Deck 12: Introduction to Binomial Trees

Question 1

In a Cox-Ross-Rubinstein binomial tree, the formula for the proportional up-movement, u, is: choose one)

A) $u = e ^ { r \Delta t }$
B) $\mathcal { U } = e ^ { r - \sqrt { \Delta t } }$
C) $u = e ^ { \sigma \Delta t }$
D) $\mathcal { U } = e ^ { \sigma \sqrt { \Delta t } }$

Answer 1

$\mathcal { U } = e ^ { \sigma \sqrt { \Delta t } }$

Question 2

Which two statements are true? choose two)

A) Delta is a measure of the volatility of an option.
B) Delta is a measure of the position in the underlying stock that should be taken to hedge an option.
C) Delta is estimated by considering two adjacent nodes on a tree at a certain time and calculating the difference in option prices divided by the difference in the stock prices.
D) The delta of a put option is positive.

Answer 2

Delta is a measure of the position in the underlying stock that should be taken to hedge an option.
Delta is estimated by considering two adjacent nodes on a tree at a certain time and calculating the difference in option prices divided by the difference in the stock prices.

Question 3

In a Cox-Ross-Rubinstein binomial tree, the relationship between the proportional down-movement, d, and the proportional up-movement, u, is: choose one)

A) d = u − 1
B) d = 1 / u
C) d = 2 − u
D) None of the above

Answer 3

B

Question 4

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.
i) What long position in the stock is necessary to hedge a short call option when the strike price is $32? Give the number of shares purchased as a percentage of the number of options that have been sold. _ _ _ _ _ _
ii) What is the value of the call option? _ _ _ _ _ _
iii) What long position in the stock is necessary to hedge a long put option when the strike price is $32? Give the number of shares purchased as a percentage of the number of options purchased. _ _ _ _ _ _
iv) What is the value of the put option? _ _ _ _ _ _
v) What is the risk-neutral probability of the stock price moving up? _ _ _ _ _ _

Question 5

American options can be valued using a binomial tree by: choose one)

A) Checking whether early exercise is optimal at all nodes where the option is in-the-money
B) Checking whether early exercise is optimal at the final nodes
C) Checking whether early exercise is optimal at the penultimate nodes and the final nodes
D) Increasing the number of time steps on the tree