Problem 1. (10 points) A European put option with strike price X = $60 and expiry in 6 months is currently trading for Pe = $10. The current stock price is S(0) = $50, the continuously compounded interest rate is r = 10% and a dividend of $6 is expected in two months. Find an arbitrage opportunity and explain clearly why it leads to arbitrage. Problem 2. (10 points) On the market there are an American put and an American call options with the same strike price X = $90 that expire in six months. The current stock price is S(0) = $90, the continuously compounded interest rate is r = 10% and the call and put prices are CA = $5 and PA = $6. Find an arbitrage opportunity and explain clearly why it leads to arbitrage. Problem 3. Consider a two-step binomial model with S(0) = 90 and returns on the stock u = 0.15 and d = -0.2. Let the risk-free return be r = 0 and suppose that the stock pays a dividend of 7 at step 1 and a dividend of 3 at step 2. a) (2 points) Draw the stock price tree. b) (8 points) Compute the risk-neutral probability and find the price of a European put with strike X = 80 and expiration at step 2