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CHAPTER 4: OPTION PRICING MODELS: THE BINOMIAL MODEL
MULTIPLE CHOICE TEST QUESTIONS
1. 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
2. In a binomial model, if the call price in the market is higher than the call price given by
the model, you should
a. sell the call and sell short the stock
b. buy the call and sell short the stock
c. buy the stock and sell the call
d. buy the call and buy the stock
e. none of the above
3. 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
4. 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
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
6. 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
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e. none of the above
7. 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
8. 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
9. 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
10. 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
11. 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.
12. What would be the call’s price if the stock goes up?
a. 3.60
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b. 8.00
c. 5.71
d. 4.39
e. none of the above
13. 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
14. What is the hedge ratio?
a. 0.429
b. 0.714
c. 0.571
d. 0.823
e. none of the above
15. 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.
16. 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
17. 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
18. What is the current value of the call?
a. 8.00
b. 7.30
c. 11.13
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d. 0.619
e. none of the above
19. 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
20. Suppose S = 70, X = 65, r = 0.05, p = 0.6, Cu = 7.17, Cd = 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
21. In a one-period binomial model with Su = 49.5, Sd = 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
22. 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
23. 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
24. 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
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b. 1.48
c. 1.40
d. 1.32
e. none of the above
25. 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
26. 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
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
28. 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
29. 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
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30. 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
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CHAPTER 4: OPTION PRICING MODELS: THE BINOMIAL MODEL
TRUE-FALSE TEST QUESTIONS
T F 1. The binomial model assumes that investors are risk neutral.
T F 2. The hedge ratio is the number of shares per call in a risk-free portfolio.
T F 3. In the binomial model, if a call is overpriced, investors should sell it and
buy stock.
T F 4. 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.
T F 5. A riskless hedge involving stock and puts requires a long position in stock
and a short position in puts.
T F 6. The up and down factors in the binomial model are analogous to the
volatility.
T F 7. In a recombining binomial model with n periods, the number of outcomes
is n + 1.
T F 8. When the hedge ratio is adjusted in the binomial model, the transactions
must be done in the option.
T F 9. The binomial probabilities are probabilities if investors were risk neutral.
T F 10. If there is one period remaining and no possibility of the option expiring
in-the-money, the hedge ratio will be zero.
T F 11. When pricing a put with the binomial model, the up and down
probabilities are reversed.
T F 12. 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.
T F 13. 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.
T F 14. Options that can be priced by considering only the payoffs at expiration
are called path-independent.
T F 15. Over a large number of periods, the up and down parameters move closer
to 1.5 and 0.5, respectively.
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T F 16. If the number of binomial periods is increased and u, d and r are not
adjusted, the value of a European call will increase.
T F 17. The binomial option pricing formula is based on the weighted average of
the next two possible values, discounted back to the present.
T F 18. 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.
T F 19. 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.
T F 20. In a multiperiod binomial model, an arbitrage profit cannot be earned until
the option expires.
T F 21. 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.
T F 22. 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.
T F 23. If the stock price adjusted for dividends at a continuous rate follows the up
and down parameters, the binomial tree will recombine.
T F 24. The binomial option pricing formula will conform to the European lower
bound.
T F 25. When calls are sold to adjust the hedge ratio, the funds must be placed in
additional shares.
T F 26. The binomial model for foreign currency options is similar to the binomial
model for stock options except the risk-free discount rate is adjusted.
T F 27. In a non-recombining binomial model with n periods, the number of
outcomes is 2n.
T F 28. 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.
T F 29. 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.
T F 30. Put-call parity holds within a two period binomial model.
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CHAPTER 4