1 Introduction
2 Approximating the call function when \(S \ne X\)
2.1 The standardized call function
2.2 Construction of the approximating functions
2.3 The approximating call functions
3 Approximating the call function when \(S=X\)
4 Approximation of the implied volatility
4.1 Case \(S \ne X\)
4.2 Case \(S=X\)
5 Numerical results
5.1 Call when \(S \ne X\)
5.1.1 On the estimation error
\(x\backslash {}\alpha \) | Absolute difference \(\bigl |E_\alpha (x)\bigr |\) | ||||
---|---|---|---|---|---|
0.25 | 0.5 | 0.75 | 1 | 1.25 | |
0.25 | 0.004525 | 0.003069 | 0.000584 | 0.000795 | 0.001040 |
0.5 | 0.001838 | 0.002137 | 0.001334 | 0.000147 | 0.001914 |
0.75 | 0.000217 | 0.000288 | 0.000226 | 0.000037 | 0.000276 |
1.25 | 0.000175 | 0.000242 | 0.000201 | 0.000050 | 0.000216 |
1.5 | 0.001262 | 0.001705 | 0.001365 | 0.000287 | 0.001461 |
1.75 | 0.003859 | 0.005034 | 0.003795 | 0.000568 | 0.004081 |
2 | 0.008332 | 0.010381 | 0.007245 | 0.000558 | 0.007817 |
2.25 | 0.014883 | 0.017547 | 0.011177 | 0.000026 | 0.012043 |
2.5 | 0.023587 | 0.026101 | 0.014991 | 0.001273 | 0.016040 |
\(x\backslash {}\alpha \) | Percent change \(\bigl |E_\alpha (x)\bigr |/\chi _\alpha (x) \) | ||||
---|---|---|---|---|---|
0.25 (%) | 0.5 (%) | 0.75 (%) | 1 (%) | 1.25 (%) | |
0.25 | 36.0435 | 27.7193 | 9.2665 | 29.3629 | 113.4827 |
0.5 | 5.0593 | 4.0755 | 2.3805 | 0.2809 | 4.2354 |
0.75 | 0.3561 | 0.2862 | 0.1798 | 0.0261 | 0.1876 |
1.25 | 0.1580 | 0.1207 | 0.0732 | 0.0147 | 0.0551 |
1.5 | 0.9298 | 0.6837 | 0.3935 | 0.0664 | 0.2883 |
1.75 | 2.4047 | 1.6914 | 0.9115 | 0.1094 | 0.6700 |
2 | 4.5021 | 3.0110 | 1.5023 | 0.0931 | 1.1220 |
2.25 | 7.1052 | 4.4929 | 2.0547 | 0.0039 | 1.5645 |
2.5 | 10.0943 | 6.0030 | 2.4937 | 0.1739 | 1.9350 |
\(\alpha \) | 0.25 | 0.5 | 0.75 | 1 | 1.25 | 1.5 |
\(G_{\alpha }\) | 0.00453 | 0.00356 | 0.00158 | 0.00091 | 0.00244 | 0.00386 |
5.1.2 Monte Carlo analysis on the estimation error
S/X | \(\bigl |\chi _\alpha (x)- \widehat{\chi }_\alpha (x)\bigr |\) | ||||||||
---|---|---|---|---|---|---|---|---|---|
\(\sigma =10\%\) | \(\sigma =20\%\) | \(\sigma =30\%\) | |||||||
1 month | 3 months | 6 months | 1 month | 3 months | 6 months | 1 month | 3 months | 6 months | |
0.5 | 1.79E−130 | 1.72E−15 | 3.13E−11 | 1.53E−13 | 3.23E−08 | 3.13E−11 | 0.00E+00 | 2.41E−09 | 8.72E−06 |
0.75 | 2.72E−13 | 3.55E−08 | 3.59E−06 | 3.05E−07 | 4.78E−05 | 3.59E−06 | 2.79E−78 | 2.12E−05 | 5.31E−05 |
1.25 | 3.62E−11 | 5.21E−07 | 1.02E−05 | 2.51E−06 | 6.58E−05 | 1.02E−05 | 4.82E−49 | 1.58E−06 | 9.43E−04 |
1.5 | 5.55E−17 | 2.49E−10 | 1.27E−07 | 4.31E−09 | 9.88E−06 | 1.27E−07 | 1.90E−150 | 1.99E−06 | 1.88E−04 |
