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Có nguồn gốc giới hạn trên và dưới giá trị của bất kỳ tuyên bố đội ngũ, chúng tôi chuyển ngay bây giờ để làm thế nào các giới hạn được sử dụng để mở rộng chức năng định giá thanh toán. Sửa chữa một tuyên bố đội ngũ Do đó N là không gian con của RS có kích thước bằng kích thước của M cộng với một và chứa M và z. | 5.4. THE EXTENSION 45 5.4 The Extension Having derived upper and lower bounds on the value of any contingent claim we turn now to how these bounds are used to extend the payoff pricing functional. Fix a contingent claim z 2 M. Define N by N z Az z 2 M and A 2 R . 5.22 Thus N is the subspace of RS that has dimension equal to the dimension of M plus one and contains M and z. It is the asset span of J 1 securities with payoffs x1 . xj and z. If there is no strong arbitrage equivalently if the payoff pricing functional q is positive then Proposition 5.3.4 implies that a finite value K can be chosen to satisfy 2 q z K rqu z . 5.23 We extend q to a linear functional on N in that we define Q N R by Q z Az q z Ak. 5.24 We now prove that Q as just defined is the desired positive extension of q. 5.4.1 Proposition If q M R is positive so is Q N R. Proof Let y 2N. Then y z Az 5.25 for some z 2 M and some A 2 R. Of the three possibilities for A suppose first that A 0. Then y 0 implies z -z. 5.26 A Applying q to both sides of 5.26 and using the implication of 5.4 that q is an increasing function there results z q z q -z . 5.27 A By Proposition 5.3.1 the functions q and q coincide on M. Since z A 2 M we have q -z A q -z A . Therefore 5.27 becomes z q z q -z . 5.28 A Since K q z 5.28 implies that K q -z 5.29 A or alternatively that q z AK 0. 5.30 Since the left-hand side of 5.30 equals Q y we obtain that Q y 0. If A 0 a similar argument but using qu and the fact that K qu z also gives Q y 0. Finally if A 0 then y z and Q y q z . The positivity of q implies that if y 0 then Q y 0. If there is no arbitrage equivalently if q is strictly positive then Proposition 5.3.5 implies that K can be chosen to satisfy q z K qu z . 5.31 Then 2 One can show that the assumption of no strong arbitrage implies that the lower and upper bounds cannot be both equal to I 1 or both equal to 1. 46 CHAPTER 5. VALUATION 5.4.2 Proposition If q M R is strictly positive so is Q N R. The proof is essentially .
