Análisis de antecedentes investigativos

Before implement the constructive complex analysis, we need to choose the break point B for decomposition of Ia,cH0,¥L, the number of terms Na of the asymptotic expansion, and the number of terms Ns of the series expansion. This is done by minimizing the error estimate of ΓΝ,qI with respect to

B given fixed Na and Ns with the working precision equal to 20 decimal digits. First of all, Na is chosen so that the local error estimate is minimum given some initial Ns. Next, Ns is selected so that the local error estimate is below the desired accuracy, i.e. 17 decimal places, given Na in the first step. After that, we may refine our choice of Na and Ns by repeating the first two step until the global minimum of error estimate is achieved. Finally, B is determined by minimizing the error estimate with respect to B given refined Na and Ns. The minimized error estimate is simply the maximal error of ΓΝ,qI.

Given the desired accuracy for ΓΝ,qI is 17 decimal places, we first determine B,NaandNs for each case.

Then, with the determined settings, we obtain the maximal error of ΓΝ,qI, the maximal error of ΓΝ,qR,

the maximal error of CHΝLHh,qL, and maximal error of Ct. From Table 4.34, the maximal error of ΓΝ,qI is indeed below 17 decimal places. The maximal error of ΓΝ,qR affected by q,ΝandB where q plays a

leading role. The accuracy of ΓΝ,qR soars to 95 decimal places as q reduces to 0.0025. However the

accuracy is poor when q is large. In seventh case, the accuracy for ΓΝ,qR is only one decimal place. The

final maximal error of Ct is determined by the largest one between the maximal errors of ΓΝ,qI and the

maximal errors of ΓΝ,qR. For example, the maximal error of ΓΝ,qI which is chosen to be just below

5´10-18 _{exceeds the maximal error of Γ}

Ν,qR for difficult cases. Since the maximal errors of ΓΝ,qR is

fixed given specific B, Na, and Ns, constructive complex analysis cannot achieve any level of accuracy.

To use constructive complex analysis for pricing, we need to specify one more parameter K0. The

parameter K0 is the number of terms of the summation in the term A∆,ΛHaL which appears in the

asymptotic expansion of Ia,cHB,¥L. Required value of K0 increases when q decreases. With any other

parameters fixed, rising K0 does not increase CPU time considerably. Hence, we use K0=40 for normal

cases, while use K0=100 for difficult cases.

To use constructive complex analysis for pricing, we need to specify one more parameter K0. The

parameter K0 is the number of terms of the summation in the term A∆,ΛHaL which appears in the

asymptotic expansion of Ia,cHB,¥L. Required value of K0 increases when q decreases. With any other

parameters fixed, rising K0 does not increase CPU time considerably. Hence, we use K0=40 for normal

cases, while use K0=100 for difficult cases.

To summarize, the maximal error of ΓΝ,qI can be controlled, while the maximal error of ΓΝ,qR is fixed

once B, Na, and Ns are determined. Therefore, the accuracy of constructive complex analysis is limited by the maximal error of ΓΝ,qR. Regarding the accuracy rather than the efficiency, this procedure can

obtain good accuracy as q is low, but works poorly as q is large, i.e. q>0.05.

