stochastic.229

# stochastic.229 - 228 1 V is given 2 Deﬁne X t 0  t  T...

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Unformatted text preview: 228 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 Xtt 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 risk-neutral 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 risk-neutral 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 risk-neutral pricing formula is valid provided there is a unique risk-neutral measure. A probability measure is said to be risk-neutral provided ...
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