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PROBLEM SET 10 1) An option strategy called the iron butterfly is illustrated in the payoff diagram below. (a) How could you generate the iron butterfly using only call and put options? (b) How could you generate the iron butterfly using only call options and bonds? (c) The premiums of some relevant options written on S are given in the table below. Draw a profit diagram for the iron butterfly (possibly in the graph above). Type X T t Premium Put 45 1 $1.876 Call 45 1 $9.070 Put 50 1 $3.730 Call 50 1 $6.168 Put 55 1 $6.331 Call 55 1 $4.013 1 (d) What market views could lead you to buy an iron butterfly? (e) What is the risk-free (continuously compounded) interest rate, r? (f) What is the current stock price, St? 2) Consider a world in which some stock, S, can either go up by 25% or down by 20% in one year and no other outcomes are possible. The continuously compunded risk-free interest, r, is 5.5% and the current price of the stock, S0, is $100. (a) What are the possible stock values in one years time, ST ? (b) What are the possible payoffs of a European call option written on stock S with a strike price, X, of $100 that expires in one years time, T = 1? (c) Suppose you want to form a portfolio, P, consisting of one short-sold unit of the call option and some number, , of the stock, such that the value of the portfolio in one years time, PT , does not depend on the value of the stock, ST . What would be the appropriate value of ? (d) What would be the (certain) value of the portfolio in one years time, PT ? (e) What is the arbitrage-free price of the portfolio today, P0? (f) What is the arbitrage-free value of the call option today, c0? (g) Define p as p = erT d u d and call this the probability that the stock price increases. What is the value of p? (h) Suppose we accept that p is indeed the probability that the stock price goes up. What is the expected value of the stock in one years time, E(ST )? (i) At what continuous rate would the stock price have to grow to end up at the expected value? (j) What would be the expected value of the call option in one years time, E(cT )? (k) At what continuous rate would the call price have to grow to end up at the expected value? 2

PROBLEM SET 10 1) An option strategy called the iron butterfly is illustrated in the payoff diagram below. (a) How could you generate the iron butterfly using only call and put options? (b) How could you generate the iron butterfly using only call options and bonds? (c) The premiums of some relevant options written on S are given in the table below. Draw a profit diagram for the iron butterfly (possibly in the graph above). Type P ut Call P ut Call P ut Call X 45 45 50 50 55 55 T t 1 1 1 1 1 1 1 P remium $1.876 $9.070 $3.730 $6.168 $6.331 $4.013 (d) What market views could lead you to buy an iron butterfly? (e) What is the risk-free (continuously compounded) interest rate, r? (f) What is the current stock price, St ? 2) Consider a world in which some stock, S, can either go up by 25% or down by 20% in one year and no other outcomes are possible. The continuously compunded risk-free interest, r, is 5.5% and the current price of the stock, S0 , is $100. (a) What are the possible stock values in one year's time, ST ? (b) What are the possible payoffs of a European call option written on stock S with a strike price, X, of $100 that expires in one year's time, T = 1? (c) Suppose you want to form a portfolio, P , consisting of one short-sold unit of the call option and some number, , of the stock, such that the value of the portfolio in one year's time, PT , does not depend on the value of the stock, ST . What would be the appropriate value of ? (d) What would be the (certain) value of the portfolio in one year's time, PT ? (e) What is the arbitrage-free price of the portfolio today, P0 ? (f) What is the arbitrage-free value of the call option today, c0 ? (g) Define p as p= erT d ud and call this the \"probability\" that the stock price increases. What is the value of p? (h) Suppose we accept that p is indeed the probability that the stock price goes up. What is the expected value of the stock in one year's time, E(ST )? (i) At what continuous rate would the stock price have to grow to end up at the expected value? (j) What would be the expected value of the call option in one year's time, E(cT )? (k) At what continuous rate would the call price have to grow to end up at the expected value? 2