How well does your neural network package handle difficult data? Find out by training a neural network for the data which follows on this page.
The data that follows consists of Black-Scholes option prices for volatility levels running from 20 percent to 200 percent, for time remaining running from 5 to 15 days, and for strike price running from 70 dollars to 130 dollars. The stock price was set to 100 dollars and the interest rate to 0 percent when generating the data.
The data is "difficult" in that, for a neural network to practically emulate Black-Scholes, a very tight fit is required. How good a fit to the data must be achieved? The R-squared should be at least 0.999999 or better, and the largest absolute error must be less than 0.05 dollars (the price increment for most options) or, better yet, less than 0.01 dollars. This kind of accuracy can be achieved by a well-designed neural network simulator, and is required in may fields of science (e.g., astronomy) as well as for neural-network-based option pricing models. If your neural network software achieves these kind of results, congratulations; if it does not, you might want to consider an upgrade or switch.
We suggest you start the test with a 3-layer network having 3 input neurons (one for volatility, one for time remaining, and one for strike), 20 middle-layer or "hidden" neurons, and one output neuron (for theoretical option price). Next, you can try a 4-layer network with 3 input neurons, 22 second-layer neurons, 8 third-layer neurons, and 1 output neuron. See how the different models perform. Take notes regarding the R-squared and largest error, as well as the root-mean-square error. The root-mean-square error can be calculated from the total sum-of-squares error (if that is what your package displays) by dividing the sum-of-squares by the number of facts, and then taking the square root.
We would greatly appreciate it if you would email us your results. Be sure to specify the neural network package you used, the size of the networks trained, and the R-squared, root-mean-square, and largest error figures achieved. If we receive sufficient data we will tabulate the results and display them on our website for the benefit of all. Of course, your privacy will be protected and no names or email addresses will ever be revealed.
