Capacitación del recurso humano

Bromwich integral involves the settings of two parameters a and x. Parameter a is set to the smallest integer greater than Max@2Ν +2, 0D so that no singularity is to the right of the contour. Parameter x

controls the truncation size of the integration. The Bromwich integral uses built-in Mathematica function "NIntegrate" to invert the Laplace transform. The precision used in internal computations of "NIntegrate" is specified by the "WorkingPrecision" option. With machine precision, Bromwich integral can produce up to 11-digit accuracy as shown in Table 4.9, and the accuracy does not decay as depicted in Figure 4.3.

Table 4.9. Results of Bromwich integration with machine precision

Bromwich : wp=MachinePrecision

Case CHP Reference x=1900 x=3500 x=5600

Accu Price Accu Price CPU Accu Price CPU Accu Price CPU

1 0.194 38 0.1931737903 10 0.19317 0.30 10 0.19317 0.30 10 0.19317 0.28 2 0.247 37 0.2464156905 10 0.24642 0.50 10 0.24642 0.50 10 0.24642 0.44 3 0.307 37 0.3062203648 10 0.30622 0.58 11 0.30622 0.51 10 0.30622 0.48 4 - 17 0.05598604154 0 0.14883 16.8 0 0.091635 20.2 0 0.069335 23.9 5 0.219 38 0.2183875466 6 0.21839 1.73 10 0.21839 1.81 11 0.21839 1.93 6 0.172 39 0.1722687410 9 0.17227 1.33 11 0.17227 1.33 11 0.17227 1.37 7 0.352 30 0.3500952190 9 0.35010 0.16 9 0.35010 0.16 9 0.35010 0.16 8 2.767 36 2.815862016 8 2.8159 1.03 10 2.8159 1.0 11 2.8159 1.0 9 2.273 38 2.310878887 10 2.3109 0.94 10 2.3109 0.90 11 2.3109 0.83 10 1.849 38 1.879023661 11 1.8790 0.83 11 1.8790 0.53 11 1.8790 0.50 11 7.779 36 7.895795199 9 7.8958 0.86 9 7.8958 0.86 9 7.8958 0.56 12 6.837 37 6.935422632 9 6.9354 0.51 10 6.9354 0.53 10 6.9354 0.50 13 5.987 38 6.070987190 9 6.0710 0.36 11 6.0710 0.36 9 6.0710 0.34 14 - 22 2.697871538 1 3.3204 25.2 2 2.7461 25.6 1 2.6300 28.5 15 - 20 1.134741432 0 2.2082 23.0 1 1.4996 25.6 1 1.2439 28.6 16 - 17 0.2853249387 0 1.3044 23.1 0 0.61838 26.1 0 0.38463 29.0 17 - 38 14.98395833 6 14.984 1.81 10 14.984 1.95 10 14.984 2.04 18 - 37 8.828758224 6 8.8288 1.67 11 8.8288 1.78 10 8.8288 1.86 19 - 38 4.696709132 6 4.6967 1.54 10 4.6967 1.59 11 4.6967 1.72

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

Bromwich: Bromwich integration

500 1000 1500 2000 2500 x 0.193174 0.193174 0.193174 0.193174 0.193174 Price

Figure 4.3. The price of Asian option Case 1 computed by Bromwich integral with machine precision

Using multi-precision, we can continue to increase the accuracy. However, we find the number of effective digits in the result computed by Bromwich integral can not accurately reflect the size of round- off error. Take the price of Case 1 with x=5600 and wp=20 in Table 4.10 as an example. The number

of effective digits in the result is 8 digits more than the accuracy. We could think the result is not constrained by the round-off error. But, this is not the case. If we increase the working precision to 40 digits with other factors fixed as in Table 4.11, it is surprising to discover that the accuracy increases from 12 significant digits to 31 significant digits. The cause of it could be associated with internal computations of built-in function “NIntegrate”. Therefore, we need to be careful when we set the working precision for Bromwich integral.

