Stochastic

# 7 calls and puts doing some algebra we have v t s 1

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Unformatted text preview: ecursion. Example 6.8. Continue now the set-up of the previous example but consider the call option (Sn 10)+ . The computations are the same but the result is boring: the optimal strategy is to always continue, so there is no di↵erence between the American and the European option. 64, 54 32, 24 A 22, 24* A A 16, 10.56 8, 4.608 6, 10.56*AA 0, 4.608*AA 4, 0.96 A 0, 0.96* AA A A 16, 6 8, 2.4 0, 2.4* AA A 4, 0 2, 0 0, 0* A A A 1, 0 197 6.5. AMERICAN OPTIONS To spare the reader the chore of doing the arithmetic we give the recursion: V2 (2) = max{0, 0} = 0 V2 (8) = max{0, 0.4(0 + 6) = 2.4} = 2.4 V2 (32) = max{22, 0.4(54 + 60 = 24} = 24 V1 (4) = max{0, 0.4(0 + 2.4) = 0.96} = 0.96 V1 (16) = max{6, 0.4(24 + 2.4) = 10.56} = 10.56 V0 (8) = max{0, 0.4(10.56 + 0.96) = 4.608} = 4.608 Our next goal is to prove that it is always optimal to continue in the case of the American call option. To explain the reason for this we formulate an abstract result. We say that g is convex if whenever 0 1 and s1 , s2 are real numbers g ( s1 + (1 )s2 ) g (s1 ) + (1 )g...
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