3 | 4.32E−320 | 1.04E−109 | 2.78E−16 | 2.59E−83 | 9.62E−12 | 2.78E−16 | 0.00E+00 | 1.96E−13 | 4.29E−08 |
6 | 0.00E+00 | 5.34E−284 | 4.01E−144 | 4.24E−214 | 5.83E−74 | 4.01E−144 | 0.00E+00 | 2.20E−97 | 4.60E−12 |
Interval of \(\sigma \) | Error | |||||
---|---|---|---|---|---|---|
Mean | Median | Maximum | Minimum | Standard deviation | Root mean squared error | |
[0.0, 1.25] | 0.000101 | 0.000000 | 0.004515 | \(-\) 0.001729 | 0.000499 | 0.000510 |
[0.0, 0.25] | 0.000025 | 0.000000 | 0.004309 | \(-\) 0.000013 | 0.000143 | 0.000145 |
[0.25, 0.5] | 0.000173 | 0.000000 | 0.004515 | \(-\) 0.000186 | 0.000505 | 0.000535 |
[0.5, 0.75] | 0.000212 | 0.000000 | 0.003899 | \(-\) 0.000586 | 0.000553 | 0.000595 |
[0.75, 1.0] | 0.000123 | 0.000000 | 0.002970 | \(-\) 0.001128 | 0.000511 | 0.000529 |
[1.0, 1.25] | \(-\) 0.000026 | 0.000000 | 0.002137 | \(-\) 0.001729 | 0.000509 | 0.000512 |
Interval of \(\sigma \) | Error | |||||
---|---|---|---|---|---|---|
Mean | Median | Maximum | Minimum | Standard deviation | Root mean squared error | |
[0.0, 1.25] | \(-\) 0.000125 | 0.000000 | 0.004506 | \(-\) 0.008262 | 0.000713 | 0.000725 |
[0.0, 0.25] | 0.000083 | 0.000000 | 0.004506 | \(-\) 0.000121 | 0.000339 | 0.000350 |
[0.25, 0.5] | 0.000198 | 0.000000 | 0.004110 | \(-\) 0.000838 | 0.000548 | 0.000586 |
[0.5, 0.75] | \(-\) 0.000005 | 0.000000 | 0.002540 | \(-\) 0.001862 | 0.000516 | 0.000519 |
[0.75, 1.0] | \(-\) 0.000296 | 0.000000 | 0.001336 | \(-\) 0.002894 | 0.000648 | 0.000718 |
[1.0, 1.25] | \(-\) 0.000605 | \(-\) 0.000005 | 0.000598 | \(-\) 0.008262 | 0.000907 | 0.001109 |
Interval of \(\sigma \) | Error | |||||
---|---|---|---|---|---|---|
Mean | Median | Maximum | Minimum | Standard deviation | Root mean squared error | |
[0.0, 1.25] | \(-\) 0.000455 | \(-\) 0.000001 | 0.004502 | \(-\) 0.037686 | 0.001340 | 0.001417 |
[0.0, 0.25] | 0.000126 | 0.000000 | 0.004502 | \(-\) 0.000359 | 0.000425 | 0.000444 |
[0.25, 0.5] | 0.000084 | 0.000000 | 0.003425 | \(-\) 0.001694 | 0.000526 | 0.000536 |
[0.5, 0.75] | \(-\) 0.000311 | 0.000000 | 0.001490 | \(-\) 0.003176 | 0.000671 | 0.000747 |
[0.75, 1.0] | \(-\) 0.000763 | \(-\) 0.000024 | 0.000470 | \(-\) 0.016198 | 0.001112 | 0.001368 |
[1.0, 1.25] | \(-\) 0.001410 | \(-\) 0.000233 | 0.000088 | \(-\) 0.037686 | 0.002200 | 0.002623 |
5.2 Call when \(S = X\)
x | \(\Theta (x)\) (%) | Fairclough (%) | Page (%) | Eidous(%) |
---|---|---|---|---|
0.01 | 0.000000 | 0.083704 | 0.114696 | 0.017498 |
0.1 | 0.000002 | 0.080783 | 0.111039 | 0.012391 |
0.25 | 0.000080 | 0.066133 | 0.092664 | 0.109891 |
0.5 | 0.001544 | 0.024474 | 0.039932 | 0.377435 |
0.6 | 0.003380 | 0.006901 | 0.017253 | 0.502277 |
