Question

The current price of a stock is $22, and at the end of one year its price will be either $29 or $15. The annual risk-free rate is 5.0%, based on daily compounding. A 1-year call option on the stock, with an exercise price of $21, is available. Based on the binominal model, what is the option's value?

Answer 1

Current price of Stock is $22

Upstate price = $ 29

Downstate Price = $15

Exercise Price = $21

Risk Free Interest rate = 5%

To identify the price of the option, lets create a portfolio of stocks with a particular weight "P"and one call option such that regardless of the upstate or downstate move, the value of the portfolio should remain the same. Suppose you purchase "P" number of shares of the underlying stock and short one call option, then in an upstate condition, the value of the payoff would be

29P - 8. i.e 29*P and you would be losing 8 on the option price (Since you have shorted).

Similarly, in a downside move, your value of portfolio would be

15d - 0. i.e 15*d and the value of the option would be zero since it would expire worthlessly.

As per our earlier statement, we are creating a portfolio where the worth of the portfolio in each state remain the same. hence 29P-8 = 15d. Solving for d, we get d = 8/14 which is equal to 0.57. Hence d = 0.57.

value of the portfolio is 15d i.e 15*0.57 which is equal to 8.57.

Present value of the portfolio given the risk free interest rate of 5% is = PV(5%,1,-8.57) (Excel formula) which is equal to $8.16.

As per our portfolio composition, the equation is 0.57 * 29 - 1*Call price = 8.16. Solving for the equation, we get the value of 1 call option price as $8.37.

Value of the call option is $8.37.