Deck 4: Option Pricing Models: The Binomial Model

Question 1

The values of u and d are which of the following?

A) the return on the stock if it goes up and down, respectively
B) the inverse of the ratio of the up and down probabilities, respectively, and the risk-free rate
C) the normal probabilities of up and down movements, respectively
D) one plus the return on the stock if it goes up and down, respectively
E) none of the above

Answer 1

D

Question 2

In the binomial model, if an option has no chance of expiring out-of-the-money, the hedge ratio will be

A) 0.5
B) infinite
C) 1
D) 0
E) none of the above

Answer 2

C

Question 3

A portfolio that combines the underlying stock and a short position in an option is called

A) a risk arbitrage portfolio
B) a hedge portfolio
C) a ratio portfolio
D) a two-state portfolio
E) none of the above

Answer 3

B

Question 4

What is the current value of the call?

A) 8.00
B) 7.30
C) 11.13
D) 0.619
E) none of the above

Question 5

If the stock pays a specific dollar dividend and the stock price, to include the dividend, follows the binomial up and down factors, which of the following will happen?

A) the binomial tree will recombine
B) the binomial tree will not recombine
C) the option will be mispriced
D) an arbitrage profit will not be possible
E) none of the above

Question 6

When the number of time periods in a binomial model is large, what happens to the binomial probability of an up move?

A) it approaches 1.0
B) it approaches zero
C) it fluctuates without pattern
D) it converges to 0.5
E) none of the above
Consider a binomial world in which the current stock price of 80 can either go up by 10 percent or down by 8 percent. The risk-free rate is 4 percent. Assume a one-period world. Answer questions 12 through 15 about a call with an exercise price of 80.

Question 7

When puts are priced with the binomial model, which of the following is true?

A) the puts must be American
B) the puts cannot be properly hedged
C) the puts will violate put-call parity
D) the hedge ratio is one throughout the tree
E) none of the above

Question 8

Suppose S = 70, X = 65, r = 0.05, p = 0.6, C_u = 7.17, C_d = 1.22 and there is one period left in an American call's life. What will the option be worth?

A) 6.83
B) 0.00
C) 4.56
D) 5.00
E) none of the above

Question 9

What is the hedge ratio?

A) 0.429
B) 0.714
C) 0.571
D) 0.823
E) none of the above

Question 10

Which of the following are not path-dependent options when the stock pays a constant dividend yield?

A) European calls and European puts
B) European calls and American puts
C) American puts and European puts
D) American puts and European calls
E) none of the above

Question 11

What is the theoretical value of the call?

A) 8.00
B) 4.39
C) 5.15
D) 5.36
E) none of the above
Now extend the one-period binomial model to a two-period world. Answer questions 16 through 18.

Question 12

In a non-recombining tree, the number of paths that will occur after three periods is

A) three
B) four
C) ten
D) eight
E) six

Question 13

In a two-period binomial world, a mispriced call will lead to an arbitrage profit if

A) the proper hedge ratio is maintained over the two periods
B) the hedge portfolio is terminated after one period
C) the option goes from over- to underpriced or vice versa
D) the option remains mispriced over both periods
E) none of the above

Question 14

When the number of time periods in a binomial model is large, a European call option value does what?

A) fluctuates around its intrinsic value
B) converges to a specific value
C) increases without limit
D) converges to the European lower bound
E) none of the above

Question 15

What would be the call's price if the stock goes up?

A) 3.60
B) 8.00
C) 5.71
D) 4.39
E) none of the above

Question 16

If the binomial model is extended to multiple periods for a fixed option life, which of the following adjustments must be made?

A) the up and down factors must be increased
B) the risk-free rate must be increased
C) the up and down factors and the risk-free rate must be decreased
D) the initial stock price must be proportionately reduced
E) none of the above

Question 17

In a binomial model, if the call price in the market is higher than the call price given by the model, you should
A) buy the stock and sell the call

A) sell the call and sell short the stock
B) buy the call and buy the stock
B) buy the call and sell short the stock
C) none of the above

Question 18

What would be the call's price if the stock goes down?

A) 8.00
B) 3.60
C) 0.00
D) 9.00
E) none of the above

Question 19

What is the value of the call if the stock goes up, then down?

A) 0.96
B) 16.80
C) 8.00
D) 0.00
E) none of the above

Question 20

What is the hedge ratio if the stock goes down one period?

A) 0.00
B) 0.0725
C) 1.00
D) 0.73
E) none of the above

Question 21

In a one-period binomial model with S_u = 49.5, S_d = 40.5, p = 0.8, r = 0.06, S = 45 and X = 50, what is a European put worth?

A) 2.17
B) 0.50
C) 9.50
D) 5.00
E) none of the above

Question 22

In a recombining binomial model with n periods, the number of outcomes is n + 1.

Question 23

Determine the value of d for a four period binomial model when the option's life is one-fourth of a year and the volatility is 0.64. Use the model for u and d that does not require the risk-free rate.

