Unformatted text preview: ecursion. Example 6.8. Continue now the setup 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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 Spring '10
 DURRETT
 The Land

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