0.62 | 0.003886 | 0.003610 | 0.012949 | 0.527200 |
0.64 | 0.004446 | 0.000427 | 0.008764 | 0.551931 |
0.66 | 0.005061 | 0.002635 | 0.004712 | 0.576396 |
0.68 | 0.005734 | 0.005562 | 0.000808 | 0.600518 |
0.7 | 0.006468 | 0.008342 | 0.002931 | 0.624222 |
0.8 | 0.011104 | 0.019689 | 0.018745 | 0.733905 |
0.9 | 0.017401 | 0.026110 | 0.028925 | 0.822677 |
1 | 0.025192 | 0.027257 | 0.032985 | 0.883538 |
1.02 | 0.026892 | 0.026874 | 0.033078 | 0.891911 |
1.04 | 0.028629 | 0.026299 | 0.032942 | 0.898954 |
1.1 | 0.033995 | 0.023502 | 0.031248 | 0.911956 |
1.5 | 0.062581 | 0.015386 | 0.007017 | 0.716576 |
2 | 0.045533 | 0.030125 | 0.027105 | 0.239427 |
2.5 | 0.012129 | 0.009982 | 0.009537 | 0.033727 |
RMSE | 0.017005 | 0.024249 | 0.033899 | 0.305931 |
5.2.1 Monte Carlo analysis on the estimation error
Interval of \(\sigma \) | Error | |||||
---|---|---|---|---|---|---|
Mean | Median | Maximum | Minimum | Standard deviation | Root mean squared error | |
[0.0, 1.25] | \(-\) 1.23405E−06 | \(-\) 5.75226E−08 | 5.42101E−20 | \(-\) 0.00002423 | 2.95917E−06 | 3.20618E−06 |
[0.0, 0.25] | \(-\) 3.31189E−10 | \(-\) 1.75199E−11 | 1.35525E−20 | \(-\) 6.31596E−09 | 7.7678E−10 | 8.44437E−10 |
[0.25, 0.5] | \(-\) 2.17191E−08 | \(-\) 7.24392E−09 | \(-\) 2.2806E−12 | \(-\) 2.17943E−07 | 3.36263E−08 | 4.00306E−08 |
[0.5, 1.75] | \(-\) 2.41594E−07 | \(-\) 1.19167E−07 | \(-\) 7.29428E−11 | \(-\) 1.78407E−06 | 3.10627E−07 | 3.93518E−07 |
[0.75, 1.0] | \(-\) 1.29135E−06 | \(-\) 7.05183E−07 | \(-\) 5.55995E−10 | \(-\) 7.96528E−06 | 1.51968E−06 | 1.99425E−06 |
[1.0, 1.25] | \(-\) 4.603E−06 | \(-\) 2.62831E−06 | \(-\) 2.35745E−09 | \(-\) 2.48933E−05 | 5.13655E−06 | 6.89723E−06 |
Time to expiration | True volatility | |||||
---|---|---|---|---|---|---|
10% | 35% | 55% | ||||
\(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | |
0.1 | NA | 6.58 | NA | 34.10 | NA | 52.40 |
0.2 | NA | 8.05 | NA | 33.62 | 43.58 | 49.87 |
0.3 | NA | 9.28 | NA | 32.75 | 54.91 | 48.10 |
0.4 | NA | 9.76 | 34.87 | 31.94 | 61.22 | 46.82 |
0.5 | NA | 9.94 | 41.03 | 31.22 | 65.98 | 45.82 |
0.6 | NA | 9.97 | 44.55 | 30.58 | 69.93 | 45.01 |
0.7 | NA | 9.97 | 47.22 | 30.00 | 73.39 | 44.33 |
0.8 | NA | 9.95 | 49.44 | 29.47 | 76.51 | 43.74 |
0.9 | NA | 9.92 | 51.39 | 28.98 | 79.36 | 43.21 |
1.0 | NA | 9.86 | 53.15 | 28.52 | 82.02 | 42.74 |
1.1 | NA | 9.80 | 54.76 | 28.09 | 84.50 | 42.29 |
1.2 | NA | 9.72 | 56.25 | 27.68 | 86.85 | 41.88 |
1.3 | NA | 13.19 | 57.66 | 27.29 | 89.08 | 41.48 |
1.4 | NA | 12.98 | 58.99 | 26.91 | 91.21 | 41.09 |
1.5 | NA | 12.76 | 60.26 | 26.54 | 93.25 | 40.71 |
Average volatility | NA | 9.41 | 50.80 | 29.85 | 75.13 | 44.63 |
