I am familiar with martingale pricing as primarily a notational abstraction which allows us to price contingent claims on $X_\tau$ by its conditional expectation. Usually, we interpret this to mean that $X_\tau$ follows a zero-drift stochastic process, such as a GBM. Such an interpretation then allows us to efficiently solve for the expectation using numerical methods, such as MC.

However, it seems to me that there may be cases where we may be agnostic to the process and its distribution, but still be able define the conditional value under some equivalent measure.

Does taking this conditional expectation require us to define a stochastic process and/or its terminal distribution?

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    $\begingroup$ You are right in that APT tells us that the discounted value of any self-financing strategy should be a martingale, hence the pricing via conditional expectations, but does not tell anything about the particular type of martingale. Now, if you observe prices in the market, you could imply parameters for different modelling assumptions (hence types of martingale processes). What model to stick with will then be relevant when pricing other contracts which you do not observe. It amounts to an arbitrage-free extrapolation assumption of sorts. Not sure it answers your question though. $\endgroup$ – Quantuple Apr 13 '18 at 8:00
  • $\begingroup$ Note: the choice of model also obviously constrains the types and number of market instruments you can fit simultaneously as well. So there is also an embedded interpolation assumption. $\endgroup$ – Quantuple Apr 13 '18 at 8:02
  • $\begingroup$ @Quantuple Your comment is helping to rephrase the question: "what modeling assumptions are absolutely necessary to price contingent claims under the conditional expectation?". For example, it seems to me that for some classes of processes, the process may not be defined by the first few moments of its terminal distribution. And further along that spectrum, there are path dependent claims in which knowledge of the terminal distribution may not be sufficient. In which cases does pricing under some equivalent expectation almost surely converge to pricing under the true measure? $\endgroup$ – David Addison Apr 13 '18 at 16:18
  • $\begingroup$ Hi David. Sorry but I don't really understand it would be worth considering an example I think. The only thing which matters IMO is that the process is indeed a (local) martingale. From a modelling perspective there are conditions on the drift and diffusion functions for their integrals with respect to the driving atomic martingales to be well defined. $\endgroup$ – Quantuple Apr 13 '18 at 16:26
  • $\begingroup$ Are you asking if there is an analogue of the QMLE in the pricing world? That's a cool thing to ask, but I feel uneasy about the question, as the EMM is defined as the density that makes the price equal the conditional expectation. Assuming that in a risk-neutral world X admits not the density that is the very essence of the risk-neutral world, but a density that has the same mean seems to me as a self-contradiction. Wouldn't you agree? $\endgroup$ – Igor Pozdeev Apr 13 '18 at 18:05

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