Root finding in Lord Kahl Method to Compute Heston Call Price (Part III)

I forgot two important points in my previous post about Lord-Kahl method to compute the Heston call price:

Strike Lord-Kahl Kahl-Jaeckel Forde-Jacquier-Lee
62.5 2.919316809400033E-34 8.405720564041985E-12 0.0
68.75 -8.923683388191852E-28 1.000266536266281E-11 0.0
75.0 -3.2319611910032E-22 2.454925152051146E-12 0.0
81.25 1.9401743410877718E-16 2.104982854689297E-12 0.0
87.5 -Infinity -1.6480150577535824E-11 0.0
93.75 Infinity 1.8277663826893331E-9 1.948392142070432E-9
100.0 0.4174318393886519 0.41743183938679845 0.4174314959743768
106.25 1.326968012594355E-11 7.575717830832218E-11 1.1186618909114702E-11
112.5 -5.205783145942609E-21 2.5307755890935368E-11 6.719872683111381E-45
118.75 4.537094156599318E-25 1.8911094912255066E-11 3.615356241778357E-114
125.0 1.006555799739525E-27 3.2365221613872563E-12 2.3126009701775733E-240
131.25 4.4339539263484925E-31 2.4794388764348696E-11 0.0

One can see negative prices and meaningless prices outside ATM. With scaling it changes to:

Strike Lord-Kahl Kahl-Jaeckel Forde-Jacquier-Lee
62.5 2.6668642552659466E-182 8.405720564041985E-12 0.0
68.75 7.156278101597845E-132 1.000266536266281E-11 0.0
81.25 7.863105641534119E-55 2.104982854689297E-12 0.0
87.5 7.073641308465115E-28 -1.6480150577535824E-11 0.0
93.75 1.8375145950924849E-9 1.8277663826893331E-9 1.948392142070432E-9
100.0 0.41743183938755385 0.41743183938679845 0.4174314959743768
106.25 1.3269785342953315E-11 7.575717830832218E-11 1.1186618909114702E-11
112.5 8.803247187972696E-42 2.5307755890935368E-11 6.719872683111381E-45
118.75 5.594342441346233E-90 1.8911094912255066E-11 3.615356241778357E-114
125.0 7.6539757567179276E-149 3.2365221613872563E-12 2.3126009701775733E-240
131.25 0.0 2.4794388764348696E-11 0.0

One can now now see that the Jacquier-Lee approximation is quickly not very good.

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