The current price of a stock is $22, and at the end of one year its price will be either $27 or $17. The annual risk-free rate is 6.0%, based on daily compounding. A 1-year call option on the stock, with an exercise price of $22, is available. Based on the binomial model, what is the option's value?

Based on the binomial model, the option's value is $3.00

Explanation:

The stock range of payoffs in one year is $27 - $17 = $10.

At expiration, if the stock price is $27, the option will be worth

= $27 - $22

= $5

And the option will be worth zero, if the stock price $17.

The range of payoffs for the stock option is $5 & $0; 0 = $5.

Equalize the range to find the number of shares of stock:

Option range/Stock range = $5/$10

= 0.5

With 0.5 shares, the stock options payoff will be either $13.5 or $8.5. The portfolio & options payoff will be

$13.5 - $5 = $8.5, or $8.5 $0; 0 = $8.5.

The present value of $8.5 at the daily compounded risk-free rate is: PV = $8.5 / (1 + (0.06/365)) 365 = $8.005.

The option price is the current value of the stock in the portfolio

minus the PV of the payoff: V = 0.5 ($22) - $8.005 = $3.00.

Therefore, Based on the binomial model, the option's value is $3.00