Part I 1. Prove that any European style derivative in an arbitrage-free binomial market can be hedged by a self-financing predictable portfolio invested in the underlying stock and the risk-free asset (max 2 points) 2. Give and explain the definition of implied volatility of European call options (max 2 points) 3. Assume that the market is arbitrage-free. Let P(t, S(t), K, T) be the price at time te [0, T] of the European put with strike K and maturity T and P(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 (0, S(0), K,T) = P(0, S(0), K,T), then (t, S(t), K, T) = P(t, S(t), K,T) for all te (0,T). (b) If the risk-free rate r is negative, then Plt, S(t), K, T) = P(t, S(t), K, T), for allt [0, T). Part I 1. Prove that any European style derivative in an arbitrage-free binomial market can be hedged by a self-financing predictable portfolio invested in the underlying stock and the risk-free asset (max 2 points) 2. Give and explain the definition of implied volatility of European call options (max 2 points) 3. Assume that the market is arbitrage-free. Let P(t, S(t), K, T) be the price at time te [0, T] of the European put with strike K and maturity T and P(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 (0, S(0), K,T) = P(0, S(0), K,T), then (t, S(t), K, T) = P(t, S(t), K,T) for all te (0,T). (b) If the risk-free rate r is negative, then Plt, S(t), K, T) = P(t, S(t), K, T), for allt [0, T)