Assuming B&S world, is it possible to price an (European) option on a general transformation $f(\cdot)$ of $X$? What kind of assumptions should we make on $f$? Is convexity sufficient to find some meaningful results?

For example for a call: $$ \max(0, f({X_T})-K) $$

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    $\begingroup$ I think that you may be looking for the Carr-Madan formula, quant.stackexchange.com/questions/27626/carr-madan-formula. There are some requirements on $f$, e.g. it must be sufficiently smooth, and it should not grow to wildly, IIRC. $\endgroup$ May 23 at 19:48
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    $\begingroup$ If you are willing to do a simple numeric integration $$\int_{-\infty}^{+\infty}\max\Big(0,f(X_0e^{\sigma\sqrt{T}x+rT-\sigma^2T/2})-K\Big)\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}dx$$ you will not need any assumption on $f$ at all. $\endgroup$
    – Kurt G.
    May 25 at 10:44
  • $\begingroup$ Thanks, do you have a paper ref for this formula? assuming something to do with FT? $\endgroup$ Jun 23 at 16:15


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