# 67 calls and puts we will now apply the theory

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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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