Chapter 21, Problem 36PS

Summary Introduction

To calculate the value of call option using the given prices

Answer to Problem 36PS

Hence, the value of call option is $\$\mathrm{13.6364.}$

Explanation of Solution

Given Information:

You are attempting to value a call option with an exercise price of $\$100$ and one year to expiration. The underlying stock pays no dividends, its current price is $\$100$ , and you believe it has a $50\%$ chance of increasing to $\$120$ and a $50\%$ chance of decreasing to $\$80$ . The risk-free rate of interest is $10\%.$

Data can be summarized as follows:

Exercise price of call option $\left(X\right)=\$100$

Time to expiration $\left(T\right)=1$ year

Dividend payout $=0$

Stock price $\left(S_0\right)=\$100$

Probability of stock price increasing to $\$120\left(u{S}_0\right)=0.5$

Probability of stock decreasing to $\$80\left(d{S}_0\right)=0.5$

Risk-free rate of interest $\left(r\right)=10\%$

Methodology to arrive at the option value is as follows:

1. Calculate the probable call values

2. Then based on the call values arrived in both the cases, i.e. in case of stock price increase

and decrease, the hedge ratio needs to be calculated

3. Calculate riskless portfolio.

4. Then the present value of stock price highly probable at the end of year can be calculated.

5. Then set the value of hedged position equal to the present value of certain pay-off.

6. Finally, value of the call option can be calculated.

Step 1 Probable Call value for stock price $\$120=u{S}_0-X$

$\begin{array}{l}{C}_U=\$120-\$100\\ =\$20\end{array}$

Probable Call value for stock price $\$80=d{S}_0-X$

$\begin{array}{l}{C}_d=\$80-\$100\\ =0\end{array}$

Therefore, $C_u=\$20$ is in-the-money and $C_d=\$0$ is out-of-the-money.

Calculate the hedge ratio.

The hedge ratio, H can be calculated by using the following formula:

$H=\frac{\left(C_u-C_d\right)}{\left(u{S}_0-d{S}_0\right)}$

Here,

Value of the call option in case of in-the-money scenario is $C_u$

Value of the call option in case of out-of-the-money scenario is $C_d$

Stock price in case stock price increases is $u{S}_0$

Stock price in case stock price decreases is $d{S}_0$

Factor by which stock increases is $u$

Factor by which stock decreases is $d$

Substitute the values in the formula as follows:

$C_u=\$20$ , $C_d=\$0$ , $u{S}_0=\$120$ and $d{S}_0=\$80$

$\begin{array}{l}H=\frac{\left(C_u-C_d\right)}{\left(u{S}_0-d{S}_0\right)}\\ =\frac{\left(\$20-\$0\right)}{\left(\$120-\$80\right)}\\ =\frac{\$20}{\$40}\\ =0.5\end{array}$

Hence, the hedge ratio is $\mathrm{0.5.}$

Create a riskless portfolio that consists of two written calls and one share of stock. The riskless

Portfolio's cost is $S_0-2C\text{ =}100-2C.$

The following table shows the payoff of the portfolio to be risk free.

Portfolio	$S=\$80$	$S=\$120$
Buy 1 share	$\$80$	$\$120$
Write 2 calls	$\$0$	$-\$40$
Total	$\$80$	$\$80$

Present value of portfolio is as follows:

Present value of stock price $=\frac{Payoff}{{\left(1+r\right)}^1}$

$\begin{array}{l}=\frac{\$80}{{\left(1+0.10\right)}^1}\\ =\frac{\$80}{1.10}\\ =\$72.7273\end{array}$

Hence, the present value of portfolio is $\$\mathrm{72.7273.}$

Step 5:

Set the value of the hedged position equal to the present value of the payoff as follows:

  $\$100-2C\text{ = }\$72.7273$

Step 6:

Calculate the value of call option $\left(C\right)$ as follows:

  $\begin{array}{l}\$100-2C\text{ = }\$72.73\\ C=\frac{100-72.7273}{2}\\ =\$13.6364\end{array}$

Hence, the value of call option is $\$\mathrm{13.6364.}$