Options and mathematics:

The number of points assigned to the tasks in this part will be based on the amount of details provided in the solution. Use your own words and explain each step in the proof, in details. Do not skip calculations!

Assume that the market is arbitrage-free. Let P(t, S(t), K, T) be the price at time t [0, T] of the European put with strike K and maturity T and Pb(t, S(t), K, T) be the price of the corresponding American put. The underlying stock pays no dividend in the interval [0, T]. Decide whether the following statements are true or false and explain your answer (max 2 points): (a) If Pb(0, S(0), K, T) = P(0, S(0), K, T), then Pb(t, S(t), K, T) = P(t, S(t), K, T) for all t [0, T]. (b) If the risk-free rate r is negative, then Pb(t, S(t), K, T) = P(t, S(t), K, T), for all t [0, T].