(a) Write down mathematical expressions for the values of European call and put options on a given security at expiry, explaining any terms that you use. Also, sketch the pay-off diagram in each case, assuming that there are transaction costs. (4 marks) (b) Sketch the pay-off diagrams for each of the following portfolios, assuming that there are transaction costs: i. Short one share and long one call and three puts both with strike price K; (3 marks) ii. Short one call with strike price 2K and short one call with strike price 4K, and long one call with strike price K; and (6 marks) iii. Short one call and short two puts both with strike price K and long one share. (3 marks) (c) A portfolio consists of the following: long: 100 shares; 300 puts with strike price 115p;200 puts with strike price 100p; and short: 400 calls with strike price 110p;210 calls with strike price 195p. Assuming that the options all have the same expiry date, find the value of the portfolio at expiry if the share price is: i. 185p; ii. 195p; iii. 200p; or iv. 215p. (4 marks) (a) Write down mathematical expressions for the values of European call and put options on a given security at expiry, explaining any terms that you use. Also, sketch the pay-off diagram in each case, assuming that there are transaction costs. (4 marks) (b) Sketch the pay-off diagrams for each of the following portfolios, assuming that there are transaction costs: i. Short one share and long one call and three puts both with strike price K; (3 marks) ii. Short one call with strike price 2K and short one call with strike price 4K, and long one call with strike price K; and (6 marks) iii. Short one call and short two puts both with strike price K and long one share. (3 marks) (c) A portfolio consists of the following: long: 100 shares; 300 puts with strike price 115p;200 puts with strike price 100p; and short: 400 calls with strike price 110p;210 calls with strike price 195p. Assuming that the options all have the same expiry date, find the value of the portfolio at expiry if the share price is: i. 185p; ii. 195p; iii. 200p; or iv. 215p. (4 marks)