Deck 18: Multiperiod Binomial Model

Question 1

Use the following data for an eight-period binomial model to answer the questions that follow.
- The stock's price S is $100.The stock price evolves according to an eight-period binomial model.
- Options mature after T = 1 year and have a strike price of K = $70.
- The continuously compounded risk-free interest rate r is 5 percent per year.
- The annualized volatility of stock price returns $\sigma$ = 0.25 or 25 percent per year.

-Today's price of a European call in this eight-period binomial model is:

A) $11.7628
B) $33.4139
C) $33.7858
D) $36.4527
E) None of these answers are correct.

Answer 1

$33.7858

Question 2

Use the following data for a two-period binomial model to answer the questions that follow.
- The stock's price S is $100.After three months,it either goes up and gets multiplied by the factor U =1.13847256,or it goes down and gets multiplied by the factor D = 0.88664332.
- Options mature after T = 0.5 year and have a strike price of K = $105.
- The continuously compounded risk-free interest rate r is 5 percent per year.
- Today's European call price is c and the put price is p.Call prices after one period are denoted by c_U in the up node and c_D in the down node.Call prices after two periods are denoted by c_UD in the "up,and then down node" and so on.Put prices are similarly defined.

-The stock price tree (in dollars)is given by:

A) S = 100;US = 113.8473 and DS = 88.6643;U ²S = 135.3238,UDS = 100,and D²S = 74.2672
B) S = 100;US = 113.8473 and DS = 88.6643;U ²S = 129.6120,UDS = 100.9419,and D²S = 78.6136
C) S F= 100;US = 113.8473 and DS = 88.6643;U ²S = 123.9862,UDS =101.5113,and D²S =83.1104
D) S = 100;US = 113.8473 and DS = 88.6643;U ²S =130.2617,UDS=101.4479,and D²S = 79.0077
E) None of these answers are correct.

Answer 2

S = 100;US = 113.8473 and DS = 88.6643;U ²S = 129.6120,UDS = 100.9419,and D²S = 78.6136

Question 3

Suppose a trader quotes a put price of $6.Then,you can make an immediate arbitrage profit of:

A) $2.41 by buying the synthetic put and selling the market-quoted put
B) $2.41 by selling the synthetic put and buying the market-quoted put
C) $7.66 by buying the synthetic put and selling the market-quoted put
D) $7.66 by selling the synthetic put and buying the market-quoted put
E) None of these answers are correct.

Answer 3

B

Question 4

To create the arbitrage-free synthetic put after one period (in the down state)you need to:

A) short sell 1 share of the stock and buy 103.70 units of the money market account
B) short sell 0.14 shares of the stock and buy 18.12 units of the money market account
C) buy 0.48 shares of the stock and short sell 42.2642 units of the money market account
D) buy 0.9343 shares of the stock and short sell 93.2677 units of the money market account
E) None of these answers are correct.

Question 5

Which of the following statements is correct regarding a binomial option pricing model?

A) In case of a dollar dividend,lower the call price by the dividend amount.
B) In case of a dollar dividend,lower the put price by the dividend amount.
C) In case of a dollar dividend paid (at an intermediate date)at the end of some time period,lower the stock price by the dividend amount.
D) In case of a dollar dividend,the binomial stock price tree recombines.
E) None of these answers are correct..

Question 6

Use the following data for an eight-period binomial model to answer the questions that follow.
- The stock's price S is $100.The stock price evolves according to an eight-period binomial model.
- Options mature after T = 1 year and have a strike price of K = $70.
- The continuously compounded risk-free interest rate r is 5 percent per year.
- The annualized volatility of stock price returns $\sigma$ = 0.25 or 25 percent per year.

-Today's price of a European put in this eight-period binomial model is:

A) $0.1107
B) $0.3719
C) $0.3910
D) $1.2798
E) None of these answers are correct.

Question 7

Use the following data for a two-period binomial model to answer the questions that follow.
- The stock's price S is $100.After three months,it either goes up and gets multiplied by the factor U = 1.13847256,or it goes down and gets multiplied by the factor
D = 0.88664332.
- Options mature after T = 0.5 year and have a strike price of K = $105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-If the stock pays a 1 percent dividend just before the end of the first three months,then today's price of a European call is:

A) $5.69
B) $5.73
C) $6.00
D) $7.96
E) None of these answers are correct.

