673_hw4_solution

673_hw4_solution - NBA673 Introduction to Derivatives I...

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1 of 5 NBA673 Introduction to Derivatives I Tibor Jánosi Spring 2006 (1 st half) Homework 4: Options Solutions Problem 1: (a) The payoff diagram of the bullish spread is drawn below using a continuous red line: (payoff of bullish spread) = (payoff of long call) + (payoff of short call) We can describe the payout analytically as follows: (1) 0 S $45 (payoff of bullish spread) = 0 + 0 = 0 (2) $45 S $55 (payoff of bullish spread) = (S – 45) + 0 = S – 45 (3) $55 S (payoff of bullish spread) = (S – 45) + (- S + 55) = 10 Notes: 1. The problem allowed you to justify your diagram by just picking one specific value out of each interval of interest; the calculation above is more general and not more complex. 2. You might have noticed that there is an overlap between the intervals given above (e.g. value $55 appears both in the second and the third interval). We could have used strict and non-strict inequalities to avoid these overlaps. We kept this form to emphasize that the payoff functions are continuous and that it does not make any difference where we include the ends of intervals. stock price (S) $45 $55 payoff $10
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2 of 5 (b) The put-call parity allows us to substitute calls with puts. Assume for now that we are on the expiration date of the options, T: c(45) – p(45) = S(T) – 45 c(55) – p(55) = S(T) – 55 From here, we get that c(45) – c(55) = [p(45) + S(T) – 45] – [p(55) + S(T) – 55] = p(45) – p(55) + 10 So our original portfolio can be replaced by a portofolio of a long put with strike price $45, a short put with strike price $55, and a holding of $10 in cash. We could write this equality using the put-call parity at time t<T (see problem 7b below). To keep the notation simple, we will use bold characters for the price of the put and the call at time t. We get: c (45) – c (55) = [ p (45) + S(t) – 45B(t,T)] – [ p (55) + S(t) – 55B(t,T)] = p (45) – p (55) + 10B(t,T) Given the formulation of the problem, any of these two formulations is acceptable. (c) This combination is called a “bullish spread” to emphasize the fact that the maximum payoff will be realized if the underlying’s price increases. Problem 2: (a) The payoff diagram of the bearish spread is drawn below using a continuous red line: Note: Since this problem is analogous to the previous one, we will just briefly present the results without commenting further. (b) - c(25) + c(35) = -[p(25) + S(T) – 25] + [p(35) + S(T) – 35] = p(35) – p(25) - 10
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This note was uploaded on 09/28/2008 for the course NBA 6730 taught by Professor Janosi,tibor during the Spring '06 term at Cornell.

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673_hw4_solution - NBA673 Introduction to Derivatives I...

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