Question 1

Which of the following is possible in a modified Cox, Ross, Rubinstein binomial tree?

Accepted Answer

A)   The interest rate and volatility can both be functions of time
B)   The interest rate or the volatility can be a function of time, but not both
C)   The interest rate can be a function of time but the volatility cannot
D)   The interest rate and volatility must be constant A

Question 2

The chapter discusses an alternative to the Cox, Ross, Rubinstein tree. In this alternative, which of the following are true?

Accepted Answer

A)   The relationship between u and d is: u=1/d
B)   The relationship between u and d is: u-1=1-d
C)   The probabilities on the tree are all 0.5
D)   None of the above C

Question 3

The values of a stock price at the end of the second time step are $80, $100, $125. The corresponding values of an option are $0, $5, and $20 respectively. What is an estimate of gamma?

Accepted Answer

A)   0.136
B)   0.146
C)   0.156
D)   0.166 C

Question 4

Which of the following cannot be valued by simulating paths through a tree in the way described in the chapter?

Accepted Answer

A)   European options
B)   American options
C)   Asian options (i.e., options on the average stock price) 
D)   An option which provides a payoff of $100 if the stock price is greater than the strike price at maturity

Question 5

Which of the following cannot be estimated from a single binomial tree?

Accepted Answer

A)   delta
B)   gamma
C)   theta
D)   vega

Question 6

What is the recommended way of making volatility a function of time in a Cox, Ross, Rubinstein tree?

Accepted Answer

A)   Make u a function of time
B)   Make p a function of time
C)   Make u and p a function of time
D)   Make the lengths of the time steps unequal

Question 7

For an option on futures, the volatility is 35%, the time step is three months, and the risk-free rate is 5%. What is the Cox, Ross, Rubinstein parameter, p?

Accepted Answer

A)   0.52
B)   0.46
C)   0.48
D)   0.50

Question 8

Which of the following can be valued without using a numerical procedure such as a binomial tree?

Accepted Answer

A)   American put options on a non-dividend paying stock
B)   American call options on a non-dividend paying stock
C)   American call options on a currency
D)   American put options on futures

Question 9

Which of the following is true for u in a Cox-Ross-Rubinstein binomial tree?

Accepted Answer

A)   It depends on the interest rate and the volatility
B)   It depends on the volatility but not the interest rate
C)   It depends on the interest rate but not the volatility
D)   It depends on neither the interest rate nor the volatility

Question 10

What is the difference between valuing an American and a European option using a tree?

Accepted Answer

A)   The value of u is higher for American options
B)   The value of u is lower for American options
C)   The time steps for American options are not equal
D)   It is necessary to do two calculations at nodes where the option is in the money

Question 11

How many different paths are there through a Cox-Ross-Rubinstein tree with four-steps?

Accepted Answer

A)   5
B)   9
C)   12
D)   16

Question 12

A binomial tree prices an American option at $3.12 and the corresponding European option at $3.04. The Black-Scholes price of the European option is $2.98. What is the control variate price of the American option?

Accepted Answer

A)   $3.06
B)   $3.18
C)   $2.90
D)   $3.08

Question 13

For an option on futures, the volatility is 35%, the time step is three months, and the risk-free rate is 5%. What is the Cox, Ross, Rubinstein parameter, u?

Accepted Answer

A)   1.34
B)   1.29
C)   1.09
D)   1.19

Question 14

Which of the following describes the way that the parameters in a binomial tree are chosen?

Accepted Answer

A)   The expected return during each time step is the risk-free rate
B)   The standard deviation of the return in each time step is, for small time steps, almost exactly equal to the volatility per annum times the square root of the length on the time step in years
C)   The tree recombines
D)   All of the above

Question 15

When the volatility of an option increases from 30% to 32% the value of the option increases from $2.00 to $2.40. What is the vega of the option?

Accepted Answer

A)   0.20 dollars per %
B)   0.50 dollars per %
C)   0.80 dollars per %
D)   2.00 dollars per %

Question 16

When we move from assuming no dividends to assuming a constant dividend yield, which of the following is true for a Cox, Ross, Rubinstein tree?

Accepted Answer

A)   The parameters u and p change
B)   p changes but u does not
C)   u changes but p does not
D)   Neither p nor u changes

Question 17

How many nodes are there at the end of a Cox-Ross-Rubinstein five-step binomial tree?

Accepted Answer

A)   4
B)   5
C)   6
D)   7

Question 18

What is the recommended way of making interest rates a function of time in a Cox, Ross, Rubinstein tree?

Accepted Answer

A)   Make u a function of time
B)   Make p a function of time
C)   Make u and p a function of time
D)   Make the lengths of the time steps unequal

Question 19

When the stock price is 20 and the present value of dividends is 2, which of the following is the recommended way of constructing a tree?

Accepted Answer

A)   Draw a tree for an initial stock price of 20 and subtract the present value of future dividends at each node
B)   Draw a tree for an initial stock price of 22 and subtract the present value of future dividends at each node
C)   Draw a tree with an initial stock price of 18 and add the present value of future dividends at each node
D)   Draw a tree with an initial stock price of 18 and add 2 at each node

Question 20

A European option on a stock with known dollar dividend is valued by setting the stock price variable equal to the stock price minus the present value of the dividend in the Black-Scholes-Merton formula. A second price can be obtained using the tree building procedure in the chapter. Which of the following is true when a very large number of time steps are used in the tree?

Accepted Answer

A)   The first price is higher than the second price
B)   The first price is lower than the second price
C)   The first price is sometimes higher and sometimes lower than the second price
D)   The two prices are almost exactly the same

Exam 18: Binomial Trees in Practice

Which of the following is possible in a modified Cox, Ross, Rubinstein binomial tree?

The chapter discusses an alternative to the Cox, Ross, Rubinstein tree. In this alternative, which of the following are true?

The values of a stock price at the end of the second time step are $80, $100, $125. The corresponding values of an option are $0, $5, and $20 respectively. What is an estimate of gamma?

Which of the following cannot be valued by simulating paths through a tree in the way described in the chapter?

Which of the following cannot be estimated from a single binomial tree?

What is the recommended way of making volatility a function of time in a Cox, Ross, Rubinstein tree?

For an option on futures, the volatility is 35%, the time step is three months, and the risk-free rate is 5%. What is the Cox, Ross, Rubinstein parameter, p?

Which of the following can be valued without using a numerical procedure such as a binomial tree?

Which of the following is true for u in a Cox-Ross-Rubinstein binomial tree?

What is the difference between valuing an American and a European option using a tree?

How many different paths are there through a Cox-Ross-Rubinstein tree with four-steps?

A binomial tree prices an American option at $3.12 and the corresponding European option at $3.04. The Black-Scholes price of the European option is $2.98. What is the control variate price of the American option?

For an option on futures, the volatility is 35%, the time step is three months, and the risk-free rate is 5%. What is the Cox, Ross, Rubinstein parameter, u?

Which of the following describes the way that the parameters in a binomial tree are chosen?

When the volatility of an option increases from 30% to 32% the value of the option increases from $2.00 to $2.40. What is the vega of the option?

When we move from assuming no dividends to assuming a constant dividend yield, which of the following is true for a Cox, Ross, Rubinstein tree?

How many nodes are there at the end of a Cox-Ross-Rubinstein five-step binomial tree?

What is the recommended way of making interest rates a function of time in a Cox, Ross, Rubinstein tree?

When the stock price is 20 and the present value of dividends is 2, which of the following is the recommended way of constructing a tree?