As I am refining a pricing model to incorporate skew, and not just ATM volatilities, I need to create random realizations of the underlying consistent with the skew-implied pdf. When searching, one ends up with the Breeden-Litzenberger formula, which states that:

$\frac{\partial^{2}C}{\partial K^{2}}=e^{-rT}g(S_{T})$

As I am slightly wary of using numerical second derivatives in my code, I looked for an alternative way of obtaining this. I came up with the idea of using correctly skew-adjusted cash-or-nothing binary put prices in order to derive the CDF of the underlying:

$P_{dig}=P_{dig,noskew}+\nu_{vanilla}*\frac{\partial \sigma}{\partial K}$

As $CDF=e^{rT}P_{dig}$, I run [0,1]-normally distributed random numbers through the inverse function $CDF^{-1}$ to get realizations of the underlying that are distributed consistent with the skew.

Can I please have your views on this / please comment if I am missing something here

  • 1
    $\begingroup$ If all you want is the CDF and not the PDF, then you could also use $\partial C / \partial K$ instead of the second order derivative. You might also find my answer to quant.stackexchange.com/questions/30749 helpful. $\endgroup$ – LocalVolatility Feb 3 at 22:30
  • $\begingroup$ So basically that would be $1+\partial C/\partial K$ $\endgroup$ – ZRH Feb 4 at 7:58
  • $\begingroup$ Yes - but don't forget to multiply the derivative by the discount factor. $\endgroup$ – LocalVolatility Feb 4 at 9:19
  • $\begingroup$ Coded this up now as proposed by you, works like a charm. For anyone who wants to try - you need to compute the CDF over a quite wide range of values in order not to truncate the pdf that is implicit in the distribution of underlying values. $\endgroup$ – ZRH Feb 4 at 16:16
  • $\begingroup$ @zrh if you have some analytic form for the smile, then you can compute the second derivative numerically to arbitrary precision - what is your worry? $\endgroup$ – will Feb 5 at 21:47

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