We suppose that the stock evolves as t = 0 t = t = 2T t = T So Sri Sarja ST uS uS uS udS S udS dS udS dS dS with S = 1, u = 1.15, d = 0.85 and T = 1. S is the stock price at time t = 0. Cash is invested/borrowed at the rate r = 5% per period. On that stock there is an option whose evolution is given by t = 0 = t = = 2T Cud C Ca Cdd At maturity T, the option value (i.e. the payoff) is given by (ST) with i your group number. The payoff functions are given below: 1 (1) 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 22 12. 13. The option pricing formula is given by 41(x)={ 42(x)={ 43(x)={ 44(x) = { 3 if x 1 0 otherwise 0 if x > 1 3 otherwise 1 if a > 1 3 otherwise 4 if x > 1 1 otherwise s(x)=0.5x2 46(x) = { 47(x) = { 48(x)={ 49(x)={ P10(x) = { 41(2)={ 412(x)={ 413(x)={ = = Cud Cdd = Cd= C= x if a 1 0 otherwise x if a >1 1 otherwise 0 if x > 1 2x otherwise x if x > 1 2 otherwise x if x > 1 2x otherwise x if x > 1 3 otherwise x if x > 1 0.5x otherwise 2x if x > 1 0.5x otherwise pCuuu + (1 - p)Cuud (1+r) pCuud + (1 - p)Cudd (1+r) pCudd + (1 p)Cddd (1+r) pCuu + (1 p)Cud (1+r) pCud + (1 - p)Cdd (1+r) pCu + (1 - p)Ca (1+r) as there are 3 periods (p = (1+r)* (p = (1+r)/n-d) 2 1. Compute the price C. The hedging ratios along the tree are given by: (3) (4) Sddd (5) (6) (7) Aud = hud = Add = hdd = A = h = Auu = huu = - Suuu Suud" Cuud - Cudd Suud-Sudd Cudd - Cadd Suud - Cud Suu-Sud Cud - Cad Ad = hd = Sud-Sdd' Cu-Ca A = h = Su-Sd' t=0 t= t = Dux A Aud Add 2. 3. Compute the deltas, the hedging ratios of the option along the tree. Suppose that you sell one option at time t = 0 and you do not hedge, you keep the premium at the rate r. What is your position at time T (depending on the stock value at that date and payoff you have to pay)? 4. You sell an option and you take the hedging position at time t = 0 and then stop hedging, that is to say, you do not modify your hedging position at times T/3 and 2T/3. What is your position at time 7' (depending on the stock value at that date and payoff you have to pay). 5. You sell an option and you take the hedging position at time t = 0 and you modify your hedging position at time T/3 but do not rebalance your portfolio at time 27/3. What is your position at time T (depending on the stock value at that date and payoff you have to pay). Pay attention to the fact that several paths can reach certain stock values at time T. 6. option. You sell an option and you hedge the option until time T. Check that you manage to hedge perfectly the