Chapter 21, Problem 1PS

a)

Summary Introduction

To determine: Value of one month call option with $40 as an exercise price.

Explanation of Solution

Given information:

Company H stock prices changes once in a month either by increase in 20% or decreases by 16.7%.

Current price level is $40 and interest rate is 1% per month.

Calculation of value of option:

First, it is necessary to find out the probabilities by using risk-neutral method.

    $\begin{array}{c}1=\left(\text{p}\times 20\right)+\left(1-\text{p}\right)\left(-16.7\right)\\ \text{p}=0.4823\\ =48\%\end{array}$

Therefore, the value of p is 48% and,

  $\begin{array}{c}\text{1}-\text{p}=1-48\%\\ =52\%\end{array}$

Therefore, there is 48% of chances that price of stock will rise by 20% and a 52% chances that option will worth of $0 when it matures.

$\begin{array}{c}\text{Value}\text{of}\text{call}\left(\text{C}\right)=\frac{\left[\left(0.48\times \$8\right)+\left(0.52\times \$0\right)\right]}{1.01}\\ =\$3.82\end{array}$

Therefore, the value of call is $3.82

b)

Summary Introduction

To determine: Value of delta.

Explanation of Solution

Calculation of value of delta:

$\begin{array}{c}\text{Delta}=\frac{\text{Spread}\text{of}\text{possible}\text{option}\text{prices}}{\text{Spread}\text{of}\text{possible}\text{share}\text{prices}}\\ =\frac{\$8-0}{\$48-\$33.32}\\ =0.545\end{array}$

Hence, the value of option delta is 0.545

c)

Summary Introduction

To determine: The way payoffs of this call option be replicated based on replicated portfolio method.

Explanation of Solution

Calculation of value of call by using replicating portfolio method:

Hence, the value of call is $3.82

d)

Summary Introduction

To determine: Value of two month call option with $40 as an exercise price.

Explanation of Solution

Calculation of value of option:

The following option possibilities are as follows,

Month 1-a:

$\begin{array}{c}\text{Call}=\frac{\left[\left(0.48\times \$0\right)+\left(0.52\times \$0\right)\right]}{\text{1}\text{.01}}\\ =0\end{array}$

Hence the value of call under month 1-a is ‘0’

Month 1-b:

$\begin{array}{c}\text{Value}\text{of}\text{call}=\frac{\left[\left(0.48\times \$17.6\right)+\left(0.52\times \$0\right)\right]}{\text{1}\text{.01}}\\ =\$8.4\end{array}$

Therefore, value of the call under month 1-b is $8.4

Month 0:

$\begin{array}{c}\text{Value}\text{of}\text{call}=\frac{\left[\left(0.48\times \$0\right)+\left(0.52\times \$8.4\right)\right]}{\text{1}\text{.01}}\\ =\$4.0\end{array}$

Therefore, value of the call under month 0 is $4.0

e)

Summary Introduction

To determine: Value of delta.

Explanation of Solution

Calculation of delta:

$\begin{array}{c}\text{Delta}=\frac{\text{Spread}\text{of}\text{possible}\text{option}\text{prices}}{\text{Spread}\text{of}\text{possible}\text{share}\text{prices}}\\ =\frac{\$8.4-0}{\$48-\$33.3}\\ =0.572\end{array}$

Hence, the delta value is 0.572