In order to use the test data, we suggest that you copy it into a spreadsheet (you can use cut-and-paste to do this) where you can remove the first column of data (the fact numbers) and the first row (the variable names) if necessary, and format the data as required by your neural network software. Many neural networks will not accept fact numbers (they will inappropriately treat the column as an input variable) and some will not accept column headers in the training data.
The "target" or dependent variable, of course, is the option price, appearing in the column labeled OPRICE in the data below.
FCTNO VLTY TIME STRIKE OPRICE 1 0.20 5.0 75.00 25.0000 2 0.20 5.0 76.00 24.0000 3 0.20 5.0 77.00 23.0000 4 0.20 5.0 78.00 22.0000 5 0.20 5.0 79.00 21.0000 6 0.20 5.0 80.00 20.0000 7 0.20 5.0 81.00 19.0000 8 0.20 5.0 82.00 18.0000 9 0.20 5.0 83.00 17.0000 10 0.20 5.0 84.00 16.0000 11 0.20 5.0 85.00 15.0000 12 0.20 5.0 86.00 14.0000 13 0.20 5.0 87.00 13.0000 14 0.20 5.0 88.00 12.0000 15 0.20 5.0 89.00 11.0000 16 0.20 5.0 90.00 10.0001 17 0.20 5.0 91.00 9.0003 18 0.20 5.0 92.00 8.0012 19 0.20 5.0 93.00 7.0043 20 0.20 5.0 94.00 6.0135 21 0.20 5.0 95.00 5.0372 22 0.20 5.0 96.00 4.0908 23 0.20 5.0 97.00 3.1976 24 0.20 5.0 98.00 2.3870 25 0.20 5.0 99.00 1.6886 26 0.20 5.0 100.00 1.1239 27 0.20 5.0 101.00 0.6992 28 0.20 5.0 102.00 0.4045 29 0.20 5.0 103.00 0.2167 30 0.20 5.0 104.00 0.1072 31 0.20 5.0 105.00 0.0488 32 0.20 5.0 106.00 0.0205 33 0.20 5.0 107.00 0.0079 34 0.20 5.0 108.00 0.0028 35 0.20 5.0 109.00 0.0009 36 0.20 5.0 110.00 0.0003 37 0.20 5.0 111.00 0.0001 38 0.20 5.0 112.00 0.0000 39 0.20 5.0 113.00 0.0000 40 0.20 5.0 114.00 0.0000 41 0.20 5.0 115.00 0.0000 42 0.20 5.0 116.00 0.0000 43 0.20 5.0 117.00 0.0000 44 0.20 5.0 118.00 0.0000 45 0.20 5.0 119.00 0.0000 46 0.20 5.0 120.00 0.0000 47 0.20 5.0 121.00 0.0000 48 0.20 5.0 122.00 0.0000 49 0.20 5.0 123.00 0.0000 50 0.20 5.0 124.00 0.0000 51 0.20 5.0 125.00 0.0000 52 0.20 10.0 75.00 25.0000 53 0.20 10.0 76.00 24.0000 54 0.20 10.0 77.00 23.0000 55 0.20 10.0 78.00 22.0000 56 0.20 10.0 79.00 21.0000 57 0.20 10.0 80.00 20.0000 58 0.20 10.0 81.00 19.0000 59 0.20 10.0 82.00 18.0000 60 0.20 10.0 83.00 17.0000 61 0.20 10.0 84.00 16.0000 62 0.20 10.0 85.00 15.0000 63 0.20 10.0 86.00 14.0001 64 0.20 10.0 87.00 13.0002 65 0.20 10.0 88.00 12.0007 66 0.20 10.0 89.00 11.0019 67 0.20 10.0 90.00 10.0048 68 0.20 10.0 91.00 9.0114 69 0.20 10.0 92.00 8.0252 70 0.20 10.0 93.00 7.0519 71 0.20 10.0 94.00 6.1002 72 0.20 10.0 95.00 5.1815 73 0.20 10.0 96.00 4.3102 74 0.20 10.0 97.00 3.5018 75 0.20 10.0 98.00 2.7715 76 0.20 10.0 99.00 2.1314 77 0.20 10.0 100.00 1.5893 78 0.20 10.0 101.00 1.1468 79 