The current price of a stock is $20. In 1 year, the price will be either $26 or $16. The annual risk-free rate is 5%. Find the price of a call option on the stock that has a strike price of $21 and that expires in 1 year.

You have to use the binomial tree model.

Let the call price be C0
The holder of the call will either stand to make a $5 gain or his option will expire worthless. ($0 gain). Today he has the choice between selling his option and gain C0 dollars or wait a year and take to risk to cash in a variable profit.

Now imagine the call holder would sell n stocks today. Today his portfolio is worth C0-20n and in a year it will be either 5-26n or 0-16n.
Therefore by putting n=1/2 he can eliminate his risk. In that case whatever the stock does his portfolio will be worth -$8.

Such a no-risk portfolio gains by definition the no-risk rate of 5%. The discounted value is therefore worth today C0-20/2 = 8/(1+5%)

We find that C0=10-8/1.05= $2.38

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