Deck 12: Binomial Option Pricing

Question 1

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the price of a 100-strike,three-month American put option?

A)$7.20
B)$7.24
C)$7.70
D)$7.74

Answer 1

$7.24

Question 2

Consider a stock index currently trading at 128.You have modeled the evolution of this index in a binomial tree and have come up with the following parameters: $u = 1.08$ , $d = 0.93$ .The gross risk-free rate per step of the binomial tree is $R = 1.03$ and the dividend yield on the index is $\delta = 0.01$ .What is the price of a one-period put option with a strike of $K = 130$ ?

A)7.93
B)7.09
C)7.02
D)6.36

Answer 2

6.36

Question 3

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the price of a 100-strike,three-month European call option?

A)$7.10
B)$7.30
C)$7.50
D)$7.70

Answer 3

$7.70

Question 4

Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } > 1$ (with probability $p$ )or down by $d < 1$ (with probability $1 - p$ ).The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns $R = \$ 1$ at the end of the period. Let $Q$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money when there are no dividends;and let $Q ^ { D }$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money when there is a dividend of size $D > 0$ between the first and second periods.Which of the following is most accurate?

A) $Q = Q ^ { D }$
Always.
B) $Q \leq Q ^ { D }$
Always.
C) $Q \geq Q ^ { D }$
Always.
D)Depending on the parameters $\mathcal { U }$
,
$d$
,and
$D$
,both
$Q > Q ^ { D }$
And
$Q \leq Q ^ { D }$
Are possible.

Question 5

Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } > 1$ (with probability $p$ )or down by $d < 1$ (with probability $1 - p$ ).The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting,the risk-neutral probability of an at-the-money two-period put finishing in-the-money is _____________ as that of a one-period at-the-money put finishing in-the-money.

A)At least as high.
B)At most as high.
C)Equal to.
D)More than one of the above is possible depending on the parameters.

Question 6

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the delta of a 100-strike,three-month European put option?

A) $- 0.4667$
B) $- 0.5000$
C) $- 0.5333$
D) $- 0.5667$

Question 7

In order for a binomial tree to recombine,the following condition is necessary and sufficient:

A)The time-step in the tree must remain constant over time.
B)The up shift factor $\mathcal { U }$
And down shift factor
$d$
Must be such that
$u \times d = 1$
)
C)The up shift factor $\mathcal { U }$
And down shift factor
$d$
Must be such that
$u \times d =$
A constant.
D)The rate of interest must be zero.

Question 8

A binomial tree setting has an up-move of $u > 1$ (with probability $p = 0.75$ )and a down move of $d < 1$ (with probability $1 - p = 0.25$ ),with $u - 1 = 1 - d$ .The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns $R = \$ 1$ at the end of the period.The risk-neutral probability of an up-move in this setting

A)Is $> \frac { 1 } { 2 }$
)
B)Is $< \frac { 1 } { 2 }$
)
C)Is equal to $\frac { 1 } { 2 }$
)
D)Is equal to $p$
)

Question 9

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the delta of a 100-strike,three-month European call option?

A)0.4667
B)0.5000
C)0.5333
D)0.5667

Question 10

Suppose you were replicating a two-period put option in a binomial tree.Then,the replicating strategy involves an initial short position in the underlying stock that

A)is held unchanged to maturity.
B)is increased in size (i.e. ,becomes more short)after one period.
C)is increased in size (i.e. ,becomes more short)if the price goes down in the first period and decreased in size (becomes less short)if the price goes up in the first period.
D)is decreased in size (i.e. ,becomes less short)if the price goes down in the first period and increased in size (becomes more short)if the price goes up in the first period.

Question 11

Consider a stock index currently trading at 128.You have modeled the evolution of this index in a binomial tree and have come up with the following parameters: $u = 1.08$ , $d = 0.93$ .The gross risk-free rate per step of the binomial tree is $R = 1.03$ and the dividend yield on the index is $\delta = 0.01$ .The risk-neutral probability of an up-move in this setting is

A)0.598
B)0.660
C)0.667
D)0.736

Question 12

Consider a binomial tree setting in which in each period the price goes up by $\mathcal { U } > 1$ (with probability $p$ )or down by $d < 1$ (with probability $1 - p$ ).The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting,let $Q _ { 1 }$ be the risk-neutral probability of a one-period at-the-money call finishing in-the-money.and let $Q _ { 2 }$ be the risk-neutral probability of a two-period at-the-money call finishing in-the-money.Which of the following is true?

