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Is there any industry consensus about the model to use for pricing exotics in equity, FX and interest rates? I assume that for vanilla options they all use Black model, but how about exotics?

Also, for those standard models applied to exotics, do they have closed form solutions like Black & Scholes or they all use Monte Carlo simulations to generate paths for the underlying and the stochastic volatility?

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In equity and FX it's LSV (local stochastic volatility) models, with each shop probably using their own LSV twist/flavour.

In rates I believe (variations on) SABR is still the standard, but more general LSV models may be catching on there as well.

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    $\begingroup$ I'd say that for exotic IR derivatives, the Libor market model is the standard. $\endgroup$
    – FunnyBuzer
    Commented Mar 5, 2019 at 21:14
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    $\begingroup$ SABR has a closed form for pricing vanilla options. Actually it's not even the exact solution, but a (very good) approximation for the implied volatility for vanilla options if the underyling dynamics were indeed SABR. You'd still Monte Carlo or PDE methods for exotics. Same goes for LMM, which is complicated by the fact that you'd probably like a correlation structure for the different Libor rates. For LSV I am not aware of any closed form expression for vanillas even. There you'd have to use PDE r MC methods even for vanilla options. $\endgroup$
    – user34971
    Commented Mar 7, 2019 at 12:25
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    $\begingroup$ Assuming there is a market implied volatility smile (ie vanilla options are traded), then for some light exotics such as digitals, even though you don't have the model closed form solution you can relate the price to the Black-Scholes price. Eg a digital is basically an infinitely tight call spread, so you can relate it to the Black-Scholes price of a digital plus a skew correction. But for general exotics you cannot even do this. The same goes for Greeks. $\endgroup$
    – user34971
    Commented Mar 8, 2019 at 8:08
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    $\begingroup$ Well for very few light exotics, such as the digital, you can do this. But for many other exotics you can't and you really need to do the PDE/MC after calibration of your LSV model. But if all you need is an indicative price then there are 'hacks' to the Black-Scholes model to incorporate stochastic volatility corrections. Google for example "vanna-volga" methods for barrier options in FX. $\endgroup$
    – user34971
    Commented Mar 8, 2019 at 9:06
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    $\begingroup$ Yes that's correct. MC or PDE depending on the type of derivative. For Greeks you could do bump and revalue or pathwise derivatives, or adjoint differentiation etc. $\endgroup$
    – user34971
    Commented Mar 11, 2019 at 11:19

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