Numerical results of the Euler method

The Euler method involves the settings of three parameters: m, n and Γ. The sum of m and n controls the

truncation error, and Γ controls the discretization error. We set Γ =18 (equivalent to A=41.4) so that

there is no discretization error within 16-digit precision. The value of m is fixed at m=15, and the value

of n is allowed to vary. It is shown in Table 4.3 that the accuracy increases consistently with the truncation size n. Using machine precision, the accuracy increases with the value of n until it reaches a limit, for example 11 digits in case 3 of Table 4.3. After that, the accuracy does not improve anymore when we continue to increase n. Figure 4.1 shows the price of Asian option Case 1 converges and remain stable .

The Euler method involves the settings of three parameters: m, n and Γ. The sum of m and n controls the

truncation error, and Γ controls the discretization error. We set Γ =18 (equivalent to A=41.4) so that

there is no discretization error within 16-digit precision. The value of m is fixed at m=15, and the value

of n is allowed to vary. It is shown in Table 4.3 that the accuracy increases consistently with the truncation size n. Using machine precision, the accuracy increases with the value of n until it reaches a limit, for example 11 digits in case 3 of Table 4.3. After that, the accuracy does not improve anymore when we continue to increase n. Figure 4.1 shows the price of Asian option Case 1 converges and remain stable .

Table 4.3. Results of the Euler method using machine precision

Euler : m=15;Γ =18; wp=MachinePrecision

Case FMW CHP Reference n=14 n=31 n=45

Accu Price Accu Price CPU Accu Price CPU Accu Price CPU 1 0.194 0.194 38 0.1931737903 6 0.19317 0.03 12 0.19317 0.03 14 0.19317 0.03 2 0.247 0.248 37 0.2464156905 6 0.24642 0.03 12 0.24642 0.03 13 0.24642 0.05 3 0.307 0.308 37 0.3062203648 5 0.30622 0.03 11 0.30622 0.05 11 0.30622 0.03 4 0.0560 0.055 17 0.05598604154 3 0.055997 11.1 8 0.055986 20.9 13 0.055986 31.1 5 0.219 0.222 38 0.2183875466 8 0.21839 0.28 12 0.21839 0.50 12 0.21839 0.75 6 0.172 0.172 39 0.1722687410 6 0.17227 0.14 12 0.17227 0.23 14 0.17227 0.33 7 0.352 0.340 30 0.3500952190 6 0.35010 0.03 11 0.35010 0.02 13 0.35010 0.03 8 - 2.808 36 2.815862016 6 2.8159 0.08 11 2.8159 0.12 12 2.8159 0.16 9 - 2.305 38 2.310878887 6 2.3109 0.05 11 2.3109 0.06 14 2.3109 0.06 10 - 1.875 38 1.879023661 7 1.8790 0.03 11 1.8790 0.03 12 1.8790 0.05 11 - 7.903 36 7.895795199 6 7.8958 0.03 11 7.8958 0.05 12 7.8958 0.05 12 - 6.942 37 6.935422632 5 6.9354 0.03 11 6.9354 0.05 13 6.9354 0.05 13 - 6.077 38 6.070987190 6 6.0710 0.03 11 6.0710 0.03 13 6.0710 0.05 14 - 2.808 22 2.697871538 4 2.6982 10.3 6 2.6979 19.3 12 2.6979 28.5 15 - 1.129 20 1.134741432 7 1.1347 9.55 9 1.1347 17.8 12 1.1347 26.0 16 - 0.278 17 0.2853249387 5 0.28532 8.91 10 0.28532 16.6 13 0.28532 24.7 17 - 15.056 38 14.98395833 6 14.984 0.31 10 14.984 0.58 12 14.984 0.87 18 - 8.964 37 8.828758224 5 8.8288 0.27 13 8.8288 0.48 12 8.8288 0.73 19 - 4.700 38 4.696709132 6 4.6967 0.22 13 4.6967 0.42 13 4.6967 0.62 wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Euler: the Euler method.

10 20 30 40 50 n

0.193174 0.193174 0.193174

Price

Figure 4.1. The price of Asian option Case 1 computed by Euler method with machine precision

In multi-precision, the accuracy constantly improves with n given that discretization error and round-off error are controlled. For normal cases in Table 4.4, the settings of n=15 and wp=20 computes results

to accuracy of at least 5 digits within 0.4 CPU second, while the settings of n=31 and wp=20 gives at

least 10 digits accuracy within 0.7 CPU second. To achieve higher accuracy of 15 digits, the working precision is set to 25 digits to eliminate the impact of the round-off error on the accuracy. The Euler method takes less than 1 second for normal cases to achieve 15-digit accuracy with n=45. For difficult

cases, the method yields less accurate result but consumes considerably more CPU time with the same settings. Table 4.5 shows the Euler method takes between 9 to 12 seconds to have 5-digit accuracy for Case 4, 14, 15 and 16 with the settings of n=22 and wp=20.

