There are only two workable principles that I know of in financial modeling.
1. Use Analogy: If you want to know the value of something that doesn’t have a price, find a (so-called replicating) portfolio consisting of more or less liquidly priced securities that collectively has the same payoffs in all future states of the world.
To carry this you need science and engineering:
- Science: Specify what you mean by all future states of the world (though you will never be able to enumerate them because they change as past experience affects the future, but, either way, they are definitely not continuous) This is the model, the stochastic description.
- Engineering: Find the replicating portfolio as best you can. You will need engineering to get the most practical one.
2. Try To Be Consistent
The replicating portfolio that values the thing that doesn’t yet have a price must also value correctly all the things that do have prices, else you’re starting from the wrong place. Thus, the parameters that specify the stochastic description of all future states (e.g. the drift etc, in the Brownian case) must be adjusted so that all liquid things get priced correctly before you value the illiquid thing that doesn’t have a price. (Thus, whatever your model, hedgeable or not, it had better reproduce the price of liquid futures, forwards, etc). This forces you to make the future states of the world be risk neutral.
These principles, in my opinion, while less rigorous, are more general than the pedantic martingale machinery that is usually taught. The equivalent martingale measures are a special very detailed instantiation of this principle, but one that loses sight of the big picture.
I like this remark by Dirac (to be a bit pedantic): I am not interested in proofs; I am interested in how nature behaves.