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An Introduction to Financial Option Valuation_11

Tham khảo tài liệu 'an introduction to financial option valuation_11', tài chính - ngân hàng, tài chính doanh nghiệp phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 23.5 FTCS andBTCS 241 T 000000000 000000000 000000000 000000000 000000000 t 000000000 000000000 000000000 000000000 000000000 0 L x Fig. 23.2. Finite difference grid jh ik N 0 N . Points are spaced at a distance of h apart in the x-direction and k apart in the t-direction. A simple method for the heat equation 23.2 involves approximating the time derivative d dt by the scaled forward difference in time k-1 At and the second order space derivative d1 dx2 by the scaled second order central difference in space h-2ỗ2. This gives the equation k-1AtUj - h-2s2ưj 0 which may be expanded as ưj 1 - ưj ưj 1 - 2ưi ư-1 k h2 A more revealing re-write is ưj 1 vưj 1 1 - 2v ưj vưj-1 23.7 where V k h2 is known as the mesh ratio. Suppose that all approximate solution values at time level i ưj N are known. Now note that ư 1 a i 1 k and ư 1 b i 1 k are given by the boundary conditions 23.4 . Equation 23.7 then gives a formula for computing all other approximate values at time level i 1 that is ưj 1 N -1. Since we 242 Finite difference methods Fig. 23.3. Stencil for FTCS. Solid circles indicate the location of values that must be known in order to obtain the value located at the open circle. are supplied with the time-zero values U0 g jh from 23.3 this means that the complete set of approximations Uj i jff0 N 0 can be computed by stepping forward in time. The method defined by 23.7 is known as FTCS which stands for forward difference in time central difference in space. Figure 23.3 illustrates the stencil for FTCS. Here the solid circles indicate the location of values Uj_ 1 Uj ri -ni 1 and Uj 1 that must be known in order to obtain the value Uj 1 located at the open circle. We may collect all the interior values at time level i into a vector U1 U2 ừ G RNx-1. 23.8 Ui UNx -1 Exercise 23.3 then asks you to confirm that FTCS may be written U 1 FU pi for0 i Nt - 1 23.9 with g h g 2h U0 G RNx-1 g Nx - 1 h 23.5 FTCS andBTCS 243 where the matrix F has the form 1 - 2v V 0. 0 F V 1 2v V 0 0 . .