Problem 4 (Required, 35 marks) The current price of an asset is . It is given that the asset will pays dividends at times 1,2, , (where < 1 < 2 < < < ) respectively. The amount of dividend, payable at time , is , where > 0 is a constant and is the asset price at time . (Note: In other words, the amount of dividend equals to units of the asset) (a) We consider a forward contract on this asset signed at time . We let and be the delivery price and delivery date of the contract respectively. Using no arbitrage pricing principle, show that the delivery price is = () (1 + 1 )(1 + 2 ) (1 + ) . (b) We consider European options on this asset. We let = ( ; , ) be the price of European call option with strike price and maturity date and = ( ; , ) be the price of European call option with strike price and maturity date . Show that the prices of these two options satisfy + () = + (1 + 1 )(1 + 2 ) (1 + ) . (c) Show that for any 1, 2 satisfying < 1 < < +1 < < + < 2 < , ( ; 1, ) < ( ; 2, ), where = (1 + +1 )(1 + +2 ) (1 + + ) and ( ; , ) denotes the current price of European call option with strike price and maturity date

. Problem 4 (Required, 35 marks) The current price of an asset is St. It is given that the asset will pays n dividends at times t, t2, ..., tn (where t < t 0 is a constant and Stk is the asset price at time tk. (Note: In other words, the amount of kth dividend equals to ok units of the asset) (a) We consider a forward contract on this asset signed at time t. We let X and T be the delivery price and delivery date of the contract respectively. Using no arbitrage pricing principle, show that the delivery price X is Ster(T-t) X= (1+1)(1+82)... (1+ on) (b) We consider European options on this asset. We let c= c(St;T,X) be the price of European call option with strike price X and maturity date T and p=p(St;T, X) be the price of European call option with strike price X and maturity date T. Show that the prices of these two options satisfy St = p + (1 +81)(1+02) ... (1+ on)" (c) Show that for any T1, T2 satisfying t < T; 0 is a constant and Stk is the asset price at time tk. (Note: In other words, the amount of kth dividend equals to ok units of the asset) (a) We consider a forward contract on this asset signed at time t. We let X and T be the delivery price and delivery date of the contract respectively. Using no arbitrage pricing principle, show that the delivery price X is Ster(T-t) X= (1+1)(1+82)... (1+ on) (b) We consider European options on this asset. We let c= c(St;T,X) be the price of European call option with strike price X and maturity date T and p=p(St;T, X) be the price of European call option with strike price X and maturity date T. Show that the prices of these two options satisfy St = p + (1 +81)(1+02) ... (1+ on)" (c) Show that for any T1, T2 satisfying t < T;