American Option

Consider the following twoperiod model. The price of a stock is $50, the simple com pounding interest rate per period is 2%. After the rst period, the price of the stock can go up to $55 or drop to $47. If the stock price jumps up in the rst period, if the holder of an American call option exercises this option, the holder will receive a dividend of $2 that is paid out by the stock. Assume that the option is exercised before dividends are paid out. After the dividend is paid out, in the second period the price can jump up to $57 or down to $48. If the stock price jumps down in the rst period, in the second period it can jump up to $48 or down to $41. Consider an American call option with strike price K = $45 that matures at the end of the second period. (a) Plot the binomial tree underlying this option. Calculate the risk neutral probability in each period. (Remember that after payout dividend, 3\" becomes 55 2 = 53) (b) Use the Snell envelop to price the option on the binomial tree. (c) Given the optimal exercise of the American call option