Table 4.34. Results of constructive complex analysis with arbitrary precision

Constructive Complex Analysis

Case B Na Ns ME of ΓΝ,qI ME of ΓΝ,qR ME of CHΝL ME of Ct K0 wp ED Accu Price CPU

1 1.5454 19 267 4.77´10-18 0.000165 0.000165 0.00476 40 30 9.40 3 0.19286 19.5 2 1.5235 19 269 4.72´10-18 0.000104 0.000104 0.00315 40 30 9.20 3 0.24621 19.6 3 1.5028 19 271 4.50´10-18 0.0000653 0.0000653 0.00209 40 30 9.00 3 0.30608 19.8 4 0.8973 145 1921 4.66´10-18 _7.05´10-95 _4.66´10-18 _3.66´10-15 _{100 260 20.2 17 0.055986 2557} 5 1.1586 22 366 4.61´10-18 _2.14´10-15 _2.14´10-15 _1.59´10-13 _{40 60 25.8 12 0.21839 57.5} 6 1.2531 23 313 4.72´10-18 _1.48´10-8 _1.48´10-8 _9.21´10-7 _{40 50 21.8 7 0.17227 39.8} 7 1.8758 20 272 4.99´10-18 _{0.0137 0.0137 0.199 40 25 5.50 0 0.33717 15.3} 8 1.4542 19 278 4.90´10-18 1.00´10-6 1.00´10-6 0.000260 40 30 8.50 4 2.8158 20.4 9 1.4827 19 277 4.44´10-18 1.92´10-6 1.92´10-6 0.000498 40 30 8.70 4 2.3108 20.3 10 1.5076 19 275 4.46´10-18 3.42´10-6 3.42´10-6 0.000891 40 30 8.90 4 1.8789 20.2 11 1.4917 19 274 4.56´10-18 8.81´10-6 8.81´10-6 0.00701 40 30 9.10 2 7.8948 19.7 12 1.5091 19 273 4.43´10-18 0.0000129 0.0000129 0.0103 40 30 9.20 2 6.9339 19.5 13 1.5239 19 271 4.85´10-18 0.0000184 0.0000184 0.0146 40 30 9.40 3 6.0689 19.3 14 0.8598 122 1712 4.87´10-18 _7.36´10-97 _4.87´10-18 _4.76´10-14 _{100 235 19.0 19 2.6979 2113} 15 0.8997 123 1726 4.33´10-18 _5.67´10-91 _4.33´10-18 _4.23´10-14 _{100 235 20.6 20 1.1347 2146} 16 0.9345 123 1731 4.23´10-18 _7.78´10-86 _4.23´10-18 _4.13´10-14 _{100 235 22.3 17 0.28532 2215} 17 1.1081 25 364 4.40´10-18 _1.04´10-14 _1.04´10-14 _4.22´10-11 _{40 60 25.3 12 14.984 50.8} 18 1.1594 25 363 4.30´10-18 1.50´10-13 1.50´10-13 6.10´10-10 40 60 25.8 11 8.8288 50.2 19 1.2014 24 359 5.00´10-18 1.35´10-12 1.35´10-12 5.49´10-9 40 60 26.3 10 4.6967 49.8 wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

It is essential to use multi-precision when implement constructive complex analysis. If we choose machine precision, this approach fails completely as shown in Table 4.35.

Table 4.35. Results of constructive complex analysis with machine precision

Constructive Complex Analysis : wp=MachinePrecision

Case Reference B Na Ns K0 Accu Price CPU

Accu Price 1 38 0.1931737903 1.5454 19 267 40 0 88.263 15.4 2 37 0.2464156905 1.5235 19 269 40 0 91.506 15.4 3 37 0.3062203648 1.5028 19 271 40 0 349.12 15.1 4 17 0.05598604154 0.8973 145 1921 100 0 -1.1436´10220 219 5 38 0.2183875466 1.1586 22 366 40 0 2.4755´1015 19.5 6 39 0.1722687410 1.2531 23 313 40 0 -7.2484´108 22.9 7 30 0.3500952190 1.8758 20 272 40 0 -15.842 12.2 8 36 2.815862016 1.4542 19 278 40 0 19 117. 17.9 9 38 2.310878887 1.4827 19 277 40 0 20 056. 17.8 10 38 1.879023661 1.5076 19 275 40 0 5756.6 17.9 11 36 7.895795199 1.4917 19 274 40 0 6904.9 9.13 12 37 6.935422632 1.5091 19 273 40 0 2267.3 9.13 13 38 6.070987190 1.5239 19 271 40 0 4315.3 8.99 14 22 2.697871538 0.8598 122 1712 100 0 4.7710´10198 169 15 20 1.134741432 0.8997 123 1726 100 0 -1.7882´10196 173 16 17 0.2853249387 0.9345 123 1731 100 0 -8.7857´10193 198 17 38 14.98395833 1.1081 25 364 40 0 5.8133´1017 18.8 18 37 8.828758224 1.1594 25 363 40 0 -1.1728´1017 18.9 19 38 4.696709132 1.2014 24 359 40 0 2.2044´1016 18.7

wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.