A) $Q _ { 1 } > q _ { 2 }$
)
B) $Q _ { 1 } < Q _ { 2 }$
)
C) $Q _ { 1 } = Q _ { 2 }$
)
D)Depending on the parameters $\mathcal { U }$
And
$d$
,more than one of the above is possible.

Question 13

Schroder's (1988)approach to binomial option pricing offers a way of

A)Obtaining recombining trees by restating cash dividends as dividend yields.
B)Obtaining recombining binomial trees even when there are cash dividends.
C)Obtaining recombining trees when dividends are stated as yields but not when they are stated as cash amounts.
D)Pricing options efficiently using non-recombining binomial trees.

Question 14

Consider a binomial tree setting in which in each period the price goes up by $u = 1.10$ (with probability $p = 0.60$ )or down by $d = 0.90$ (with probability $1 - p = 0.40$ ).The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting,the risk-neutral probability of a two-period call with strike $K = 95$ finishing in-the-money is

A)0.25
B)0.36.
C)0.75
D)0.84

Question 15

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.05$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the price of a 100-strike,two-month European put option?

A)$2.36
B)$3.36
C)$4.36
D)$5.36

Question 16

Consider a two-period binomial tree setting in which the up and down moves are given,respectively,by $\mathcal { U } > 1$ and $d < 1$ .The risk-free gross rate of interest per time step is $R$ .Suppose there is a dividend $D > 0$ paid after the first period of the tree. Let $C _ { 1 }$ denote the value of a one-period call that is initially at-the-money.Let $C _ { 2 } ^ { A } , C _ { 2 } ^ { E }$ denote the values of two-period American and European calls,respectively,that too are initially at-the-money.
Which of the following statements is always true regardless of the specific parameter values?

A) $C _ { 2 } ^ { A } > C _ { 2 } ^ { F } > C _ { 1 }$
)
B) $C _ { 2 } ^ { S } \geq C _ { 1 }$
)
C) $C _ { 2 } ^ { A } \geq C _ { 1 }$
)
D) $C _ { 2 } ^ { A } > C _ { 1 } > C _ { 2 } ^ { F }$
)

Question 17

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is(a)the price of a 100-strike,three-month European put option,and(b)the price of a 100-strike,two-month European put option?

A)$7.20 and $5.41,respectively.
B)$7.20 and $5.08,respectively.
C)$5.08 and $7.70,respectively.

Question 18

A binomial tree setting has an up-move of $u > 1$ (with probability $p = 0.75$ )and a down move of $d < 1$ (with probability $1 - p = 0.25$ ),with $u d = 1$ .The risk-free interest rate per time step is zero,so a dollar invested at the beginning of the period returns a dollar at the end of the period. In this setting,the risk-neutral probability of an up-move

A)Is $> \frac { 1 } { 2 }$
)
B)Is $< \frac { 1 } { 2 }$
)
C)Is equal to $\frac { 1 } { 2 }$
)
D)Is equal to $p$
)

Question 19

A stock is currently trading at $100.In each period of a binomial tree,the stock will either increase in price by a factor of $u = 1.05$ or fall by a factor of $d = 0.90$ .The risk-free rate per period of the binomial tree is 0.1668%,i.e. ,an investment of a dollar at the risk-free rate returns $1.001668 after one period.What is the risk-neutral probability of the stock finishing in the money on a 100-strike,two-month call option?

A)0.46
B)0.50
C)0.89
D)0.90

Question 20

Which of the following statements is valid?

A)It is not possible to build a recombining binomial tree if volatility is changing over time.
B)A recombining binomial tree with $n$
Periods has
$n$
Nodes at the end of the last period on the tree.
C)A non-recombining binomial tree with $n$
Periods has
$n ^ { 2 }$
Nodes at the end of the last period on the tree.
D)None of the above.