Average error | NA | \(-\) 0.59 | 15.80 | \(-\) 5.15 | 20.13 | \(-\) 10.37 |
Std. Dev. error | NA | 0.91 | 7.41 | 2.37 | 14.11 | 3.30 |
Time to expiration | True volatility | |||||
---|---|---|---|---|---|---|
75% | 95% | 125% | ||||
\(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | |
0.1 | 40.82 | 68.64 | 66.50 | 83.93 | 94.13 | 106.24 |
0.2 | 66.63 | 64.88 | 86.37 | 79.63 | 114.63 | 102.02 |
0.3 | 76.60 | 62.78 | 97.30 | 77.56 | 127.47 | 100.40 |
0.4 | 83.70 | 61.42 | 105.57 | 76.35 | 137.48 | 99.56 |
0.5 | 89.48 | 60.44 | 112.46 | 75.53 | 145.87 | 99.02 |
0.6 | 94.48 | 59.69 | 118.47 | 74.93 | 153.19 | 98.57 |
0.7 | 98.93 | 59.08 | 123.85 | 74.43 | 159.70 | 98.15 |
0.8 | 102.99 | 58.56 | 128.75 | 73.99 | 165.59 | 97.71 |
0.9 | 106.74 | 58.10 | 133.26 | 73.57 | 126.79 | 97.25 |
1.0 | 110.22 | 57.67 | 137.46 | 73.17 | 126.69 | 96.75 |
1.1 | 113.50 | 57.27 | 141.38 | 72.76 | 126.65 | 96.21 |
1.2 | 116.59 | 56.88 | 145.07 | 72.33 | 126.65 | 95.63 |
1.3 | 119.52 | 56.48 | 96.08 | 71.89 | 126.69 | 95.01 |
1.4 | 122.32 | 56.09 | 95.98 | 71.42 | 126.77 | 94.34 |
1.5 | 124.98 | 55.68 | 95.90 | 70.93 | 126.89 | 93.63 |
Average volatility | 97.83 | 59.58 | 112.29 | 74.83 | 132.35 | 98.03 |
Average error | 22.83 | \(-\) 15.42 | 17.29 | \(-\) 20.17 | 7.35 | \(-\) 26.97 |
Std. Dev. error | 22.54 | 3.48 | 22.02 | 3.34 | 17.35 | 3.12 |
5.3 Implied volatility
5.3.1 Case when \(S \ne X\)
Time to expiration | True volatility | |||
---|---|---|---|---|
5% | 10% | |||
\(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | |
0.1 | NA | 6.35 | NA | 11.92 |
0.2 | NA | 6.34 | NA | 10.76 |
0.3 | NA | 6.15 | 9.49 | 10.00 |
0.4 | NA | 5.95 | 9.74 | 9.44 |
0.5 | NA | 5.77 | 9.84 | 9.01 |
0.6 | NA | 5.60 | 9.89 | 8.66 |
0.7 | NA | 5.45 | 9.91 | 8.36 |
0.8 | 4.41 | 5.32 | 9.93 | 8.11 |
0.9 | 4.62 | 5.19 | 9.94 | 7.89 |
1 | 4.73 | 5.08 | 9.95 | 7.69 |
1.1 | 4.80 | 4.97 | 9.96 | 7.51 |
1.2 | 4.84 | 4.87 | 9.96 | 7.35 |
1.3 | 4.87 | 4.78 | 9.97 | 7.20 |
1.4 | 4.90 | 4.70 | 9.97 | 7.06 |
1.5 | 4.91 | 4.61 | 9.97 | 6.94 |
Average volatility | 4.76 | 5.41 | 9.89 | 8.53 |
Average error | \(-\) 0.24 | 0.41 | \(-\) 0.11 | \(-\) 1.47 |
Std. Dev. error | 0.16 | 0.57 | 0.13 | 1.41 |
5.3.2 Case when \(S = X\)
Time to expiration | True volatility | |||||
---|---|---|---|---|---|---|
10% | 35% | 55% | ||||
\(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | |
0.1 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.2 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.3 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.4 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.5 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.6 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.7 | 10.00 | 10.00 | 35.00 | 35.00 | 55.00 | 55.00 |
