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Unformatted text preview: Introduction to derivatives, Fall 2011 BUSI 588, Case 6 solutions Solutions to Shostakovich and binomial trees In the spirit of parsinomia I shall draw trees a-la-Excel in what follows. In particular, the statements in the problem suggest the NYPO stock can fluctuate as follows over the following three months. 100 125 156.25 195.31 80 100 125.00 64 80.00 51.20 Note: the way to read the graph is to say that the stock can go up to the cell on the East (right), and down to the cell on the South-East (down-right). For example, in the dd state, when the stock trades at $64, it can further go up to $80 or down to $51.20. 1. (*) The most direct way to approach this problem is via risk-neutral probabilities (i.e. nor- malized state prices). In our problem we had p u = r- d u- d = 1 . 01- . 8 1 . 25- . 8 = 0 . 4667; p d = 0 . 5333 . Starting at expiration we note that by basic principles the values of the put will be given by the following table. ? ? ? ? ? ? 10 38.8 The next step is to work backwards through the tree calculating at each node what is the value of the option using the basic theorem from the Ronaldinho case. For example, one period before maturity, if the stock is standing at $64, the value of the put should be given by P dd = p u 10 + p d 38 . 80 1 . 01 = 25 . 1089 Similarly, when the NYPO stock is at $100 at date 2 ( ud state), the price of the put should be P ud = p u 0 + p d 10 1 . 01 = 5 . 2805 In the case where the stock went up twice in the first two periods the put stands no chance of being exercised next period, so its value should be zero. This argument gives us the values of the put at date t = 2. ? ? 0.00 ? 5.28 25.11 10 38.8 c Diego Garc a, Kenan-Flagler Business School Page 1 of 4 Introduction to derivatives, Fall 2011 BUSI 588, Case 6 solutions The way to interpret the above calculations is to say that one would need to have $25.11 in the state dd if one were to replicate (buy) the payoffs from the put; $5.28 in the state ud ; and $0 in the uu state....
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