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nif:isString """13.6 Empirical study using barcode generation for none control to generate, create none image in none applications. source code for asp.net projects with barcodes information manner. databar N1 2 M 1 V1 = M 1 i,i =1 Ji Ji e 1i 1i + 2 Gii + 2 Gi i +Gii + 1 1 1i. 1i + 1i e 1i + 2 Gii + N2 2 M 2 V2 e 1i + 2 Gi i + (13.38). 2j + 2j = M2. j ,j =a Jj Jj e 2j 2j + 1 Gjj + 1 Gj 2 2 +Gjj + e 2j + 2 Gjj + and e 2j + 1 Gj 2 (13.39). M1 M2 V12 = M1 M2 i,i =1 Ji Ji e 1i 1i + 1i + 2i 1 1 + 2 Gii + 2 Gi i +Gii + 1i + 1i 2i + 2i 1 e 1i + + 2 Gii + e 1i + 1 + 2 Gi i + 1j + 1j (13.40). 2j + 2j M1 M2 V22 = M1 M2 j ,j =a Jj Jj e 2j 2j 1j + 2j 1 1 + + 2 Gjj + 2 Gj +Gjj + e 2j + M1 M2 V1 V2 = M1 M2 1 + 2 Gjj + e 2j 1 + + 2 Gj (13.41). N1 N2 i=1 j =a Ji Jj e 1i 2j + 1i + 2i 1 1 + 2 Gii + 2 Gjj +Gij + 1i + 1j + 2i + 2j e 1i + 1 + 2 Gii + e 2j + 1 + 2 Gjj + 1j + 2j (13.42). Furthermore, M1 and M2 none none are not the coupon bond option prices if one consistently uses the market drift. Different from the one derived in [9], one has Di = 1 M i Vi Mi 1 Mi Vi2 Di2 ; Ai = Mi 13.6 Empirical study The correlation and auto-correlation of coupon bond options have been evaluated for both martingale and nonmartingale evolution of the bond forward interest rates (13. 43) i = 1, 2 (13.44). Correlation of coupon b ond options Table 13.1 Evaluating I = t0 dt m2 dx m3 dx M(t, x, x ) for different limits of integration. When an entry has the value of t, it means it is not a xed value and depends on the t integration. I m1 m2 m3 d1 d2. i m1 d1 d2 Gij t t t Ti Tj Gii t1 t1 t1 Ti Ti Gjj t2 t2 t2 Tj Tj ij t1 t1 t2 Ti Tj t t t t Ti t1 t t1 t1 Ti t1 t t1 t2 Ti t1 t t2 t1 Tj t1 t t2 t2 Tj by using analytical tec none none hniques; these results are now studied empirically. The structure of the ZCYC data and computational procedures discussed in 12 carry over to the empirical study of this section. The analytical results show that all the computations nally boil down to a set of three-dimensional integrations on M(x, x ; t) M(x, x ; t) = (t, x)D (x, x ; t) (t, x ) with various integration limits. A general form for all the integrations, similar to Eq. (12.13), is given by. m1 d1 d2 t0 m2 dx M(x, x ; t). (13.45). with the limits of inte grations being listed in the Table 13.1. The evaluation I , given in Eq. (13.45), is similar to the computation carried out in Section 12.7, except now, as can be seen from Table 13. 1, the limits on I are more complicated. A set of two-dimensional integrations on (t, x) is required for evaluating market drift and which has the general form. m1 d1 t0 m2 dx (t, x). (13.46). with the limits of inte none for none gration being listed in Table 13.2. As discussed in Section 4. 9 swaptions are equivalent to a special class of coupon bond options. Swaption data will be used to empirically study the formulae obtained for the correlation of coupon bond options. A swaption s price is equivalent to a coupon bond option, which will be taken to mature at t ; the indices i and j run from 1 to N , with the last payment being made at TN . For the correlation, swaptions will. 13.6 Empirical study Ta ble 13.2 Limits of integration for evaluating D = t0 dt m2 dx (t, x). When m2 has the value of t, it means it is not a xed value and depends on the t integration. D m1 m2 d1 t1 t t1. m1 d1 t2 t t2 1i t1 t1 Ti 2j t2 t2 Tj mature at two different none for none times t2 t1 and the two indices i and j have the following rages: i = 1, 2, . . . , N 1 and j = a, a + 1, . . . , N 2; the last payments are made at TN1 and TN2 respectively. Since M(x, x ; t) = 1 f (t, x) f (t, x ). (13.47). one has, similar to Eq. (12.16), the following t0 m2 dx f (t, x) f (t, x ). (13.48). A shift on integration none for none domain, as discussed in Section 12.7, converts integration on future data into a summation on current and past data as follows (t = t0 + tk ; tk = k ). tk d1 tk d2 tk m2 tk d1 tk m3 tk d2 tk dy M(y + tk , y + tk ; t0 + tk ) dy M(y, y ; t0 ). m2 tk d1 tk m3 tk d2 tk f (t0 , y)dy m2 tk m3 tk f (t0 , y )dy (13.49). The shift in the integr none none ation over future calendar time is shown in Figure 12.2. In order to use the ZCYC data directly, the operator t0 needs to be taken outside the integration. As can be seen from the list of integration limits given in Table 13.1, one of the lower limits of integration can be the integration variable t; hence, unlike the case analyzed in Section 12.7, two different results are possible and are shown below. ."""^^xsd:string;
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