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3. St(w) denotes the share price at time t for a given path w. The share S satisfies the following filtration (P+)o, where T = 2 years, the interest rate r = 0 and the prices are in USD. Note that these share prices assume no dividend will be paid at any time. 3533 W1 W2 W3 W4 HH S(0) S(ly) S(2y) 4 8 16 4 8 4. 4 2 4. 4 2 1 NN o (i) A single dividend of 40 cents per share is announced to be paid after 18 months. Calculate the premium of both the European and American call options with strike K = 3 and expiration in 2 years. Explain why the American call option has a greater premium than the European call option. (ii) Typically the size of the dividend payment is not announced in advance. Consider the American call option struck at 3 and let D be the unknown dividend amount in USD. Show that for at least one path early exercise is always optimal for any dividend payment $0