0.8 | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
0.9 | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
1.0 | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
1.1 | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
1.2 | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
1.3 | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
1.4 | 10.00 | 10.00 | 35.00 | 35.00 | 55.02 | 55.00 |
1.5 | 10.00 | 10.00 | 35.00 | 35.00 | 55.02 | 55.00 |
Average volatility | 10.00 | 10.00 | 35.00 | 35.00 | 55.01 | 55.00 |
Average error | 0.00 | 0.00 | 0.00 | 0.01 | 0.00 | 0.00 |
Std. Dev. error | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
Time to expiration | True volatility | |||||
---|---|---|---|---|---|---|
75% | 95% | 125% | ||||
\(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | \(\widehat{\sigma }_{L}\) (%) | \(\widehat{\sigma }\) (%) | |
0.1 | 75.00 | 75.00 | 95.00 | 95.00 | 125.00 | 125.00 |
0.2 | 75.00 | 75.00 | 95.00 | 95.00 | 125.02 | 125.00 |
0.3 | 75.00 | 75.00 | 95.01 | 95.00 | 125.04 | 125.00 |
0.4 | 75.01 | 75.00 | 95.02 | 95.00 | 125.08 | 125.00 |
0.5 | 75.01 | 75.00 | 95.03 | 95.00 | 125.13 | 125.00 |
0.6 | 75.01 | 75.00 | 95.05 | 95.00 | 125.19 | 125.00 |
0.7 | 75.02 | 75.00 | 95.06 | 95.00 | 125.26 | 125.00 |
0.8 | 75.02 | 75.00 | 95.08 | 95.00 | 125.35 | 125.00 |
0.9 | 75.03 | 75.00 | 95.11 | 95.00 | 125.45 | 125.00 |
1.0 | 75.04 | 75.00 | 95.13 | 95.00 | 125.57 | 125.00 |
1.1 | 75.05 | 75.00 | 95.16 | 95.00 | 125.70 | 125.00 |
1.2 | 75.06 | 75.00 | 95.20 | 95.00 | 125.85 | 125.00 |
1.3 | 75.07 | 75.00 | 95.23 | 95.00 | 126.02 | 125.00 |
1.4 | 75.08 | 75.00 | 95.27 | 95.00 | 126.21 | 125.00 |
1.5 | 75.09 | 75.00 | 95.31 | 95.00 | 126.42 | 125.00 |
Average volatility | 75.03 | 75.00 | 95.11 | 95.00 | 125.49 | 125.00 |
Average error | 0.03 | 0.00 | 0.11 | 0.00 | 0.49 | 0.00 |
Std. Dev. error | 0.03 | 0.00 | 0.10 | 0.00 | 0.44 | 0.00 |
True volatility | ||||||
---|---|---|---|---|---|---|
15% | 35% | 55% | 75% | 95% | 125% | |
Statistics | Implied volatility estimation error for \(\widehat{\sigma }_{L}\) | |||||
Mean | 0.00000983 | 0.00068779 | 0.00672967 | 0.03275487 | 0.11169245 | 0.48689149 |
Median | 0.00000761 | 0.00053019 | 0.00515269 | 0.02481224 | 0.08326543 | 0.34881310 |
Minimum | 0.00000012 | 0.00000822 | 0.00007887 | 0.00037279 | 0.00121955 | 0.00484083 |
Maximum | 0.00002678 | 0.00187959 | 0.01850733 | 0.09098208 | 0.31488411 | 1.42410046 |
Std Dev | 0.00000846 | 0.00059359 | 0.00584033 | 0.02867370 | 0.09902358 | 0.44499502 |