0.20 10.0 102.00 0.7995 80 0.20 10.0 103.00 0.5377 81 0.20 10.0 104.00 0.3486 82 0.20 10.0 105.00 0.2177 83 0.20 10.0 106.00 0.1308 84 0.20 10.0 107.00 0.0757 85 0.20 10.0 108.00 0.0421 86 0.20 10.0 109.00 0.0226 87 0.20 10.0 110.00 0.0116 88 0.20 10.0 111.00 0.0058 89 0.20 10.0 112.00 0.0028 90 0.20 10.0 113.00 0.0013 91 0.20 10.0 114.00 0.0006 92 0.20 10.0 115.00 0.0002 93 0.20 10.0 116.00 0.0001 94 0.20 10.0 117.00 0.0000 95 0.20 10.0 118.00 0.0000 96 0.20 10.0 119.00 0.0000 97 0.20 10.0 120.00 0.0000 98 0.20 10.0 121.00 0.0000 99 0.20 10.0 122.00 0.0000 100 0.20 10.0 123.00 0.0000 101 0.20 10.0 124.00 0.0000 102 0.20 10.0 125.00 0.0000 103 0.20 15.0 75.00 25.0000 104 0.20 15.0 76.00 24.0000 105 0.20 15.0 77.00 23.0000 106 0.20 15.0 78.00 22.0000 107 0.20 15.0 79.00 21.0000 108 0.20 15.0 80.00 20.0000 109 0.20 15.0 81.00 19.0000 110 0.20 15.0 82.00 18.0000 111 0.20 15.0 83.00 17.0001 112 0.20 15.0 84.00 16.0002 113 0.20 15.0 85.00 15.0005 114 0.20 15.0 86.00 14.0012 115 0.20 15.0 87.00 13.0029 116 0.20 15.0 88.00 12.0063 117 0.20 15.0 89.00 11.0130 118 0.20 15.0 90.00 10.0254 119 0.20 15.0 91.00 9.0472 120 0.20 15.0 92.00 8.0838 121 0.20 15.0 93.00 7.1419 122 0.20 15.0 94.00 6.2304 123 0.20 15.0 95.00 5.3590 124 0.20 15.0 96.00 4.5384 125 0.20 15.0 97.00 3.7790 126 0.20 15.0 98.00 3.0898 127 0.20 15.0 99.00 2.4776 128 0.20 15.0 100.00 1.9465 129 0.20 15.0 101.00 1.4967 130 0.20 15.0 102.00 1.1256 131 0.20 15.0 103.00 0.8273 132 0.20 15.0 104.00 0.5941 133 0.20 15.0 105.00 0.4166 134 0.20 15.0 106.00 0.2852 135 0.20 15.0 107.00 0.1906 136 0.20 15.0 108.00 0.1243 137 0.20 15.0 109.00 0.0791 138 0.20 15.0 110.00 0.0492 139 0.20 15.0 111.00 0.0299 140 0.20 15.0 112.00 0.0177 141 0.20 15.0 113.00 0.0102 142 0.20 15.0 114.00 0.0058 143 0.20 15.0 115.00 0.0032 144 0.20 15.0 116.00 0.0017 145 0.20 15.0 117.00 0.0009 146 0.20 15.0 118.00 0.0005 147 0.20 15.0 119.00 0.0002 148 0.20 15.0 120.00 0.0001 149 0.20 15.0 121.00 0.0001 150 0.20 15.0 122.00 0.0000 151 0.20 15.0 123.00 0.0000 152 0.20 15.0 124.00 0.0000 153 0.20 15.0 125.00 0.0000 154 0.40 5.0 75.00 25.0000 155 0.40 5.0 76.00 24.0000 156 0.40 5.0 77.00 23.0000 157 0.40 5.0 78.00 22.0000 158 0.40 5.0 79.00 21.0000 159 0.40 5.0 80.00 20.0000 160 0.40 5.0 81.00 19.0001 161 0.40 5.0 82.00 18.0003 162 0.40 5.0 83.00 17.0006 163 0.40 5.0 84.00 16.0014 164 0.40 5.0 85.00 15.0030 165 0.40 5.0 86.00 14.0060 166 0.40 5.0 87.00 13.0115 167 0.40 5.0 88.00 12.0212 168 0.40 5.0 89.00 11.0375 169 0.40 5.0 90.00 