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Reading the literature I see that quite an effort is made to price derivatives in an incomplete setting. I see stuff like efficient hedging, indifference pricing, choosing $\mathbb{Q}$ by considering some metric to $\mathbb{P}$ etc. However I cannot think of a situation where on would use this. I imagine the derivatives pricing pipeline as follows:

  1. Choose an arbitrage free model in its risk neutral form
  2. Calibrate the free parameters to market price
  3. Price your derivative of interest

So after step 2 we have exactly one risk neutral measure $\mathbb{Q}$. In which scenario would we have to deal with multiple rn measures?

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From a model risk perspective, banks are required to re-evaluate their model-dependent products (think: exotics) using more than one model and - if there are free parameters - by varying those parameters. This will result in multiple theoretical values, and hence with multiple risk neutral densities. Note that, usually, all vanillas will be priced correctly.

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    $\begingroup$ Yes exactly. Generally exotics trade in incomplete markets, that is some parameters are marked by the desk instead of calibrated to quotes, like correlation in spread options or mean-reversion for Bermudans. Different desks and banks might mark the parameter differently, each implying a different risk-neutral distribution. $\endgroup$ Sep 10 at 17:49

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