In multi-precision, the accuracy constantly improves with n given that discretization error and round-off error are controlled. For normal cases in Table 4.4, the settings of n=15 and wp=20 computes results

to accuracy of at least 5 digits within 0.4 CPU second, while the settings of n=31 and wp=20 gives at

least 10 digits accuracy within 0.7 CPU second. To achieve higher accuracy of 15 digits, the working precision is set to 25 digits to eliminate the impact of the round-off error on the accuracy. The Euler method takes less than 1 second for normal cases to achieve 15-digit accuracy with n=45. For difficult

cases, the method yields less accurate result but consumes considerably more CPU time with the same settings. Table 4.5 shows the Euler method takes between 9 to 12 seconds to have 5-digit accuracy for Case 4, 14, 15 and 16 with the settings of n=22 and wp=20.

We suggest the working precision should be set such that the result has several more effective digits than its accuracy, say 5 more digits. If the difference between the accuracy and the effective digits is less than one digit, it will imply that the round-off error may dominate. In this case, increasing the working precision usually improves the accuracy.

Table 4.4. Results of the Euler method with arbitrary precision

Euler : m=15;Γ =18

Case Reference

n=14; wp=20 n=31; wp=20 n=45; wp=25 Accu Price ED Accu Price CPU ED Accu Price CPU ED Accu Price CPU 1 38 0.1931737903 16.5 6 0.19317 0.08 16.5 12 0.19317 0.12 21.5 15 0.19317 0.16 2 37 0.2464156905 16.5 6 0.24642 0.08 16.5 12 0.24642 0.12 21.5 15 0.24642 0.16 3 37 0.3062203648 16.5 5 0.30622 0.09 16.5 11 0.30622 0.12 21.5 16 0.30622 0.17 4 17 0.05598604154 16.5 3 0.055997 9.17 16.5 8 0.055986 17.1 21.5 12 0.055986 26.2 5 38 0.2183875466 16.5 8 0.21839 0.34 16.5 12 0.21839 0.59 21.5 17 0.21839 0.86 6 39 0.1722687410 16.5 6 0.17227 0.22 16.5 12 0.17227 0.36 21.5 16 0.17227 0.51 7 30 0.3500952190 16.5 6 0.35010 0.06 16.5 11 0.35010 0.08 21.5 14 0.35010 0.09 8 36 2.815862016 16.5 6 2.8159 0.16 16.5 11 2.8159 0.22 21.5 15 2.8159 0.33 9 38 2.310878887 16.5 6 2.3109 0.09 16.5 11 2.3109 0.16 21.5 16 2.3109 0.20 10 38 1.879023661 16.5 7 1.8790 0.08 16.5 11 1.8790 0.14 21.5 16 1.8790 0.19 11 36 7.895795199 16.5 6 7.8958 0.09 16.5 11 7.8958 0.12 21.5 15 7.8958 0.19 12 37 6.935422632 16.5 5 6.9354 0.09 16.5 11 6.9354 0.12 21.5 15 6.9354 0.17 13 38 6.070987190 16.5 6 6.0710 0.09 16.5 11 6.0710 0.12 21.5 15 6.0710 0.17 14 22 2.697871538 16.5 4 2.6982 8.61 16.5 6 2.6979 16.0 21.5 12 2.6979 23.7 15 20 1.134741432 16.5 7 1.1347 7.91 16.5 9 1.1347 14.8 21.5 13 1.1347 22.7 16 17 0.2853249387 16.5 5 0.28532 7.33 16.5 10 0.28532 13.7 21.5 12 0.28532 20.3 17 38 14.98395833 16.5 6 14.984 0.37 16.5 10 14.984 0.64 21.5 16 14.984 0.97 18 37 8.828758224 16.5 5 8.8288 0.33 16.5 12 8.8288 0.58 21.5 15 8.8288 0.84 19 38 4.696709132 16.5 6 4.6967 0.30 16.5 14 4.6967 0.53 21.5 16 4.6967 0.76 wp: working precision; ED: effective number of significant digits; Accu: accuracy measured by the number of significant digits; CPU: computing time in seconds.

Euler: the Euler method.

Table 4.5. Results of the Euler method on the difficult cases with enhanced parameter settings

Euler : m=15;Γ =18

Case Reference n=22; wp=20

Accu Price ED Accu Price CPU

4 17 0.05598604154 17.1 5 0.055986 12.2

14 22 2.697871538 17.1 5 2.6979 11.4

15 20 1.134741432 17.1 6 1.1347 10.5

16 17 0.2853249387 17.1 6 0.28533 9.73

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

Euler: the Euler method.