10.0638 170 0.40 5.0 91.00 9.1047 171 0.40 5.0 92.00 8.1657 172 0.40 5.0 93.00 7.2537 173 0.40 5.0 94.00 6.3761 174 0.40 5.0 95.00 5.5410 175 0.40 5.0 96.00 4.7564 176 0.40 5.0 97.00 4.0295 177 0.40 5.0 98.00 3.3665 178 0.40 5.0 99.00 2.7717 179 0.40 5.0 100.00 2.2475 180 0.40 5.0 101.00 1.7938 181 0.40 5.0 102.00 1.4087 182 0.40 5.0 103.00 1.0880 183 0.40 5.0 104.00 0.8262 184 0.40 5.0 105.00 0.6168 185 0.40 5.0 106.00 0.4526 186 0.40 5.0 107.00 0.3263 187 0.40 5.0 108.00 0.2313 188 0.40 5.0 109.00 0.1611 189 0.40 5.0 110.00 0.1103 190 0.40 5.0 111.00 0.0742 191 0.40 5.0 112.00 0.0491 192 0.40 5.0 113.00 0.0319 193 0.40 5.0 114.00 0.0204 194 0.40 5.0 115.00 0.0129 195 0.40 5.0 116.00 0.0080 196 0.40 5.0 117.00 0.0048 197 0.40 5.0 118.00 0.0029 198 0.40 5.0 119.00 0.0017 199 0.40 5.0 120.00 0.0010 200 0.40 5.0 121.00 0.0006 201 0.40 5.0 122.00 0.0003 202 0.40 5.0 123.00 0.0002 203 0.40 5.0 124.00 0.0001 204 0.40 5.0 125.00 0.0001 205 0.40 10.0 75.00 25.0003 206 0.40 10.0 76.00 24.0005 207 0.40 10.0 77.00 23.0010 208 0.40 10.0 78.00 22.0018 209 0.40 10.0 79.00 21.0031 210 0.40 10.0 80.00 20.0054 211 0.40 10.0 81.00 19.0091 212 0.40 10.0 82.00 18.0149 213 0.40 10.0 83.00 17.0237 214 0.40 10.0 84.00 16.0369 215 0.40 10.0 85.00 15.0560 216 0.40 10.0 86.00 14.0832 217 0.40 10.0 87.00 13.1208 218 0.40 10.0 88.00 12.1719 219 0.40 10.0 89.00 11.2397 220 0.40 10.0 90.00 10.3280 221 0.40 10.0 91.00 9.4408 222 0.40 10.0 92.00 8.5821 223 0.40 10.0 93.00 7.7563 224 0.40 10.0 94.00 6.9672 225 0.40 10.0 95.00 6.2188 226 0.40 10.0 96.00 5.5142 227 0.40 10.0 97.00 4.8562 228 0.40 10.0 98.00 4.2468 229 0.40 10.0 99.00 3.6872 230 0.40 10.0 100.00 3.1780 231 0.40 10.0 101.00 2.7187 232 0.40 10.0 102.00 2.3083 233 0.40 10.0 103.00 1.9450 234 0.40 10.0 104.00 1.6262 235 0.40 10.0 105.00 1.3493 236 0.40 10.0 106.00 1.1108 237 0.40 10.0 107.00 0.9074 238 0.40 10.0 108.00 0.7356 239 0.40 10.0 109.00 0.5917 240 0.40 10.0 110.00 0.4723 241 0.40 10.0 111.00 0.3742 242 0.40 10.0 112.00 0.2942 243 0.40 10.0 113.00 0.2296 244 0.40 10.0 114.00 0.1778 245 0.40 10.0 115.00 0.1368 246 0.40 10.0 116.00 0.1044 247 0.40 10.0 117.00 0.0791 248 0.40 10.0 118.00 0.0596 249 0.40 10.0 119.00 0.0445 250 0.40 10.0 120.00 0.0331 251 0.40 10.0 121.00 0.0244 252 0.40 10.0 122.00 0.0179 253 0.40 10.0 123.00 0.0130 254 0.40 10.0 124.00 0.0094 255 0.40 10.0 125.00 0.0068 256 0.40 15.0 75.00 25.0039 257 0.40 15.0 76.00 24.0062 258 0.40 15.0 77.00 23.0097 259 0.40 15.0 78.00 22.0149 260 0.40 15.0 79.00 