Hull: Options, Futures, and Other Derivatives, Ninth Edition
Chapter 13: Binomial Trees
Multiple Choice Test Bank: Questions with
Answers(Binomial trees)
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.
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
The formula for the risk-neutral probability of an up movement is
erT - d
p=
u- d
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.
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
The formula for the risk-neutral probability of an up movement is
, Hull: Options, Futures, and Other Derivatives, Ninth Edition
Chapter 13: Binomial Trees
Multiple Choice Test Bank: Questions with
Answers(Binomial trees)
erT - d
p=
u- d
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.)
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.
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
The formula for the risk-neutral probability of an up movement is
erT - d
p=
u- 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
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
Chapter 13: Binomial Trees
Multiple Choice Test Bank: Questions with
Answers(Binomial trees)
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.
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
The formula for the risk-neutral probability of an up movement is
erT - d
p=
u- d
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.
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
The formula for the risk-neutral probability of an up movement is
, Hull: Options, Futures, and Other Derivatives, Ninth Edition
Chapter 13: Binomial Trees
Multiple Choice Test Bank: Questions with
Answers(Binomial trees)
erT - d
p=
u- d
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.)
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.
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
The formula for the risk-neutral probability of an up movement is
erT - d
p=
u- 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
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
