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3. (10 marks) Assume a stock with current price (0) = 62, T = 2 years, and the price of European call and put options at time 0 are given as follows: Strike Price 40 100 120 European Call Price 24 6 3 6 3 European Put Price 2 8 8 20 Assume prices for American Options on the same underlying are as follows: Strike Price 40 120 American Call 24 10 American Put 3 21 a) (2 marks) Assume no dividend payments and r = 0. Construct an arbitrage oppor- tunity using the put-call parity relation for European options with strike price K = 120. Show it indeed leads to an arbitrage. b) (3 marks) Assume no dividend payments and r > 0. Prove that there is no value X leading to an arbitrage opportunity in a portfolio of: long one call with strike price $40, short 4 calls with strike price $100, lend $3, and short X calls with strike price $120. c) (3 marks) Assume no dividend payments and r = 0. Compute the best lower bound available for the price of an American call option with strike price K = 55. d) (2 marks) Assume a dividend payment with present value D = 7, r = 0, and compute a lower bound for a portfolio consisting of a long call with strike price K = 55 and a short put with strike price K = 55. 3. (10 marks) Assume a stock with current price (0) = 62, T = 2 years, and the price of European call and put options at time 0 are given as follows: Strike Price 40 100 120 European Call Price 24 6 3 6 3 European Put Price 2 8 8 20 Assume prices for American Options on the same underlying are as follows: Strike Price 40 120 American Call 24 10 American Put 3 21 a) (2 marks) Assume no dividend payments and r = 0. Construct an arbitrage oppor- tunity using the put-call parity relation for European options with strike price K = 120. Show it indeed leads to an arbitrage. b) (3 marks) Assume no dividend payments and r > 0. Prove that there is no value X leading to an arbitrage opportunity in a portfolio of: long one call with strike price $40, short 4 calls with strike price $100, lend $3, and short X calls with strike price $120. c) (3 marks) Assume no dividend payments and r = 0. Compute the best lower bound available for the price of an American call option with strike price K = 55. d) (2 marks) Assume a dividend payment with present value D = 7, r = 0, and compute a lower bound for a portfolio consisting of a long call with strike price K = 55 and a short put with strike price K = 55

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