is the below equation true irrespective of if that 2nd derivative turns out to be negative or >1 , (ie even if theres an arbitrage) ?

the reason i ask is that i am writing a single asset montecarlo (numerical integration) based on a market implied distribution. i want to see the affect of changing the vol smile slope on pricing. i am using linear normal vol smile. (i know that normal vols are arbitrageable). when i make slope more than 0.5 (change in vol per 1bp change in strike) , then my convergence to market price is very bad. What is the mathematical reason for that? I guess it may be because of probabilities outside 0-100% , but mathematically, i am not sure why thats not allowed, as all we are doing here really is a numeric integration , which is like reverse engineering. how do i know what limits to do my numerical integration - is it enough to go out to the 99.99% point?

here is the equation i am talking about (copied from word, sorry about formatting) :

$c_k=∫_k^∞(x-k) (d^2 c_k)/(dx^2 ) dx$

here are some more details of what i have done: i have done this all in excel. i could have used some VBA, but apparently, for this task , VBA would be slower than excel, perhaps because VBA is not 'native'.

  1. i calc vol, simply at normal vol(K) = given atm vol + given slope * (K-F). 2, i use bachelier to get price.
  2. i do this for small strike increments of 1bp apart for a big range (from delta% 0.01 - 99.99)
  3. then i calc 1-sided dV/dK = CDF.
  4. then i have a new table of %iles from 0.005 to 99.995 going up by 0.01% , where i lookup the CDF- and get value.
  5. in step 5, i initially did with step function interp , then i did linear interp , that did not help at all
  • $\begingroup$ Hi, it would be beneficial if you added your simulation code and/or the corresponding equations. Thanks $\endgroup$ Commented Aug 10, 2021 at 8:35
  • $\begingroup$ its in excel. i suppose i could share that, but my post is clear enough $\endgroup$
    – Randor
    Commented Aug 10, 2021 at 10:48
  • 1
    $\begingroup$ Sorry, I cannot fully understand what you are asking. It would be necessary to know how you come up with the density, how are you calibrating it, what is your simulated product, how are you performing the density inversion/Integration? $\endgroup$ Commented Aug 10, 2021 at 11:23
  • $\begingroup$ i wrote that there. i wrote the density is based on a linear normal vol smile. normal vol is linear in strike. $\endgroup$
    – Randor
    Commented Aug 10, 2021 at 11:57
  • 3
    $\begingroup$ Hi Randor, if you'd add your steps in some detail that would be helpful. It would be helpful if you added code (R, python, VBA) or at least the mathematics behind it with some proper formatting. That would be quite helpful to our understanding of your problem! $\endgroup$ Commented Aug 10, 2021 at 13:49


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