Hull: Fundamentals of Futures and Options Markets,Ninth Edition
Chapter 12: Introduction to Binomial Trees
Multiple Choice Test Bank
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 0.4 shares for each call option sold
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
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
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
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
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
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
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
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%
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
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
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 si the option price when
u = 1.1 and d = 0.9.
A. $1.29
B. $1.49
C. $1.69
D. $1.89
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. Each step is 3 months,
the risk free rate is 8%, and u = 1.1 and d = 0.9.
A. $2.24
B. $2.44
C. $2.64
D. $2.84
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
15. If the volatility of a 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
16. If the volatility of a 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
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
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
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 be replaced by the excess of the domestic risk-free rate over the
foreign risk-free rate for discounting
D. The risk-free rate be replaced by the excess of the domestic risk-free rate over the
foreign risk-free rate when p is calculated
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%.
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