7.15. Let Bt be 1-dimensional and define + F(w) = (Br(W) - K) = where K > 0, T> 0 are constants. By the Itô representati

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7 15 Let Bt Be 1 Dimensional And Define F W Br W K Where K 0 T 0 Are Constants By The Ito Representati 1 (165.97 KiB) Viewed 40 times

7.15. Let Bt be 1-dimensional and define + F(w) = (Br(W) - K) = where K > 0, T> 0 are constants. By the Itô representation theorem (Theorem 4.3.3) we know that there exists E V(0,T) such that T F(w) = E[F] + ott, wdB! - 0 How do we find o explicitly? This problem is of interest in mathe- matical finance, where o may be regarded as the replicating portfolio for the contingent claim F (see Chapter 12). Using the Clark-Ocone formula (see Karatzas and Ocone (1991) or Øksendal (1996)) one can deduce that olt,w) = E(X[K, ..) (BT)| Ft] ; = t<T. (7.5.3) Use (7.5.3) and the Markov property of Brownian motion to prove that for t < T we have 1 Ꮖ øſt, w) = = - PC exp (C - B (w))2 dx . 2(T – t) 2) (7.5.4) 27(T-t) K