Using multi-precision, we can continue to increase the accuracy. However, we find the number of effective digits in the result computed by Bromwich integral can not accurately reflect the size of round- off error. Take the price of Case 1 with x=5600 and wp=20 in Table 4.10 as an example. The number

of effective digits in the result is 8 digits more than the accuracy. We could think the result is not constrained by the round-off error. But, this is not the case. If we increase the working precision to 40 digits with other factors fixed as in Table 4.11, it is surprising to discover that the accuracy increases from 12 significant digits to 31 significant digits. The cause of it could be associated with internal computations of built-in function “NIntegrate”. Therefore, we need to be careful when we set the working precision for Bromwich integral.

Table 4.10. Results of Bromwich integration with 20-digit precision

Bromwich

Case Reference

x=1900; wp=20 x=3500; wp=20 x=5600; wp=20 Accu Price ED Accu Price CPU ED Accu Price CPU ED Accu Price CPU 1 38 0.1931737903 20.0 15 0.19317 4.73 20.0 14 0.19317 4.65 20.0 12 0.19317 4.32 2 37 0.2464156905 20.0 13 0.24642 3.67 20.0 13 0.24642 5.85 20.0 11 0.24642 5.58 3 37 0.3062203648 20.0 13 0.30622 3.99 20.0 13 0.30622 6.24 20.0 12 0.30622 6.07 4 17 0.05598604154 20.0 0 0.14883 58.8 20.0 0 0.091635 65.4 20.0 0 0.069335 136 5 38 0.2183875466 20.0 6 0.21839 7.13 20.0 10 0.21839 11.8 20.0 16 0.21839 19.0 6 39 0.1722687410 20.0 9 0.17227 9.91 20.0 13 0.17227 9.89 20.0 13 0.17227 9.64 7 30 0.3500952190 20.0 12 0.35010 1.72 20.0 12 0.35010 2.70 20.0 10 0.35010 2.57 8 36 2.815862016 20.0 12 2.8159 5.26 20.0 11 2.8159 8.44 20.0 15 2.8159 8.39 9 38 2.310878887 20.0 12 2.3109 4.88 20.0 13 2.3109 7.99 20.0 11 2.3109 7.72 10 38 1.879023661 20.0 14 1.8790 4.48 20.0 12 1.8790 6.24 20.0 15 1.8790 6.18 11 36 7.895795199 20.0 12 7.8958 4.87 20.0 12 7.8958 7.91 20.0 14 7.8958 6.36 12 37 6.935422632 20.0 13 6.9354 3.81 20.0 13 6.9354 6.15 20.0 11 6.9354 6.04 13 38 6.070987190 20.0 14 6.0710 3.06 20.0 13 6.0710 4.85 20.0 11 6.0710 4.73 14 22 2.697871538 20.0 1 3.3204 43.5 20.0 2 2.7461 71.9 20.0 1 2.6300 77.5 15 20 1.134741432 20.0 0 2.2082 44.4 20.0 1 1.4996 68.7 20.0 1 1.2439 71.9 16 17 0.2853249387 20.0 0 1.3044 61.5 20.0 0 0.61838 66.6 20.0 0 0.38463 71.9 17 38 14.98395833 20.0 6 14.984 7.35 20.0 10 14.984 12.4 20.0 15 14.984 12.7 18 37 8.828758224 20.0 6 8.8288 11.5 20.0 11 8.8288 11.7 20.0 15 8.8288 11.9 19 38 4.696709132 20.0 6 4.6967 11.2 20.0 10 4.6967 11.3 20.0 15 4.6967 11.5 wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Bromwich: Bromwich integration.

