Assume the price of a stock is $50 now and it only could be $60 or $40 in a month. Buy a European call option of the stock at with strike price $50 and 1 month maturity. And the risk-free annual interest rate is 12%. How much should the premium for this call option be?

1. Payoff when the stock price is $60 = Max(Stock Price - Strike Price,0) = Max(60 - 50,0) = $10

2. Payoff when the stock price is $40 = Max(Stock Price - Strike Price,0) = Max(40 - 50,0) = $0

3. Delta of Call Option = (Payoff when stock price is $60 - Payoff when stock price is $40) / (Up Stock Price - Down Stock Price)

Delta of Call Option = (10 - 0) / (60 - 40)

Delta of Call Option = 0.50 Shares

4. Price of Call Option = Delta * Current Stock Price  - Portfolio value at the end of 1 months * e^{-(Time * Interest Rate)}
Price of Call Option = 0.50 * $50 - Down Stock Price * Delta * e^{-(Time * Interest Rate)}
Price of Call Option = $25 - 40 * 0.50 * e^{-(1/12 * 12%)}
Price of Call Option = $25 - $20 * 0.99005
Price of Call Option = $5.20