Chapter 21, Problem 1P

The current price of Estelle Corporation stock is $25. In each of the next two years, this stock price will either go up by 20% or go down by 20%. The stock pays no dividends. The one-year risk-free interest rate is 6% and will remain constant. Using the Binomial Model, calculate the price of a one-year call option on Estelle stock with a strike price of $25.

Summary Introduction

To determine: The price of a one year call option on ET stock.

Introduction: A binomial model portrays the development of irregular variables over a progression of time steps, relegating specified probabilities to increase or decrease in the variable. The binomial option pricing model makes the improving supposition that, toward the finish of every period, the price of stocks has just two conceivable values.

Answer to Problem 1P

The price of one year call option on ET stock is $3.07.

Explanation of Solution

Determine the increase or decrease in the stock price.

$\begin{array}{c}{S}_u=\left[S\times \left(1+Percentageofincrease\right)\right]\\ =\left[\$25\times \left(1+20\%\right)\right]\\ =\$30\end{array}$

$\begin{array}{c}{S}_d=\left[S\times \left(1-Percentageofdecrease\right)\right]\\ =\left[\$25\times \left(1-20\%\right)\right]\\ =\$20\end{array}$

Therefore, the stock price either increases to $30 or decreases to $20

Here

S – Denotes the current stock price

K – Denotes the strike price

C – Denotes the call price

B – Denotes the risk-free investment or initial investment in the portfolio

S_u – Denotes the probability of increase in stock price (the price to go up)

S_d – Denotes the probability of decrease in stock price next period (the price to go down)

r_f – Denotes the risk-free rate

C_u – Denotes the price of the call option if the stock price increases (the price to go up)

C_d – Denotes the price of the call option if the stock price decreases (the price to go down)

Δ – Denotes the shares of stock in the portfolio or the sensitivity of option price to stock price

Determine the option payoff:

Using Excel function =MAX, the option payoff is determined as follows:

Excel spreadsheet:

Excel workings:

Therefore, the option payoff either increases to $5 or decreases to $0.

Determine the shares of stock in the replicating portfolio:

$\begin{array}{c}\Delta =\left[\frac{C_u-C_d}{S_u-S_d}\right]\\ =\left[\frac{\$5-\$0}{\$30-\$20}\right]\\ =\left[\frac{\$5}{\$10}\right]\\ =0.5\end{array}$

Therefore, the share of stock in the replicating portfolio is 0.5.

Determine the risk-free investment or initial investment in the portfolio:

$\begin{array}{c}B=\left[\frac{C_d-S_d\Delta }{\left(1+r_f\right)}\right]\\ =\left[\frac{0-\$20\times 0.5}{\left(1+6\%\right)}\right]\\ =\left[\frac{-\$10}{1.06}\right]\\ =-9.4340\end{array}$

Therefore, the risk-free investment or initial investment in the portfolio is -9.4340.

Determine the price of a one year call option on ET stock:

$\begin{array}{c}C=\left[S\Delta +B\right]\\ =\left[\$25\times 0.5+\left(-9.4340\right)\right]\\ =\left[\$12.5-9.4340\right]\\ =\$3.0660or\$3.07\end{array}$

Therefore, the price of a one year call option on ET stock is $3.07.