Question

The following tree depicts the price evolution of a stock. Each sub-period is nine months and the entire period is therefore 18 months. The annual interest rate (continuous compounding) is 3.5%. $18.95 $17.05 $13.35 $15.00 $12.50 $10.85 A. Use the simple binomial tree method (i.e., form a risk-neutral portfolio to find the optimal combination at each node as we did in class) to find the value of an American call with an exercise price of $12.5. B. Use the three ending prices and today's price to calculate the return volatility. The following hints may be useful to you. First, calculate the three percentage returns (e.g., 15/13.35 \text{-} 1 = 0.1235955). Then, annualize the return if necessary (e.g., 0.1235955 (12/18) = 0.082397). Finally, use the three annualized returns to calculate the standard deviation or volatility according to the standard statistical formula: $\sigma^2 = \frac{1}{n \text{-} 1} \sum_{i=1}^{n} (r_i - \bar{r})^2$ where $n$ is the number of observations (3 in our case) and $\bar{r}$ is the mean which is the simple average of the three returns in our case. Please keep six decimal places in calculations. C. Use the volatility obtained from Part B and the formulas for $u$, $d$ and $p$ to build a binomial tree (based on the current stock price of $13.35) and value the same option in Part A. Compare the two values.

          The following tree depicts the price evolution of a stock. Each sub-period is nine months
and the entire period is therefore 18 months. The annual interest rate (continuous
compounding) is 3.5%.
$18.95
$17.05
$13.35
$15.00
$12.50
$10.85
A.
Use the simple binomial tree method (i.e., form a risk-neutral portfolio to find the
optimal combination at each node as we did in class) to find the value of an
American call with an exercise price of $12.5.
B.
Use the three ending prices and today's price to calculate the return
volatility. The following hints may be useful to you. First, calculate the three
percentage returns (e.g., 15/13.35 \text{-} 1 = 0.1235955). Then, annualize the return if
necessary (e.g., 0.1235955 (12/18) = 0.082397). Finally, use the three annualized
returns to calculate the standard deviation or volatility according to the standard
statistical formula: $\sigma^2 = \frac{1}{n \text{-} 1} \sum_{i=1}^{n} (r_i - \bar{r})^2$
where $n$ is the number of observations
(3 in our case) and $\bar{r}$ is the mean which is the simple average of the three
returns in our case. Please keep six decimal places in calculations.
C.
Use the volatility obtained from Part B and the formulas for $u$, $d$ and $p$ to build a
binomial tree (based on the current stock price of $13.35) and value the same
option in Part A. Compare the two values.

The following tree depicts the price evolution of a stock. Each sub-period is nine months
and the entire period is therefore 18 months. The annual interest rate (continuous
compounding) is 3.5%.
18.9517.05
13.3515.00
12.5010.85
A.
Use the simple binomial tree method (i.e., form a risk-neutral portfolio to find the
optimal combination at each node as we did in class) to find the value of an
American call with an exercise price of 12.5.
B.
Use the three ending prices and today's price to calculate the return
volatility. The following hints may be useful to you. First, calculate the three
percentage returns (e.g., 15/13.35 - 1 = 0.1235955). Then, annualize the return if
necessary (e.g., 0.1235955 (12/18) = 0.082397). Finally, use the three annualized
returns to calculate the standard deviation or volatility according to the standard
statistical formula:σ^2 = (1)/(n - 1) ∑i=1^n (ri - r̅)^2wherenis the number of observations
(3 in our case) andr̅is the mean which is the simple average of the three
returns in our case. Please keep six decimal places in calculations.
C.
Use the volatility obtained from Part B and the formulas foru,dandpto build a
binomial tree (based on the current stock price of13.35) and value the same
option in Part A. Compare the two values.

Added by Joshua F.

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