Mean, Variance and Risk-adjusted Returns of Stock and Option Positions. Consider a non-dividend-paying stock and consider the following portfolios - long 1 stock (unhedged long stock position) - long 1 stock, long 1 put option with strike K1 (protective put). - long 1 stock, short 1 call option with strike K2 (covered call position). Suppose that the initial stock price is $100, the risk-free rate of interest for the period is 5%, the options mature at time T and suppose that the time- T stock price is ST=8090100110120withprobability101withprobability101withprobability101withprobability103withprobability104 Suppose that K1=95 and K2=105 (a) For each portfolio, write the payoff as a function of ST and plot these payoff functions on the same set of axes. (b) For each portfolio, compute the mean and standard deviation of the payoff. (c) Suppose the set of risk-neutral probabilities is (q1,q2,q3,q4,q5)=(101,101,103,102,103). That is, the risk-neutral probability that ST=80 is q1, the risk-neutral probability that ST=90 is q2, etc.. i. Compute the call and put option prices. ii. For each portfolio, write the profit as a function of ST and plot these profit functions on the same set of axes. iii. For each portfolio, compute the mean and standard deviation of the profit. iv. For each portfolio, compute the risk-adjusted return. Which portfolio do you prefer? Why? (d) Repeat the above for K1=85 and K2=115. (e) Briefly discuss the results and compare the results from the different sets of strike prices. Mean, Variance and Risk-adjusted Returns of Stock and Option Positions. Consider a non-dividend-paying stock and consider the following portfolios - long 1 stock (unhedged long stock position) - long 1 stock, long 1 put option with strike K1 (protective put). - long 1 stock, short 1 call option with strike K2 (covered call position). Suppose that the initial stock price is $100, the risk-free rate of interest for the period is 5%, the options mature at time T and suppose that the time- T stock price is ST=8090100110120withprobability101withprobability101withprobability101withprobability103withprobability104 Suppose that K1=95 and K2=105 (a) For each portfolio, write the payoff as a function of ST and plot these payoff functions on the same set of axes. (b) For each portfolio, compute the mean and standard deviation of the payoff. (c) Suppose the set of risk-neutral probabilities is (q1,q2,q3,q4,q5)=(101,101,103,102,103). That is, the risk-neutral probability that ST=80 is q1, the risk-neutral probability that ST=90 is q2, etc.. i. Compute the call and put option prices. ii. For each portfolio, write the profit as a function of ST and plot these profit functions on the same set of axes. iii. For each portfolio, compute the mean and standard deviation of the profit. iv. For each portfolio, compute the risk-adjusted return. Which portfolio do you prefer? Why? (d) Repeat the above for K1=85 and K2=115. (e) Briefly discuss the results and compare the results from the different sets of strike prices
