Deck 21: Basic Numerical Procedures

Question 1

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

Answer 1

A
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 2

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

Answer 2

C
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 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?

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

Answer 3

C
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 4

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

Question 5

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

Question 6

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

Question 7

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

Question 8

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

Question 9

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?

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

Question 10

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

Question 11

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

Question 12

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

Question 13

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

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

Question 14

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

Question 15

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

Question 16

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

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

Question 17

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

Question 18

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

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

Question 20

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