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174%202011%20Summer%20Quiz_2

# 174%202011%20Summer%20Quiz_2 - 174 Quiz 2 Solution Gabe...

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174 Quiz 2 Solution Gabe Merton August 19, 2011 The first thing to realize is the underlying asset is a futures contract and not a stock. The first step is to create a binomial tree for this asset. Recall that we model the value of a futures contract via the formula F 0 = S 0 e ( r - δ ) t where t is the time to expiration of the contract. Furthermore, for futures contracts we set u = e σ h and d = e - σ h . This is equivalent to modeling a stock where r = δ . With these parameters, we also have p * = 1 - d u - d . Thus, F 0 = 50 e (0 . 08 - 0 . 04) 52 . 0405 u = e 0 . 3 0 . 25 1 . 1618 d = e - 0 . 3 0 . 25 0 . 8607 p * = 1 - 0 . 8607 1 . 1618 - 0 . 8607 0 . 46257 From here on I’ll just use = when I should use . One constructs the tree for the futures contract by multiplying F 0 by combinations of u and d as usual. The tree is below. 1

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52.0405 44.7917 60.4625 38.5526 52.0405 70.2474 33.18 44.7917 60.4625 81 . 6158 For example, the calculation of the up, up, down node is: (52 . 0405) u 2 d = 60 . 4625 The next step is to calculate the four possible call values at expiration. Since this is a call option, the payoff is max(0 , F - K ) where F is the value of the future contract and K is the strike, 55. Thus, C uuu = max(0 , 81 . 6518 - 55) = 26 . 6518 C uud = max(0 , 60 . 4625 - 55) = 5 . 4625 C udd = max(0 , 44 . 7917 - 55) = 0 C ddd = max(0 , 33 . 18 - 55) = 0 Some of you tried to simply discount these four values over three periods; this would
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