which can then be symbolically differentiated, to produce new interpolating objects (of less accuracy).For delta and gamma, we can take raw divided differences but this breaks down sooner or later.
Now, if we use the array of V(j) prices we can do cubic spline interpolation is what MM seems to do. And there is "spline on spline" in which the second spline interpolates on delta to get a better gamma.
/* S exact delta CN delta Spline delta exact gamma CN gamma Spline gamma
59 0.330935233 0.330702564 0.330478248 0.040967715 0.04098453 0.041068871
60 0.372482798 0.372232265 0.372093554 0.042043376 0.042074873 0.042161739
61 0.41484882 0.414593361 0.414541127 0.042603904 0.042647319 0.042733408
62 0.457519085 0.457270132 0.457302102 0.042654217 0.042706223 0.042788541
63 0.499993235 0.499760129 0.499871257 0.042216737 0.04227377 0.042349769
64 0.541801022 0.541590678 0.541773643 0.041328768 0.041387329 0.041455004
65 0.582515647 0.582332471 0.582578239 0.040039354 0.040096257 0.040154187
66 0.62176381 0.621609821 0.621908227 0.038405882 0.038458444 0.038505789
67 0.659232357 0.659107471 0.659447781 0.036490702 0.036536856 0.03657332
68 0.694671633 0.694574052 0.694945475 0.034357967 0.034396306 0.034422069
69 0.727895851 0.727822509 0.728214631 0.032070838 0.032100608 0.032116243
70 0.758780913 0.758727922 0.759131055 0.029689185 0.029710218 0.029716604
71 0.787260172 0.787223244 0.787628682 0.027267796 0.027280425 0.027278649*/
Table 22.1 Output from Steps 1-4; data is K = 65, T = 0.25, r = b = 0.8, v = 0.3,NS = 325, NT = 1000
Ahlberg, Nielson and Walsh 1967.