21.0225 261 0.40 15.0 80.00 20.0332 262 0.40 15.0 81.00 19.0481 263 0.40 15.0 82.00 18.0685 264 0.40 15.0 83.00 17.0958 265 0.40 15.0 84.00 16.1319 266 0.40 15.0 85.00 15.1788 267 0.40 15.0 86.00 14.2387 268 0.40 15.0 87.00 13.3141 269 0.40 15.0 88.00 12.4078 270 0.40 15.0 89.00 11.5224 271 0.40 15.0 90.00 10.6608 272 0.40 15.0 91.00 9.8259 273 0.40 15.0 92.00 9.0205 274 0.40 15.0 93.00 8.2472 275 0.40 15.0 94.00 7.5084 276 0.40 15.0 95.00 6.8063 277 0.40 15.0 96.00 6.1425 278 0.40 15.0 97.00 5.5184 279 0.40 15.0 98.00 4.9350 280 0.40 15.0 99.00 4.3928 281 0.40 15.0 100.00 3.8917 282 0.40 15.0 101.00 3.4315 283 0.40 15.0 102.00 3.0112 284 0.40 15.0 103.00 2.6298 285 0.40 15.0 104.00 2.2856 286 0.40 15.0 105.00 1.9769 287 0.40 15.0 106.00 1.7017 288 0.40 15.0 107.00 1.4578 289 0.40 15.0 108.00 1.2429 290 0.40 15.0 109.00 1.0547 291 0.40 15.0 110.00 0.8907 292 0.40 15.0 111.00 0.7487 293 0.40 15.0 112.00 0.6265 294 0.40 15.0 113.00 0.5218 295 0.40 15.0 114.00 0.4326 296 0.40 15.0 115.00 0.3571 297 0.40 15.0 116.00 0.2934 298 0.40 15.0 117.00 0.2401 299 0.40 15.0 118.00 0.1956 300 0.40 15.0 119.00 0.1587 301 0.40 15.0 120.00 0.1282 302 0.40 15.0 121.00 0.1031 303 0.40 15.0 122.00 0.0826 304 0.40 15.0 123.00 0.0659 305 0.40 15.0 124.00 0.0524 306 0.40 15.0 125.00 0.0415 307 0.60 5.0 75.00 25.0006 308 0.60 5.0 76.00 24.0011 309 0.60 5.0 77.00 23.0020 310 0.60 5.0 78.00 22.0035 311 0.60 5.0 79.00 21.0059 312 0.60 5.0 80.00 20.0097 313 0.60 5.0 81.00 19.0156 314 0.60 5.0 82.00 18.0243 315 0.60 5.0 83.00 17.0371 316 0.60 5.0 84.00 16.0554 317 0.60 5.0 85.00 15.0811 318 0.60 5.0 86.00 14.1162 319 0.60 5.0 87.00 13.1634 320 0.60 5.0 88.00 12.2256 321 0.60 5.0 89.00 11.3061 322 0.60 5.0 90.00 10.4082 323 0.60 5.0 91.00 9.5357 324 0.60 5.0 92.00 8.6924 325 0.60 5.0 93.00 7.8820 326 0.60 5.0 94.00 7.1078 327 0.60 5.0 95.00 6.3732 328 0.60 5.0 96.00 5.6810 329 0.60 5.0 97.00 5.0333 330 0.60 5.0 98.00 4.4318 331 0.60 5.0 99.00 3.8775 332 0.60 5.0 100.00 3.3707 333 0.60 5.0 101.00 2.9110 334 0.60 5.0 102.00 2.4973 335 0.60 5.0 103.00 2.1282 336 0.60 5.0 104.00 1.8015 337 0.60 5.0 105.00 1.5147 338 0.60 5.0 106.00 1.2650 339 0.60 5.0 107.00 1.0493 340 0.60 5.0 108.00 0.8646 341 0.60 5.0 109.00 0.7076 342 0.60 5.0 110.00 0.5753 343 0.60 5.0 111.00 0.4646 344 0.60 5.0 112.00 0.3728 345 0.60 5.0 113.00 0.2972 346 0.60 5.0 114.00 0.2354 347 0.60 5.0 115.00 0.1853 348 0.60 5.0 116.00 0.1449 349 0.60 5.0 117.00 0.1126 350 0.60 5.0 118.00 0.0870 351 0.60 5.0 119.00 0.0668 352 