A) 0.85
B) 1.17
C) 2.56
D) 0.90
E) none of the above

Question 24

All of the following are variables used to determine a call option's price except

A) the risk-free rate
B) the probability of stock price movement
C) the exercise price
D) the possible future stock prices at expiration
E) none of the above

Question 25

The binomial probabilities are probabilities if investors were risk neutral.

Question 26

When the hedge ratio is adjusted in the binomial model, the transactions must be done in the option.

Question 27

Pricing a put with the binomial model is the same procedure as pricing with a call, except that the

A) underlying stock must not pay dividends
B) binomial model cannot account for expiration payoffs
C) value of the underlying must be discounted back to the current time period
D) expiration payoffs reflect the fact that the option is the right to sell the underlying stock
E) none of the above

Question 28

The hedge ratio is the number of shares per call in a risk-free portfolio.

Question 29

The binomial model assumes that investors are risk neutral.

Question 30

If there is one period remaining and no possibility of the option expiring in-the-money, the hedge ratio will be zero.

Question 31

The binomial option pricing model will converge to what value as the number of periods increases?

A) a random value
B) the Black-Scholes-Merton value of the option
C) the intrinsic volatility of the option
D) the true value of the underlying
E) none of the above

Question 32

The up and down factors in the binomial model are analogous to the volatility.

Question 33

Which of the following statements about the binomial model is incorrect?

A) it converges to the Black-Scholes-Merton model
B) it can accommodate early exercise
C) it allows only two stock prices at expiration
D) it can be extended to a large number of time periods
E) none of the above

Question 34

Determine the value of u for a three period binomial problem when the option's life is one-half a year and the volatility is 0.48. Use the model for u that does not require the risk-free rate.

A) 1.22
B) 1.48
C) 1.40
D) 1.32
E) none of the above

Question 35

A riskless hedge involving stock and puts requires a long position in stock and a short position in puts.

Question 36

In the binomial model, if a call is overpriced, investors should sell it and buy stock.

Question 37

All of the following are practical applications of the binomial model except

A) choices regarding real options
B) options regarding executive incentive plans
C) models in which the stock price can go up, down, or remain constant in the next period
D) embedded options within debt securities
E) none of the above

Question 38

Which of the following statements about the binomial option pricing model is not always true?

A) it can capture the effect of early exercise
B) it can accommodate a large number of possible stock prices at expiration
C) it reflects the effects of the stock price, exercise price, risk-free rate, volatility and time to expiration
D) it gives the price at which the option will trade in the market.
E) none of the above

Question 39

One way to model an option with dividends in the binomial framework is for the stock price minus the present value of the dividends to grow by the up and down factors.

Question 40

A stock priced at 50 can go up or down by 10 percent over two periods. The risk-free rate is 4 percent. Which of the following is the correct price of an American put with an exercise price of 55?

A) 7.88
B) 3.38
C) 4.00
D) 5.00
E) 1.65

Question 41

Put-call parity holds within a two period binomial model.

Question 42

Options that can be priced by considering only the payoffs at expiration are called path-independent.

Question 43

If a call is underpriced and you buy the call and sell short the stock, it is equivalent to investing money at more than the risk-free rate.

Question 44

The single period binomial hedge ratio for stock call options could be computed by equating the two future cash flows -- from a portfolio of long h shares of stock and short one call -- and solve for the number of underlying stocks to hold.

Question 45

The formula for a hedge ratio of a put is the same as that of the call, except that put prices are used instead of call prices.

Question 46

In a non-recombining binomial model with n periods, the number of outcomes is 2n.

Question 47

When calls are sold to adjust the hedge ratio, the funds must be placed in additional shares.

Question 48

In a multiperiod binomial model, an arbitrage profit cannot be earned until the option expires.

Question 49

If the number of binomial periods is increased and u, d and r are not adjusted, the value of a European call will increase.

Question 50

When pricing a put with the binomial model, the up and down probabilities are reversed.

Question 51

The binomial model will give a higher price for an American call on a stock that pays no dividends than if that call is European.

Question 52

Over a large number of periods, the up and down parameters move closer to 1.5 and 0.5, respectively.

Question 53

The binomial option pricing formula is based on the weighted average of the next two possible values, discounted back to the present.

Question 54

If the stock price adjusted for dividends at a continuous rate follows the up and down parameters, the binomial tree will recombine.

Question 55

The binomial model for foreign currency options is similar to the binomial model for stock options except the risk-free discount rate is adjusted.

Question 56

If the binomial model is used with a specific dollar dividend and the stock price follows the up and down parameters, the tree will explode and end up with far more outcomes than time periods.

Question 57

If a call is overpriced and you buy the call and sell short the stock, it is equivalent to investing money at less than the risk-free rate.

Question 58

When pricing an American put with the binomial model, you must check for early exercise at each time point and stock price except the current one.

Question 59

The binomial option pricing formula will conform to the European lower bound.

Question 60

If the binomial model describes the real world, the combined actions of all investors will cause the market price to converge to the binomial price.