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Unformatted text preview: (s2 )
(6.22)
Geometrically the line segment from (s1 , g1 (s)) to (s2 , g2 (s)) always lies above
the graph of the function g . This is true for the call g (x) = (x K )+ and the
put g (x) = (K x)+ . However only the call satisﬁes all of the conditions in the
following result.
Theorem 6.8. If g is a nonnegative convex function with g (0) = 0 then for
the American option with payo↵ g (Sn ) it is optimal to wait until the end to
exercise.
Proof. Since Sn /(1 + r)n is a martingale under P ⇤
◆◆
✓✓
Sn+1
⇤
g (Sn ) = g En
1+r
Under the risk neutral probability
Sn (a) = p⇤ (a)
n
Using (6.22) with Sn+1 (aH )
+ (1
1+r p⇤ (a))
n Sn+1 (aH )
1+r = p⇤ (a) it follows that
n
g ✓ ⇤
En ✓ Sn+1
1+r ◆◆ ⇤
En ✓✓
◆◆
Sn+1
g
1+r Using (6.22) with s2 = 0 and g (0) = 0 we have g ( s1 ) g (s1 ), so we have
⇤
En ✓✓
◆◆
Sn+1
1
g E ⇤ g (Sn+1 )
1+r
1+r n Combining three of the last four equations we have
g (Sn ) 1
E ⇤ g (Sn+1 )
1+r n This shows that if we were to stop at time n for some outcome a, it would be
better to continue. Using (6.21) now the desired result follows. 198 CHAPTER 6. MATHEMATICAL FINANCE...
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 Spring '10
 DURRETT
 The Land

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