0.60 5.0 120.00 0.0510 353 0.60 5.0 121.00 0.0387 354 0.60 5.0 122.00 0.0292 355 0.60 5.0 123.00 0.0219 356 0.60 5.0 124.00 0.0164 357 0.60 5.0 125.00 0.0121 358 0.60 10.0 75.00 25.0275 359 0.60 10.0 76.00 24.0385 360 0.60 10.0 77.00 23.0531 361 0.60 10.0 78.00 22.0723 362 0.60 10.0 79.00 21.0970 363 0.60 10.0 80.00 20.1285 364 0.60 10.0 81.00 19.1682 365 0.60 10.0 82.00 18.2176 366 0.60 10.0 83.00 17.2783 367 0.60 10.0 84.00 16.3520 368 0.60 10.0 85.00 15.4407 369 0.60 10.0 86.00 14.5463 370 0.60 10.0 87.00 13.6707 371 0.60 10.0 88.00 12.8159 372 0.60 10.0 89.00 11.9839 373 0.60 10.0 90.00 11.1766 374 0.60 10.0 91.00 10.3957 375 0.60 10.0 92.00 9.6430 376 0.60 10.0 93.00 8.9198 377 0.60 10.0 94.00 8.2275 378 0.60 10.0 95.00 7.5671 379 0.60 10.0 96.00 6.9395 380 0.60 10.0 97.00 6.3454 381 0.60 10.0 98.00 5.7849 382 0.60 10.0 99.00 5.2583 383 0.60 10.0 100.00 4.7655 384 0.60 10.0 101.00 4.3058 385 0.60 10.0 102.00 3.8789 386 0.60 10.0 103.00 3.4839 387 0.60 10.0 104.00 3.1198 388 0.60 10.0 105.00 2.7854 389 0.60 10.0 106.00 2.4795 390 0.60 10.0 107.00 2.2007 391 0.60 10.0 108.00 1.9476 392 0.60 10.0 109.00 1.7186 393 0.60 10.0 110.00 1.5121 394 0.60 10.0 111.00 1.3267 395 0.60 10.0 112.00 1.1608 396 0.60 10.0 113.00 1.0127 397 0.60 10.0 114.00 0.8811 398 0.60 10.0 115.00 0.7646 399 0.60 10.0 116.00 0.6617 400 0.60 10.0 117.00 0.5711 401 0.60 10.0 118.00 0.4916 402 0.60 10.0 119.00 0.4221 403 0.60 10.0 120.00 0.3616 404 0.60 10.0 121.00 0.3089 405 0.60 10.0 122.00 0.2633 406 0.60 10.0 123.00 0.2238 407 0.60 10.0 124.00 0.1899 408 0.60 10.0 125.00 0.1607 409 0.60 15.0 75.00 25.1178 410 0.60 15.0 76.00 24.1503 411 0.60 15.0 77.00 23.1898 412 0.60 15.0 78.00 22.2375 413 0.60 15.0 79.00 21.2946 414 0.60 15.0 80.00 20.3622 415 0.60 15.0 81.00 19.4417 416 0.60 15.0 82.00 18.5345 417 0.60 15.0 83.00 17.6419 418 0.60 15.0 84.00 16.7652 419 0.60 15.0 85.00 15.9059 420 0.60 15.0 86.00 15.0654 421 0.60 15.0 87.00 14.2449 422 0.60 15.0 88.00 13.4457 423 0.60 15.0 89.00 12.6689 424 0.60 15.0 90.00 11.9158 425 0.60 15.0 91.00 11.1872 426 0.60 15.0 92.00 10.4840 427 0.60 15.0 93.00 9.8070 428 0.60 15.0 94.00 9.1568 429 0.60 15.0 95.00 8.5338 430 0.60 15.0 96.00 7.9385 431 0.60 15.0 97.00 7.3709 432 0.60 15.0 98.00 6.8312 433 0.60 15.0 99.00 6.3192 434 0.60 15.0 100.00 5.8347 435 0.60 15.0 101.00 5.3774 436 0.60 15.0 102.00 4.9468 437 0.60 15.0 103.00 4.5423 438 0.60 15.0 104.00 4.1633 439 0.60 15.0 105.00 3.8091 440 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