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Unformatted text preview: money not
in the stock in the money market account which pays interest r per period our
wealth satisﬁes the recursion:
Wn+1 = n Sn+1 + (1 + r)(Wn n Sn ) (6.14) Theorem 6.3. If W0 = V0 and we use the investment strategy in (6.13) then
we have Wn = Vn .
In words, we have a trading strategy that replicates the option payo↵s.
Proof. We proceed by induction. By assumption the result is true when n = 0.
Let a be a string of H and T of length n. (6.14) implies
Wn+1 (aH ) = n (a)Sn+1 (aH ) = (1 + r)Wn (a) + + (1 + r)(Wn (a)
n (a)[Sn+1 n (a)Sn (a)) (1 + r)Sn (a)] ⇤
By induction the ﬁrst term = (1 + r)Vn (a). Letting qn (a) = 1 p⇤ (a), (6.10)
(1 + r)Sn (a) = p⇤ (a)Sn+1 (aH ) + qn (a)Sn+1 (aT )
n 186 CHAPTER 6. MATHEMATICAL FINANCE Subtracting this equation from Sn+1 (aH ) = Sn+1 (aH ) we have
Sn+1 (aH ) ⇤
(1 + r)Sn (a) = qn (a)[Sn+1 (aH ) Sn+1 (aT )] Using (6.13) now, we have
n (a)[Sn+1 ⇤
(1 + r)Sn (a)] = qn (a)[Vn (aH ) Vn+1 (aT )] Combining our results then using (6.12)
Wn+1 (aH ) = (1 + r)Vn (a) + qn (a)[Vn (aH ) Vn+1 (aT )]...
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