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?

Accepted Answer

C

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?

Accepted Answer

The 95% confidence interval for the stock price in 3 months is given by:$$100 * e^{0.40 * \sqrt{3} * (-1.96, 1.96)} = (67.6, 148.0)$$

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 expected continuously compounded return on the stock for one month?

Accepted Answer

The expected continuously compounded return for one month can be calculated by dividing the annual expected return by 12 (since there are 12 months in a year). Given an annual expected return (mean) of 0.10 (or 10%), the monthly expected return would be 0.10/12 = 0.00833 or 0.833%.

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 standard deviation of the continuously compounded return on the stock for one month?

Accepted Answer

The answer of Suppose returns on a stock are lognormally...

Question 5

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

Accepted Answer

When the logarithm of a variable (in this case, $\ln x$) is normally distributed, the variable itself (in this case, $x$) is said to be lognormally distributed. This is the definition of a lognormal distribution.

Question 6

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?

Accepted Answer

In the Jarrow-Rudd (JR) binomial model, the risk-neutral probability $p$ is given by the formula $p = \frac{1}{2} + \frac{1}{2} \cdot \frac{r - \frac{\sigma^2}{2}}{\sigma \sqrt{\Delta t}}$, where $r$ is the risk-free rate, $\sigma$ is the volatility, and $\Delta t$ is the time step. Given $\sigma = 0.2$, $r = 0.02$, and $\Delta t = \frac{1}{12}$ (for one month), substituting these values into the formula simplifies to $p = 0.5$ because the term $\frac{r - \frac{\sigma^2}{2}}{\sigma \sqrt{\Delta t}}$ becomes 0 when $r = 0.02$ and $\sigma = 0.2$, leading to $p = \frac{1}{2} + \frac{1}{2} \cdot 0 = 0.5$.

Question 7

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?

Accepted Answer

The expected monthly return can be found using the formula:
E(monthly return) = e^((E(annual return) - 0.5*SD^2)/12) - 1
= e^((0.10 - 0.5*0.20^2)/12) - 1
= e^(0.08333) - 1
= 0.0855 or 8.55%
So, the expected simple return on the stock for one month is 1 + 0.0855 = 1.0855 or 8.55%. Rounded to two decimal places, this is 1.01.

Question 8

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:

Accepted Answer

For a binomial tree representation to be recombining, the time step on the tree must be the same each period and the volatility of the stock must be constant each period. The interest rate may be different each period without affecting the recombining property of the tree. Therefore, Choice B is the best answer.

Question 9

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

Accepted Answer

Recursive programming can often lead to slower performance due to the extra overhead involved in calling and returning from functions. However, the other statements are not necessarily true. A recursive program may require less lines of code and/or less computer memory depending on the implementation. The binomial tree method itself involves exponential complexity, so even a non-recursive loop-driven program would likely run in exponential time.

Question 10

While stock returns are commonly modeled as lognormal, bond returns are less ideally modeled as lognormal because

Accepted Answer

Bond returns are less ideally modeled as lognormal because both bond return volatility may not always be increasing in maturity, and bond yields can be negative, which is not well-represented by a lognormal distribution that only takes positive values.

Question 11

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:

Accepted Answer

The given equation implies that the returns are normally distributed with mean $\mu T$ and variance $\sigma^2 T$. This means that the returns are independent and identically distributed over time, as the mean and variance are constant over time.

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?

Accepted Answer

The answer of In the Cox-Ross-Rubinstein (CRR) binomial model, the...

Question 13

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?

Accepted Answer

The answer of Assume that a stock has lognormal returns...

Question 14

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

Accepted Answer

When a variable $x$ is normally distributed, the transformation $y = e^x$ results in $y$ being lognormally distributed. This is because the logarithm of $y$, which is $x$, follows a normal distribution.

Question 15

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?

Accepted Answer

The CRR binomial tree model, as the number of steps increases, indeed converges to the Black-Scholes formula, which is a continuous model for option pricing. This convergence can be oscillatory for even and odd numbers of steps, meaning the solution can slightly overshoot and undershoot as the number of steps changes. This behavior illustrates that the convergence is not strictly monotonic but can still lead to the same limit result, the Black-Scholes formula, demonstrating the robustness of the CRR model in approximating continuous processes.

Question 16

Suppose that the stock price is stochastic (say, lognormal) with constant volatility, and that the interest rate, though not stochastic changes from period to period on the tree. Which of the following statements is valid?

Accepted Answer

A CRR binomial tree assumes a constant interest rate, but allows for a stochastic stock price with constant volatility. The tree is also recombining, meaning that the same set of possible future outcomes can be reached by different paths. The JR binomial tree, on the other hand, assumes a constant interest rate and a stochastic stock price with variable volatility. Therefore, option B is not valid.

Question 17

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

Accepted Answer

The answer of Suppose the returns on a stock are...

Question 18

Consider a binomial tree in which the stock moves up by a factor $2 i$ 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:

Accepted Answer

The variance of log-returns in a binomial model is calculated as the expected value of the squared deviation from the mean log-return. The correct formula is $p ( 1 - p ) [ \ln ( u d ) ] ^ { 2 }$, which accounts for the probability-weighted variance between the two possible log-returns, capturing the spread in outcomes due to the up and down movements.

Quiz 13: Implementing the Binomial Model

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?

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?

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?

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?

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

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?

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?

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

While stock returns are commonly modeled as lognormal, bond returns are less ideally modeled as lognormal because

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?

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?

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

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?

Suppose that the stock price is stochastic (say, lognormal) with constant volatility, and that the interest rate, though not stochastic changes from period to period on the tree. Which of the following statements is valid?

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

Consider a binomial tree in which the stock moves up by a factor $2 i$ 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:

Quiz 13: Implementing the Binomial Model

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?

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

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?

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?

If ln⁡x\ln xlnx is normally distributed with mean μ\muμ and variance σ2\sigma ^ { 2 }σ2 , then xxx is

In the Jarrow-Rudd (JR) binomial model, the volatility is given as σ=0.2\sigma = 0.2σ=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?

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?

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

While stock returns are commonly modeled as lognormal, bond returns are less ideally modeled as lognormal because

In the Cox-Ross-Rubinstein (CRR) binomial model, the volatility is given as σ=0.2\sigma = 0.2σ=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?

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

If xxx is normally distributed with mean μ\muμ and variance σ2\sigma ^ { 2 }σ2 , then y=exy = e ^ { x }y=ex is

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?

Suppose that the stock price is stochastic (say, lognormal) with constant volatility, and that the interest rate, though not stochastic changes from period to period on the tree. Which of the following statements is valid?

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

Consider a binomial tree in which the stock moves up by a factor 2i2 i2i and down by a factor ddd , respectively with probabilities ppp and 1−p1 - p1−p . The variance of log-returns per time step is given by the following formula:

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?

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

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?

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?

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?

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

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

Consider a binomial tree in which the stock moves up by a factor $2 i$ 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: