The second derivative of the call price at K gives the probability of that strike (implied probability density).

In practice, what adjustments or acknowledgements (if any) need to be made to produce a IPD with puts?


By call-put parity, the second derivative of a European call option price with respect to strike is strictly equivalent to that of a European put.

So, yes: the result, known as the Breeden-Litzenberger identity, stays unchanged.

  • $\begingroup$ Are there any results for American options? $\endgroup$ – Jared Oct 22 '16 at 19:46
  • $\begingroup$ No, because an American option does not only depend on a terminal probability density. $\endgroup$ – Quantuple Oct 22 '16 at 19:50
  • $\begingroup$ Is it actually the same for both? Normally you see different implied forwards from p/c parity at different strikes (I believe due to different margining costs for puts and calls, but not totally sure). $\endgroup$ – will Oct 23 '16 at 22:12
  • $\begingroup$ Theoretically (under the usual assumptions) then yes it should. There is only one conditional pdf $q(s)=d\Bbb{Q}(S_T\leq s)/ds$ after all. $\endgroup$ – Quantuple Oct 24 '16 at 7:08

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