Deck 13: Implementing the Binomial Model

Question 1

Suppose returns on a stock are lognormally distributed with expected (annualized)mean of of 0.10 and standard deviation of 0.20.What is the standard deviation of simple return on the stock for one month?

A)0.10
B)0.34
C)0.58
D)0.67

Answer 1

0.58

Question 2

Stock ABC is currently trading at 100.The stock has lognormal returns with with $\mu = 0$ and $\sigma = 0.40$ .What is the 95% confidence interval for the stock price in 3 months?

A) $( 21.6,178.4 )$
B) $( 67.6,148.0 )$
C) $( 45.7,219.0 )$
D)Cannot be calculated from the given information.

Answer 2

$( 67.6,148.0 )$

Question 3

Suppose returns on a stock are lognormally distributed with expected (annualized)mean of of 0.10 and standard deviation of 0.20.What is the standard deviation of the continuously compounded return on the stock for one month?

A)1.77%
B)3.33%
C)5.77%
D)7.33%

Answer 3

5.77%

Question 4

Suppose returns on a stock are lognormally distributed with expected (annualized)mean of of 0.10 and standard deviation of 0.20.What is the expected simple return on the stock for one month?

A)0.83
B)1.01
C)1.08
D)1.13

Question 5

Suppose the returns on a stock are lognormally distributed with $\mu = 0$ and $\sigma = 0.2$ .The expected three-month simple returns on the stock are

A)0.
B)0.25%
C)0.50%
D)1.01%

Question 6

If $\ln x$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$ ,then $x$ is

A)Normally distributed.
B)Lognormally distributed.
C)Exponentially distributed.
D)None of the above.

Question 7

Assume that a stock has lognormal returns with mean $\mu = 0.10$ and standard deviation $\sigma = 0.20$ .The current stock price is $50.What is a 95% confidence interval for the stock price in six months?

A)37.90,65.97
B)37.81,73.08
C)39.84,69.35
D)40.12,60.24

Question 8

Let $S _ { T }$ denote the time- $T$ price of a stock and $S _ { 0 }$ its current price.Suppose that for any $T$ , $\ln \left( \frac { S _ { T } } { S _ { 0 } } \right) : N \left( \mu _ { T } , \sigma ^ { 2 } T \right)$ for constant annual parameters $\mu$ and $\sigma$ .What does this imply about the returns process? Pick the most accurate of the following alternatives:

A)The returns are independent and identically distributed over time.
B)The returns are independent over time.
C)The returns are normally distributed.
D)None of the above.

Question 9

Which of the following statements is most valid for the recursive programming of a binomial tree for pricing options?

A)The recursive program requires more lines of code than a non-recursive loop-driven program.
B)The recursive program requires less computer memory than a non-recursive loop-driven program.
C)The recursive program runs slower than a non-recursive loop-driven program.
D)The recursive program runs in polynomial time whereas a non-recursive loop-driven program runs in exponential time.

Question 10

If $x$ is normally distributed with mean $\mu$ and variance $\sigma ^ { 2 }$ ,then $y = e ^ { x }$ is

A)Normally distributed.
B)Lognormally distributed.
C)Exponentially distributed.
D)None of the above.

Question 11

Consider a binomial tree in which the stock moves up by a factor $u$ and down by a factor $d$ ,respectively with probabilities $p$ and $1 - p$ .The variance of log-returns per time step is given by the following formula:

A) $p ( 1 - p ) \left[ \ln ( u / d ] ^ { 2 } \right.$
B) $p ( \ln u ) ^ { 2 } + ( 1 - p ) ( \ln d ) ^ { 2 }$
C) $[ p \ln u + ( 1 - p ) \ln d ] ^ { 2 }$
D) $p ( 1 - p ) ( u / d ) ^ { 2 }$

Question 12

In the Cox-Ross-Rubinstein (CRR)binomial model,the volatility is given as $\sigma = 0.2$ .The risk-free rate of interest is 2%.What is the risk-neutral probability of an up move on a binomial tree with a time step of one month?

A)0.45
B)0.50
C)0.55
D)0.60

Question 13

As the number of steps in the CRR binomial tree increases (keeping maturity fixed),the solution "converges" to a limit result.Which of the following statements characterizes this convergence best?

A)The solution results in the Black-Scholes formula.
B)The convergence may be oscillatory for even and odd number of steps in the tree.
C)The convergence may be monotonic for even and odd number of steps in the tree.
D)All of the above.

Question 14

Suppose you are modeling the price evolution of a stock on a tree using a general version of the CRR model.The stock price is stochastic (lognormal),but the rate of interest each time step may not be the same,and the time step itself may be different across periods.The following is sufficient for a binomial tree representation of the stock price process to be recombining:

A)The volatility of the stock is constant each period,and the time step and interest rate are different each period.
B)The volatility of the stock is constant each period,the time step on the tree is the same each period,and the interest rate may be different each period.
C)The volatility of the stock is constant,the time step on the tree is the different each period,and the up and down probabilities are equal.
D)The volatility of the stock is different each period,the time step on the tree is the same each period,and the interest rate is the same each period.

Question 15

In the Jarrow-Rudd (JR)binomial model,the volatility is given as $\sigma = 0.2$ .The risk-free rate of interest is 2%.What is the risk-neutral probability of an up move on a binomial tree with a time step of one month?

A)0.45
B)0.50
C)0.55
D)0.60

Question 16

Suppose returns on a stock are lognormally distributed with expected (annualized)mean of of 0.10 and standard deviation of 0.20.What is the expected continuously compounded return on the stock for one month?

A)0.100%
B)0.333%
C)0.833%
D)1.667%