $30. Which of the following hedges the investor’s 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 0.4 shares for each call option sold

Answer 34

A

Question 35

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. 7N(0.2)−20N(0.1) D. 19.7N(0.1)−20N(0.2)

Answer 35

B

Question 36

One-year European call and put options on an asset are worth $3 and $4 respectively when the strike price is $20 and the one-year risk-free rate is 5%. What is the one-year forward price of the asset if there are no arbitrage opportunities? A. –$1 B. $17.15 C. $18.95 D. $20.45

Answer 36

C

Question 37

A 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? A. Long position in 250 options B. Short position in 250 options C. Long position in 20 options D. Short position in 20 options

Answer 37

A

Question 38

Assume interest rate is zero. In a shout call option the strike price is $30. The holder shouts when the asset price is $40. What is the payoff from the option if the final asset price is $35? A. $0 B. $5 C. $10 D. $15

Answer 38

C