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I know that (time changed) Lévy processes are actively researched in the academic world, including tools such as minimal entropy martingale pricing measures and fast Fourier transforms. To what extent are such topics used in financial engineering, i.e. in trading desks of options market makers?

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Levy processes are not used for pricing derivatives and are useless in practice. When the task at hand is to price a derivative, i.e., working in the risk neutral measure, then using Levy processes is worse than useless, it is dangerous and should actively be avoided. You can add entropy risk measures, FFTs and other (practically) useless concepts from academia to that list.

If implemented properly, and there is a big emphasis on "properly", then they may be useful for risk management, i.e., working in the statistical measure. I would not use them for risk management either, but they could be used in this context for benchmarking purposes.

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    $\begingroup$ This answer is materially incorrect ('levy, FFT practically useless') and the tone strangely derisive. Perhaps lack of appreciation for these 'useless concepts' stems from narrow field of problems encountered? $\endgroup$ – James Spencer-Lavan Nov 24 '18 at 8:15
  • $\begingroup$ James: In what context have you seen Levy processes and/or FFT used? Thanks $\endgroup$ – Rodolfo Oviedo Nov 24 '18 at 23:08
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    $\begingroup$ Hmm - So I guess there is a distinction between the fact that many useful processes are Levy, and whether or not Levy process specific techniques are used in practice in their analysis. @Rodolfo for example jump diffusions used for pricing eg quanto CDS - pure brownians cannot hit mkt prices. Don't think myself that I've seen FFT used much for years (>decade) but again, particular to a given area, so wouldn't seek to generalise $\endgroup$ – Mehness Nov 25 '18 at 16:47
  • $\begingroup$ @Mehness Thanks for the noting the use of jump-diffusions and your not having seen FFT used to solve it. May that be because there are almost closed form solutions for affine processes? That about gamma variance processes, which have infinitely many jumps per period? $\endgroup$ – Rodolfo Oviedo Nov 25 '18 at 23:45
  • $\begingroup$ @Rodolfo, no probs. So, cpl of things, yes affine processes super useful, particularly for gap risk-sensitive payoffs, when frankly, pure brownians cannot expose the salient risk characteristics. A friend of mine at another bank was using variance gamma a long time ago for credit vol products however not sure how widely it caught on. As a general point, not testing exotics against a suite of models and dynamics, including levy processes, is frankly remiss from a trading / model approval perspective, and anywhere I've been forms part of a rigorous model validation process. $\endgroup$ – Mehness Nov 26 '18 at 13:39

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