Question 1

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

Accepted Answer

The control variate price is 3.12+(2.98-3.04)= 3.06.

Question 2

Which of the following can be valued without using a numerical procedure such as a binomial tree?
A) American put options on a non-dividend paying stock
B) American call options on a non-dividend paying tock
C) American call options on a currency
D) American put options on futures

Accepted Answer

The answer is B because an American call on a non-dividend paying stock is worth the same as the corresponding European call,which can be valued with BSM.

Question 3

Which of the following cannot be valued by Monte Carlo simulation
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

Accepted Answer

American options cannot be valued in a simple way using Monte Carlo simulation because Monte Carlo simulation works forward from the beginning of the life of an option and we do not know whether the option should be exercised when a particular time is reached.

Question 4

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

Accepted Answer

When valuing American options it is necessary to calculate at each in-the-money node how much the option is worth if it is exercised (i.e.,the intrinsic value at the node)and how much it is worth if it is not exercised (i.e.,the roll back value)

Question 5

How many nodes are there at the end of a Cox-Ross-Rubinstein five-step binomial tree?
A) 4
B) 5
C) 6
D) 7

Accepted Answer

The number of nodes at the end of one time step is 2; the number of nodes at the end of two time steps is 3; and so on.

Question 6

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

Accepted Answer

In a Cox-Ross-Rubinstein binomial tree, the up factor (u) is determined by the volatility of the underlying asset, not the interest rate.

Question 7

A European option on a stock with a 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?
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

Accepted Answer

The two prices are exactly the same because they are based on treating dividends in the same way.

Question 8

The standard deviation of the values of an option calculated using 10,000 Monte Carlo trials is 4.5.The average of the values is 20.What is the standard error of this as an estimate of the option price?
A) 4.5
B) 0.45
C) 0.045
D) 0.0045

Accepted Answer

The standard deviation of 20 as an estimate of the price is the standard deviation of the calculated values divided by the square root of the number of trials.In this case the square root of the number of trials is 100 and so the standard error of the price estimate is 4.5/100 or 0.045.

Question 9

Which of the following cannot be estimated from a single binomial tree?
A) delta
B) gamma
C) theta
D) vega

Accepted Answer

To calculate vega it is necessary to increase volatility slightly and construct a new tree.The other three can be estimated from a single tree.

Question 10

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

Accepted Answer

The two delta estimates are (5−0)/(100-80)=0.25 and (20-5)/125-100)=0.6.The mid points of the ranges over which these are calculated are 90 and 112.5.The gamma estimate is therefore (0.6-0.25)/(112.5-90)= 0.156

Question 11

Which of the following describes the way that the parameters in a binomial tree are chosen?
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 of the time step in years
C) The tree recombines
D) All of the above

Accepted Answer

A,B,and C are all true.

Question 12

How many different paths are there through a Cox-Ross-Rubinstein tree with four-steps?
A) 5
B) 9
C) 12
D) 16

Accepted Answer

There are two choices for the path chosen at the initial node,two choices at the node reached at the end of the first time step,and so on.The total number of paths is 2⁴ = 16.

Question 13

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

Accepted Answer

To make volatility a function of time we make the lengths of time steps inversely proportional to variance.

Question 14

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

Accepted Answer

Both the F1F1F1S1F1F1F10t-period interest rate and the volatility can be made functions of time as described in the text.

Question 15

Which of the following is true?
A) The implicit finite difference method relates prices at one node to three prices at nodes at a later time
B) The implicit finite difference method relates prices at one node to three prices at nodes at an earlier time
C) The implicit finite difference method relates prices at one node to three prices at nodes at the same time
D) None of the above

Accepted Answer

The implicit finite difference relates prices at time t to three prices at nodes at time t−F1F1F1S1F1F1F10t

Question 16

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

Accepted Answer

u does not depend on interest rates but p does.The impact of the interest rate being a function of time is therefore to make p a function of time.

Question 17

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

Accepted Answer

Instead of using a degree of freedom to set u=1/d (as is done in CRR),the alternative method uses a degree of freedom to set both the up and down probabilities to 0.5.

Question 18

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?
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

Accepted Answer

We first subtract the present value of dividends from the initial stock price.We then draw the tree and then add back the present value of future dividends at each node

Question 19

Which of the following is true?
A) The implicit finite difference method is equivalent to using a trinomial tree
B) The explicit finite difference method is equivalent to using a trinomial tree
C) Both methods are equivalent to using a trinomial tree
D) Neither method is equivalent to using a trinomial tree

Accepted Answer

The explicit finite difference method is equivalent to using a trinomial tree.The implicit finite difference method is equivalent to using a multinomial tree.

Question 20

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?
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

Accepted Answer

The formula for u does not change but the formula for p does change.

Quiz 21: Basic Numerical Procedures

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?

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

Which of the following cannot be valued by Monte Carlo simulation

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

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

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

The standard deviation of the values of an option calculated using 10,000 Monte Carlo trials is 4.5.The average of the values is 20.What is the standard error of this as an estimate of the option price?

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

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

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

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

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

Which of the following is true?

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

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

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?

Which of the following is true?

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?