1. We use an additive model to predict the price of AAPL (Apple Inc.). The price of one share of AAPL is $150 at t = 0. For each day, the price increases $1 with probability p and drop $1 with probability 1 p. Denote SN to be the price of one share of AAPL after N days (i.e, t = N), which is considered as a discrete random variable.

(a) (2 mark) If p = 0.5 and N = 4, compute the standard derivation of S4.

(b) (3 marks) Compute the pdf of SN.

(c) (2 marks) Compute E[SN]. Please justify your answer.

2. Consider a 12-month forward contract on a stock when the stock price is $50. We assume that the risk-free rate of interest (compound quarterly) is 8%. We also assume that (cash) dividends of $0.75 per share are expected after 3 months, 6 months, 9 months, 12 months, etc. (That is, a cash dividend is paid at the end of each quarter.) A cash dividend is a payment made by a company to its stockholders in the form of periodic distributions of cash.

(a) Write down the cash flow of 12 months.

(b) Determine the theoretic price of this forward contract (2 decimal places)