Problem 2. Arbitrage-Free Strike Price of Put Options Consider a market that involves a stock A, which has a price of $1 per share today. The price of A tomorrow is a random variable S, which can take two values $2 or $0.5. The market also has a put option" P, which is a financial product that gives the owner the right to sell one share of A at a strike price K tomorrow. Each share of P has a price of $0.1 today and has a payoff max{K - S,0} tomorrow. We use the notation (K - $)+ = max{K - S,0} throughout this problem. We want to determine the optimal amounts of stocks A and put options Op to purchase, given by the linear program below! min 0A + 0.10p s.t. 20 A +(K-2)+ep 20 0.50 A + (K -0.5)+ep > 0 (2) DA, Op free The payoff of an option P with strike price K is (K-S). If K > S then we can buy a stock at price S and sell it immediately at price K and earn a profit, if KS then we lose money by selling the good at price K, so we gain nothing. The modern financial market allows people to sell stocks and options "short", meaning that one can sell items without actually having any to start with; this corresponds to a negative value for A or Op. Here constraint (1) means that we must have a nonnegative payoff if the first scenario occurs, and (2) means that we must have a nonnegative payoff if the second scenario occurs. A market has an arbitrage opportunity if there is a way to earn a strictly positive payoff by buying and selling assets in the market today, with nonnegative payoffs in the future in other words, guaranteed free money, regardless of which outcome happens). A market with no arbitrage opportunity is called arbitrage free. b) Suppose K = 0.7 (Note (K-2)+ = 0 and (K - 0.5)+ = 0.2) and answer the questions below (the simplex method should not be necessary). (b1) Write the dual program. (b2) Is the dual feasible? What is the dual optimal solution? (63) Argue that if the dual program is feasible then the market is arbitrage free (so there is no portfolio with negative cost today, with a nonnegative payoff tomor- row.) Problem 2. Arbitrage-Free Strike Price of Put Options Consider a market that involves a stock A, which has a price of $1 per share today. The price of A tomorrow is a random variable S, which can take two values $2 or $0.5. The market also has a put option" P, which is a financial product that gives the owner the right to sell one share of A at a strike price K tomorrow. Each share of P has a price of $0.1 today and has a payoff max{K - S,0} tomorrow. We use the notation (K - $)+ = max{K - S,0} throughout this problem. We want to determine the optimal amounts of stocks A and put options Op to purchase, given by the linear program below! min 0A + 0.10p s.t. 20 A +(K-2)+ep 20 0.50 A + (K -0.5)+ep > 0 (2) DA, Op free The payoff of an option P with strike price K is (K-S). If K > S then we can buy a stock at price S and sell it immediately at price K and earn a profit, if KS then we lose money by selling the good at price K, so we gain nothing. The modern financial market allows people to sell stocks and options "short", meaning that one can sell items without actually having any to start with; this corresponds to a negative value for A or Op. Here constraint (1) means that we must have a nonnegative payoff if the first scenario occurs, and (2) means that we must have a nonnegative payoff if the second scenario occurs. A market has an arbitrage opportunity if there is a way to earn a strictly positive payoff by buying and selling assets in the market today, with nonnegative payoffs in the future in other words, guaranteed free money, regardless of which outcome happens). A market with no arbitrage opportunity is called arbitrage free. b) Suppose K = 0.7 (Note (K-2)+ = 0 and (K - 0.5)+ = 0.2) and answer the questions below (the simplex method should not be necessary). (b1) Write the dual program. (b2) Is the dual feasible? What is the dual optimal solution? (63) Argue that if the dual program is feasible then the market is arbitrage free (so there is no portfolio with negative cost today, with a nonnegative payoff tomor- row.)