Question 8

For a stock price following a binomial process,the up factor U = 1.1,the down factor D= 0.9,the dollar return (1 +R)= 1.05 percent (per period),and the initial stock price is 100.The probability that the stock will have 18 up movements and 2 down movements is:

A) 0.0556
B) 0.0669
C) 0.075
D) 0.10
E) None of these answers are correct.

Question 9

The arbitrage-free price of a put option is:

A) $2.00
B) $6.45
C) $8.41
D) $15.03
E) None of these answers are correct..

Question 10

To create the arbitrage-free synthetic call today,you need to:

A) buy 0.8585 shares of the stock and short sell 85.5777 units of the money market account
B) buy 0.8585 shares of the stock and short sell 42.2642 units of the money market account
C) buy 0.9343 shares of the stock and short sell 93.2677 units of the money market account
D) buy 0.4827 shares of the stock and short sell 42.2642 units of the money market account
E) None of these answers are correct.

Question 11

Which set of arbitrage-free put prices (in dollars)is correct?

A) p = 2.00,p_U = 0,and p_D = 4.06
B) p = 8.41,p_U = 2.00,and p_D = 15.03
C) p =15.03,p_U = 4.06,and p_D = 26.39
D) p = 8.41,p_U F=0,and p_D = 26.39
E) None of these answers are correct.

Question 12

Which of the following statements about the multiperiod binomial option pricing model is INCORRECT?

A) The pricing technique is described as forward induction.
B) The pricing technique is described as backward induction.
C) When the model's parameters are properly set and the model is run over many small intervals,then it gives an answer that approaches the Black-Scholes-Merton model value.
D) The binomial model can be used to price exotic options.
E) Strictly speaking,the binomial model is not a separate model but an approximation method.

Question 13

Suppose a trader quotes a call price of $4.50.Then,you can make an immediate arbitrage profit of:

A) $1.50 by buying the synthetic call and selling the market-quoted call
B) $1.50 by selling the synthetic call and buying the market-quoted call
C) $7.66 by buying the synthetic call and selling the market-quoted call
D) $7.66 by selling the synthetic call and buying the market-quoted call
E) None of these answers are correct.

Question 14

Use the following data for a two-period binomial model to answer the questions that follow.
- The stock's price S is $100.After three months,it either goes up and gets multiplied by the factor U = 1.13847256,or it goes down and gets multiplied by the factor
D = 0.88664332.
- Options mature after T = 0.5 year and have a strike price of K = $105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-If the stock pays a $1 dividend just before the end of the first three months,then today's price of a European put is:

A) $2.44
B) $5.73
C) $9.12
D) $16.03
E) None of these answers are correct.

Question 15

Which of the following statements about option pricing in a multiperiod framework using synthetic construction is INCORRECT?

A) The two assets,a stock and a money market account,dynamically complete the market by matching all possible option values at maturity.
B) The replicating portfolio is self-financing.
C) The replicating portfolio is arbitrage-free.
D) Option prices obtained for different period setups are different.
E) The pricing model estimates real-world probabilities and uses them for the expected payoff computations.

Question 16

To create the arbitrage-free synthetic put today,you need to:

A) buy 0.48 shares of the stock and short sell 42.2642 units of the money market account
B) short sell 1 share of the stock and buy 103.70 units of the money market account
C) short sell 0.14 shares of the stock and buy 18.12 units of the money market account
D) short sell 0.52 shares of the stock and buy 60.14 units of the money market account
E) None of these answers are correct.

Question 17

Use the following data for a two-period binomial model to answer the questions that follow.
- The stock's price S is $100.After three months,it either goes up and gets multiplied by the factor U =1.13847256,or it goes down and gets multiplied by the factor D = 0.88664332.
- Options mature after T = 0.5 year and have a strike price of K = $105.
- The continuously compounded risk-free interest rate r is 5 percent per year.
- Today's European call price is c and the put price is p.Call prices after one period are denoted by c_U in the up node and c_D in the down node.Call prices after two periods are denoted by c_UD in the "up,and then down node" and so on.Put prices are similarly defined.

