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1. V is given, 2. Deﬁne X t; 0 t T , by (3.3) (not by (3.1) or (3.2), because we do not yet have t). 3. Construct t so that (3.2) (or equivalently, (3.1)) is satisﬁed by the
deﬁned in step 2. X t; 0 t T , To carry out step 3, we ﬁrst use the tower property to show that Xtt deﬁned by (3.3) is a martingale
f
under I . We next use the corollary to the Martingale Representation Theorem (Homework Problem
P
4.5) to show that d X t e
t = t dB t (3.4) for some proecss . Comparing (3.4), which we know, and (3.2), which we want, we decide to
deﬁne t = t t :
S t
Then (3.4) implies (3.2), which implies (3.1), which implies that X t; 0
the portfolio process t; 0 t T . (3.5) t T , is the value of From (3.3), the deﬁnition of X , we see that the hedging portfolio must begin with value
f V ;
X 0 = IE T and it will end with value
V
f V
X T = T IE T F T = T T = V:
Remark 22.1 Although we have taken r and to be constant, the riskneutral pricing formula is
still “valid” when r and are processes adapted to the ﬁltration generated by B . If they depend on
e
either B or on S , they are adapted to the ﬁltration generated by B . The “validity” of the riskneutral
pricing formula means:
f V ;
X 0 = IE T
then there is a hedging portfolio t; 0 t T , such that X T = V ; 1. If you start with 2. At each time t, the value X t of the hedging portfolio in 1 satisﬁes X t = IE V F t :
f
t
T Remark 22.2 In general, when there are multiple assets and/or multiple Brownian motions, the
riskneutral pricing formula is valid provided there is a unique riskneutral measure. A probability
measure is said to be riskneutral provided ...
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 Fall '15
 mr.somebody
 Probability theory, riskneutral pricing formula

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