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Just forget you ever heard the words "risk-neutral probability", and insteadjust call p~ and q~ foo and bar and be done with it. These are no probabilities,well except for the fact that they happen to add up to one and fit into thedefinition of a probability measure. Okay maybe this is exaggerating a little.But here's the deal. You have the real world with some real probabilities.And some binomial model of the stock market. According to a bunch oftheorems that have been proven mathematically (no magic there at all),if you want to price derivatives you dump the real probabilities and usethese so-called risk neutral ones instead. That's all there is to it. It's justa tool. Simple (well simple after the fact of course) maths show that thereal ones don't matter.If this is an issue and it feels wrong somehow, that has nothing to do withany probabilities but rather the model we started out with in the first place.

The idea of using risk neutrality came with the wish to explain why the underlying of the BS price is the imaginary stock. By the original broad definition a derivative should take its value from the real stock while BS developed formula shows that underlying is something from another world. There are 2 absolute different approaches to highlight the phenomena. Probably initial one used wrong interpretation of the change measure techniques. They thought that starting with real world the change measure techniques leads to risk neutral world where the real stock replaced real expected return on risk free one. In this approach they simply lost the density that come up with the measure change. In essence it demonstrates fail of mathematical background. The error could be easy comprehend if we recall definite integral and imagine that one makes a change variable t = f ( x ) then change integrand and forget that dt= df(x) and wrote dt = dx.In other approach they used real world stock equation on risk-neutral world that was chosen such that on real world this equation will get risk free expected return. This approach is incorrect because real stock equation is defined on the real probability space and consider it on risk neutral world rather stupid than incorrect. In either way the risk-neutral word works for interpretation of the BS formula and does not deal with option pricing itself.

Here is my intuitive understanding of why risk-neutral pricing works (please don't mistake it for a rigorous detailed argument).In a market where buyers of a risky asset expect a positive risk premium and sellers expect a negative risk premium, the market will likely settle on a price where the present value of the expected risk premium is zero (i.e. it is already priced in). The same argument applies to any derivative of the asset. Hence, subtracting the actual risk premium at any future date and discounting at the risk-free rate will give the current market value of either the derivative or the underlying. Since the risk premium of the derivative stems from the risk premium of the underlying at any future time, subtracting the risk premium for the derivative can be achieved by subtracting the risk premium of the underlying.

Fermion, the argument relates to the option pricing underlying my argument relates to risk neutral interpretation of the given BS equation and its solution. That is different. Concerning pricing if you look at the arguments you introduced you might note a number of if that in formal interpretation are assumptions. These assumptions if 100% are accepted then the argument looks make sense. On the other hand if you ask why for instance given random stock has a spot price is its mean then all constructions applied for pricing might be failed. Though without doubt the real mean not neutral is the one of two most important risk parameters.

Risk neutral probability is a theory which requires a risk neutral portfolio to exist (which in turn requires a great mean other things to exist in the markets which do not necessarily ... the assumptions of the model). Given such a thing doesn't exist in the real world (? discuss ?) its meaning is model dependent. Hence the original question posted years ago by a newbie is a fantastic question and is incredibly deep one and difficult to answer ... i.e. what does risk neutral actually mean in the real markets rather than in a text book?