Table 4.11. Results of Bromwich integration with 40-digit precision

Bromwich

Case Reference

x=1900; wp=40 x=3500; wp=40 x=5600; wp=40 Accu Price ED Accu Price CPU ED Accu Price CPU ED Accu Price CPU 1 38 0.1931737903 40.0 15 0.19317 16.2 40.0 22 0.19317 16.8 40.0 31 0.19317 17.5 2 37 0.2464156905 40.0 13 0.24642 19.7 40.0 22 0.24642 20.7 40.0 31 0.24642 21.8 3 37 0.3062203648 40.0 13 0.30622 21.1 40.0 21 0.30622 21.7 40.0 29 0.30622 23.5 4 17 0.05598604154 40.0 0 0.14883 178 40.0 0 0.091635 197 40.0 0 0.069335 11 082 5 38 0.2183875466 40.0 6 0.21839 37.1 40.0 10 0.21839 41.6 40.0 16 0.21839 2329 6 39 0.1722687410 40.0 9 0.17227 31.9 40.0 13 0.17227 37.6 40.0 20 0.17227 37.6 7 30 0.3500952190 40.0 20 0.35010 11.1 40.0 30 0.35010 13.4 40.0 30 0.35010 12.3 8 36 2.815862016 40.0 12 2.8159 29.8 40.0 20 2.8159 35.5 40.0 28 2.8159 34.8 9 38 2.310878887 40.0 12 2.3109 26.6 40.0 20 2.3109 27.8 40.0 27 2.3109 27.9 10 38 1.879023661 40.0 14 1.8790 24.4 40.0 20 1.8790 21.9 40.0 29 1.8790 22.7 11 36 7.895795199 40.0 12 7.8958 25.8 40.0 20 7.8958 26.8 40.0 28 7.8958 24.3 12 37 6.935422632 40.0 13 6.9354 21.0 40.0 20 6.9354 22.3 40.0 29 6.9354 22.6 13 38 6.070987190 40.0 14 6.0710 17.1 40.0 21 6.0710 18.4 40.0 29 6.0710 18.9 14 22 2.697871538 40.0 1 3.3204 204 40.0 2 2.7461 230 40.0 1 2.6300 245 15 20 1.134741432 40.0 0 2.2082 188 40.0 1 1.4996 209 40.0 1 1.2439 225 16 17 0.2853249387 40.0 0 1.3044 177 40.0 0 0.61838 195 40.0 0 0.38463 211 17 38 14.98395833 40.0 6 14.984 42.0 40.0 10 14.984 39.5 40.0 15 14.984 44.0 18 37 8.828758224 40.0 6 8.8288 36.5 40.0 11 8.8288 38.0 40.0 15 8.8288 42.2 19 38 4.696709132 40.0 6 4.6967 35.0 40.0 10 4.6967 36.5 40.0 15 4.6967 39.4 wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Bromwich: Bromwich integration.

For difficult cases, Bromwich integral requires much bigger truncation size. To achieve an accuracy of at least 5 digits, the truncation size is set to 29000 as in Table 4.12. Our code rounds all approximate real numbers to exact number before put them into "NIntegrate", and use the "WorkingPrecision" option to control the output precision. This gets rid of the annoying warning message when the internal precision is less than the working precision. However, the computing time increases dramatically as a result. The Bromwich integral takes an order of magnitude more computational time than the Euler method to get a similar accuracy.

For difficult cases, Bromwich integral requires much bigger truncation size. To achieve an accuracy of at least 5 digits, the truncation size is set to 29000 as in Table 4.12. Our code rounds all approximate real numbers to exact number before put them into "NIntegrate", and use the "WorkingPrecision" option to control the output precision. This gets rid of the annoying warning message when the internal precision is less than the working precision. However, the computing time increases dramatically as a result. The Bromwich integral takes an order of magnitude more computational time than the Euler method to get a similar accuracy.

Table 4.12. Results of Bromwich integration on the difficult cases with enhanced parameter settings

Bromwich

Case Reference x=29 000; wp=20

Accu Price ED Accu Price CPU

4 17 0.05598604154 20.0 5 0.055986 110

14 22 2.697871538 20.0 7 2.6979 119

15 20 1.134741432 20.0 6 1.1347 102

16 17 0.2853249387 20.0 7 0.28532 95.6

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

Bromwich: Bromwich integration