Statistics | Implied volatility estimation error for \(\widehat{\sigma }\) | |||||
Mean | \(-\) 0.00000002 | \(-\) 0.00000131 | \(-\) 0.00001338 | \(-\) 0.00006825 | \(-\) 0.00024386 | \(-\) 0.00111527 |
Median | \(-\) 0.00000001 | \(-\) 0.00000100 | \(-\) 0.00001004 | \(-\) 0.00005016 | \(-\) 0.00017529 | \(-\) 0.00077766 |
Minimum | \(-\) 0.00000005 | \(-\) 0.00000362 | \(-\) 0.00003744 | \(-\) 0.00019455 | \(-\) 0.00070786 | \(-\) 0.00331424 |
Maximum | 0.00000000 | \(-\) 0.00000002 | \(-\) 0.00000015 | \(-\) 0.00000070 | \(-\) 0.00000231 | \(-\) 0.00000928 |
Std Dev | 0.00000002 | 0.00000114 | 0.00001180 | 0.00006121 | 0.00022249 | 0.00104102 |
5.3.3 Empirical results on SPDR S&P 500 ETF TRUST
S/X | S | Call price | X | Days | SPX True Vol (%) | \(\widehat{\sigma _B}\) (%) | \({\widehat{\sigma _L}}\)(%) | \(\widehat{\sigma }\) (%) |
---|---|---|---|---|---|---|---|---|
1.23 | 287.970 | 53.240 | 235 | 30 | 33.00 | 81.23 | NA | 35.90 |
1.20 | 287.970 | 48.300 | 240 | 30 | 31.48 | 73.82 | NA | 34.25 |
1.18 | 287.970 | 43.400 | 245 | 30 | 30.33 | 66.54 | NA | 32.87 |
0.98 | 287.970 | 2.155 | 295 | 30 | 14.43 | 17.22 | 14.55 | 13.48 |
0.93 | 287.970 | 0.055 | 310 | 30 | 11.77 | 33.61 | NA | 13.72 |
0.91 | 287.970 | 0.025 | 315 | 30 | 12.63 | 41.11 | NA | 14.16 |
1.23 | 287.970 | 53.840 | 235 | 72 | 33.38 | 53.61 | NA | 28.68 |
1.20 | 287.970 | 49.040 | 240 | 72 | 31.58 | 49.10 | NA | 27.64 |
1.18 | 287.970 | 44.260 | 245 | 72 | 29.73 | 44.64 | NA | 26.40 |
0.98 | 287.970 | 4.615 | 295 | 72 | 15.12 | 15.93 | 14.73 | 10.91 |
0.93 | 287.970 | 0.485 | 310 | 72 | 11.56 | 22.54 | NA | 12.44 |
0.91 | 287.970 | 0.220 | 315 | 72 | 14.24 | 26.92 | NA | 12.35 |
1.23 | 287.970 | 54.880 | 235 | 135 | 28.90 | 40.64 | NA | 24.89 |
1.20 | 287.970 | 50.320 | 240 | 135 | 27.12 | 37.69 | 24.23 | 24.20 |
1.18 | 287.970 | 45.795 | 245 | 135 | 26.07 | 34.79 | 24.11 | 23.30 |
0.98 | 287.970 | 7.780 | 295 | 135 | 15.69 | 16.17 | 15.46 | 9.64 |
0.93 | 287.970 | 1.980 | 310 | 135 | 12.56 | 18.60 | 10.96 | 12.28 |
0.91 | 287.970 | 1.105 | 315 | 135 | 11.93 | 20.93 | NA | 12.17 |
5.4 Computational performance of the algorithm: hybrid method versus closed form solution
Numerical methods | Analytical approximations | ||||
---|---|---|---|---|---|
Formula | LW | OT | Matlab | Equation (4.3) | Li |
Implied Vol. | 0.2160 | 0.1000 | 0.0999 | 0.1191 | NA |
Numeric algorithm (a) | Closed form solution (b) | \(\Delta \) Performance \(\left( \frac{a}{b}-1\right) \) (%) | |
---|---|---|---|
ATM | 1.0927734 | 0.2968750 | 268.092 |
ITM | 1.2275391 | 0.2812500 | 336.458 |
OTM | 1.1992188 | 0.2929688 | 309.333 |