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PS8Solutions - Finance 100 Problem Set 8 Alternative...

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Finance 100: Problem Set 8 Alternative Solutions Note: Where appropriate, the “final answer” for each problem is given in bold italics for those not interested in the discussion of the solution. All payoff diagrams include the initial cost of the security, however they do not take into account the time value of this cost by computing the future value. Each payoff diagram is drawn with the future price of the security on the horizontal axis and the future payoff to the position on the vertical axes. Strictly speaking, when computing the future payoff to any position, we should calculate the future value of the money paid (or received) today when factoring this into the net position. This is not done here because the dollar impact is small and it adds little to the problem. I. Formulas This section contains the formulas you might need for this homework set: 1. Payoff to a long position in a call option: Payoff = max { S T - K, 0 } (1) where S T is the price of the underlying security at expiration of the contract (time T ) and K is the strike price. 2. Payoff to a short position in a call option: Payoff = - max { S T - K, 0 } = min { K - S T , 0 } (2) 3. Payoff to a long position in a put option: Payoff = max { K - S T , 0 } (3) 1
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4. Payoff to a short position in a put option: Payoff = - max { K - S T , 0 } = min { S T - K, 0 } (4) 5. Put-Call Parity (PCP): C = P + Se - dT - Ke - rT , (5) where C is the price of a call option, P is the price of an otherwise identical put option, S is the price of the underlying security, d is the dividend yield, T is the time to maturity (in years), K is the strike price and r is the risk-free rate. II. Problems 1. 1.a The payoff at maturity (net of the initial cost of the option) to the call buyer is given by max [0 , S T - K ] - FV ( C ) , (6) where S T is the price of the underlying stock at maturity, K is the strike price and FV ( C ) is the future value of the call premium (price), C. The net payoff at maturity (net of the initial cost of the option) to the buyer of the put option at is, mathematically, max [0 , K - S T ] - FV ( P ) , (7) where FV ( P ) is the future value of the put premium, P. Graphically, these payoffs are depicted in Figures 1 and 2 below. 1.b The payoff at maturity (net of the initial cost of the option) to the call seller is given by - ( max [0 , S T - K ] - FV ( C )) = min (0 , K - S T ) + FV ( C ) , (8) where S T is the price of the underlying stock at maturity, K is the strike price and FV ( C ) is the future value of the call premium (price), C. The payoff at maturity (net of the initial cost of the option) to the put buyer is given by - ( max [0 , K - S T ] - FV ( P )) = min (0 , S T - K ) + FV ( P ) , (9) where FV ( P ) is the future value of the put premium, P. Graphically, these payoffs are depicted in Figures 3 and 4 below. 2
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Figure 1: Net Payoff to a Long Call Po- sition with $50 Strike Price Figure 2: Net Payoff to a Long Put Po- sition with $50 Strike Price -10 0 10 20 30 40 50 0 20 40 60 80 100 Stock Price at Maturity Net Payoff at Maturity -10 0 10 20 30 40 50 0 20 40 60 80 100 Stock Price at Maturity Net Payoff at Maturity 1.c All of these graphs were done in Excel. The names of each position are provided upon request. You need not know them.
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