-Call prices (in dollars)are given by:

A) c = 8.81,c_U=17.84,c_UU = 36.11,and zero at other nodes
B) c = 7.68,c_U =15.09,c_D = 0.47,c_UU =29.61,c_UD = 0.94,c_DD = 0
C) c =4.78,c_U = 9.69,c_UU = 19.61,and zero at other nodes
D) c = 6.00,c_U =12.16,c_UU =24.61,and zero at other nodes
E) None of these answers are correct.

Question 18

Use the following data for a two-period binomial model to answer the questions that follow.
- The stock's price S is $100.After three months,it either goes up and gets multiplied by the factor U = 1.13847256,or it goes down and gets multiplied by the factor
D = 0.88664332.
- Options mature after T = 0.5 year and have a strike price of K = $105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-If the stock pays a $1 dividend just before the end of the first three months,then today's price of a European call is:

A) $4.86
B) $5.73
C) $6.00
D) $11.59
E) None of these answers are correct.

Question 19

The formula for pricing options by repeated application of risk-neutral pricing is given by which of the following formulas,where $\pi$ = [(1 + R)- D]/(U - D);U and D are the up and down factors,respectively; (1 + R)is the dollar return;option_U is the option price at the next node in the up state;and option_D is the option price at the next node in the down state?

A) option price= [ $\pi$ * option _U + (1 - $\pi$ )* option_D] * (1 + R)
B) option price = [ $\pi$ * option _U + (1 - $\pi$ )* option_D] / R
C) option price = [ $\pi$ * option _U +(1 - $\pi$ )*option_D] / (1 +R)
D) option price = [ $\pi$ ² * option _U + (1 - $\pi$ )²*option_D] * (1 +R)
E) None of these answers are correct.

Question 20

To create the arbitrage-free synthetic call after one period (in the up state)you need to:

A) buy 0.8585 shares of the stock and short sell 85.5777 units of the money market account
B) buy 0.4827 shares of the stock and short sell 42.2642 units of the money market account
C) buy 0.8585 shares of the stock and short sell 42.2642 units of the money market account
D) buy 0.9343 shares of the stock and short sell 93.2677 units of the money market account
E) None of these answers are correct.

Question 21

Use the following data for a two-period binomial model to answer the questions that follow.
- The stock's price S is $100.After three months,it either goes up and gets multiplied by the factor U = 1.13847256,or it goes down and gets multiplied by the factor
D = 0.88664332.
- Options mature after T = 0.5 year and have a strike price of K = $105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-If the stock pays a 1 percent dividend just before the end of the first three months,then today's price of a European put is:

A) $8.41
B) $9.09
C) $9.12
D) $10.03
E) None of these answers are correct.

Question 22

Today's price of an American call option is:

A) $4.86
B) $5.73
C) $6.00
D) $11.59
E) None of these answers are correct.

Question 23

Consider the following exotic option whose payoff at maturity is given by the stock price squared less a strike price if it has a positive value,zero otherwise,that is: max[S(1)² - K,0].
Using the above data except for assuming a new strike price is $5,today's arbitrage-free price of this exotic option is:

A) $210.13
B) $438.85
C) $786.63
D) $888.60
E) None of these answers are correct.

Question 24

Which of the following statements is INCORRECT? A binomial model may be used for valuing:

A) European options in the presence of known dollar dividends
B) European options in the presence of a known dividend yield
C) an exotic option whose payoff depends on some function of the current stock price and the strike price
D) a European option in an incomplete market
E) an American option

Question 25

Today's price of an American put option is:

A) $8.41
B) $9.05
C) $9.12
D) $10.03
E) None of these answers are correct.

Question 26

Consider the following exotic option whose payoff at maturity is given by the square root of the stock price less the strike price if it has a positive value,zero otherwise,that is: max[ $\surd$ S(2)- K,0].
Using the above data except for assuming a new strike price is $5,today's arbitrage-free price of this exotic option is:

A) $0.21
B) $0.29
C) $0.36
D) $0.70
E) None of these answers are correct.