Stochastic

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Unformatted text preview: but it is no more di cult to consider path dependent options, so we derive the basic formulas in that generality. Given a sequence a of heads and tails of length n let gn (a) be the value if we exercise at time n. Our ﬁrst goal is to compute the value function Vn (a) 195 6.5. AMERICAN OPTIONS for the N -period problem. To simplify some statements, we will suppose with essentially no loss of generality that gN (a) 0, so VN (a) = gN (a). To work backwards in time note that at time n we can exercise the option or let the game proceed for one more step. Since we will stop or continue depending on which choice gives the better payo↵: ⇢ 1 ⇤ [p⇤ (a)Vn+1 (aH ) + qn (a)Vn+1 (aT )] (6.20) Vn (a) = max gn (a), 1+r n ⇤ where p⇤ (a) and qn (a) = 1 p⇤ (a) are the risk-neutral probabilities which make n n the underlying stock a martingale. 64, 0 32, 0 0, 0* A 16, 0.96 0, 0.96* 8, 2.874 A A A 2, 2.784*A A A 4, 6 6*,4.16 A 16, 0 A A 4, 6 A A 1, 9 8, 2.4 2, 2.4* A A A A A 2, 8 8*, 6 A Example 6.7. For a concrete example, suppose as we did in Example 6.3 that the stock price follows the binomial model with S0 = 8, u = 2, d = 1/2, r = 1/...
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## This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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