Question 21

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the early-exercise premium of a 100-strike,six-month American call option when a dividend of $1 is paid at the end of each month? (Assume that if the option is exercised,it is done just before the dividends are paid. )

A)$0.00
B)$0.25
C)$0.30
D)$0.35

Question 22

A stock is currently trading at $50.In each month,the stock will either increase in price by a factor of $u = 1.2$ or fall by a factor of $d = 0.9$ .The risk-free rate of interest is 0.0834% per month in simple terms,i.e. ,an investment of $1 today returns $1.00834 after one period.Consider a 52-strike,three-month European put option when a dividend of $0.5 is paid at the end of each month.Assume that the company just announces a cancellation of future dividends.Ceteris paribus,by how much does the option price change on this announcement? (Assume that if the option is exercised,it is done before dividends are paid. )

A)The option price drops by $0.39
B)The option price drops by $0.69
C)The option price rises by $0.39
D)The option price rises by $0.69

Question 23

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.If there are no dividends,what is the early-exercise premium of a 100-strike,six-month American call option?

A)$0
B)$0.15
C)$0.25
D)$0.50

Question 24

The current price of a stock is $50.Every month,the stock price will rise by a factor of 1.2 or fall by a factor of 0.8;and a $1 risk-free investment will be worth $1.03.A dividend as a proportion 0.01 of the stock price is paid each period.What is the price of a six-month American put option at a strike of $51?

A)$7.51
B)$7.91
C)$8.11
D)$8.51

Question 25

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the price of a 100-strike,six-month European call option when a dividend of $1 is paid at the end of each month? (Assume that if the option is exercised,it is done just before a dividend payment. )

A)$7.20
B)$7.50
C)$7.70
D)$8.20

Question 26

The current price of a stock is $50.Every month,the stock price will rise by a factor of 1.2 or fall by a factor of 0.8;and a $1 risk-free investment will be worth $1.03.Dividends are paid as a proportion of 0.02 of the stock price.What can you say about the the 49-strike,European and American six-month calls if the risk-less investment grows to $1.05 (instead of $1.03)each period,and the dividend rate increases to 0.04?

A)The European call rises in value and the American call rises in value.
B)The European call rises in value and the American call falls in value.
C)The European call falls in value and the American call rises in value.
D)The European call falls in value and the American call falls in value.

Question 27

The current price of a stock is $50.Every month,the stock price will rise by a factor of 1.2 or fall by a factor of 0.8;and a $1 risk-free investment will be worth $1.03.Dividends are paid as a proportion of the stock price.Which of the following $51-strike,six-month options has the highest value?

A)An American put when the stock pays no dividends.
B)A European put when the stock pays dividends at proportion 0.01 of the stock price.
C)An American call when the stock pays no dividends.
D)A European call when the stock pays dividends at proportion 0.01 of the stock price.

Question 28

For a call option on a stock that pays dividends,which of the following statements is valid?

A)The European call is worth the same as the American call.
B)The European call always decreases in value as dividends increase,but the American call always increases in value as dividends increase.
C)The American call is worth at least as much as the European call.
D)The American call is less likely to be exercised as dividends increase.

Question 29

A stock is currently trading at $100.In each month,the stock will either increase in price by a factor of $u = 1.10$ or fall by a factor of $d = 0.90$ .The risk-free rate of interest per month is 0.1668% in simple terms,i.e. ,an investment of $1 at the risk-free rate returns $1.001668 after one month.What is the early-exercise premium of a 100-strike,six-month American put option?

A)$0.15
B)$0.17
C)$0.89
D)$1.12

Question 30

The current price of a stock is $50.Every month,the stock price will rise by a factor of 1.2 or fall by a factor of 0.8;and a $1 risk-free investment will be worth $1.03.Dividends are paid as a proportion of 0.02 of the stock price.What is the early-exercise premium on the 49-strike,six-month American call?

A)$0.00
B)$0.02
C)$0.06
D)$0.11

Question 31

The current price of a stock is $50.This includes the value of a dividend of $2 that will be paid three months from now.In three months,the net of dividend stock price will rise by a factor of 1.2 or fall by a factor of 0.8.The risk-free rate of interest is 0.0834% per month (simple interest).Using the technique of Schroder (1988),what is the price of a six-month American call option at a strike of $51?

A)$3.62
B)$4.12
C)$4.62
D)$5.12