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What is the risk-neutral probability?
You have actually asked several questions, so I think what I'll do is give you an intuition about risk-neutral pricing to get you started.  Then I think the answer to many of your questions will become clear. Physical Probability There is some probability of every event out there actually occurring, including the price of a stock going up.  That's what we call the physical probability.  It's very intuitive, but not directly useful for finding the price of something because price is not the weighted average of future outcomes.   For example, if you have a stock that is highly correlated with the market and has 50% chance of being worth $20 dollars tomorrow and a 50% chance of being worth $10, it's value today is not $15.  It will be worth less, because it's a risky stock and must earn a premium.  When you are dealing with physical probabilities, if you want to compute value you have to take the probability-weighted average of all the prices it could have tomorrow and then add in some kind of compensation for risk, which may be hard to compute. Risk-Neutral Probability Finance theory has shown that instead of computing values this way, we can embed risk-compensation into our probabilities.  That is, we can create a new set up "probabilities" by adjusting the probability of good market outcomes downward and increasing the probability of bad market outcomes.  This may sound crazy because these probabilities are no longer physical, but it has the desirable property that we then use this set of probabilities to price of every asset out there: all of them (equity, options, bonds, savings accounts, etc.).  We call these adjusted probabilities that risk-neutral probabilities.  When I say price I mean that you can multiply every outcome by its risk-neutral probability and discount at the risk-free rate to find its correct price.    To be clear, we have changed the probability of the market going up and down, not our probability of a particular stock moving independent of the market.  Because moves that are independent of the market do not affect prices, we don't have to adjust the probabilities of them happening in order to get risk-neutral probabilities. Anyway, the best way to think of risk-neutral probabilities is as a set of bogus probabilities that consistently give the correct price of every asset in the economy without having to add a risk premium. If we just take the risk-neutral probability-weighted average of all outcomes and discount at the risk-free rate, we get the price.  Very handy if you have them. Risk-Neutral Pricing We can't get risk-neutral probabilities from research about how likely a stock is to actually go up or down.  That would be the physical probability.  Instead, we can figure out the risk-neutral probabilities from prices. If a stock has only two possible prices tomorrow, U and D, and the risk-neutral probability of U is q, then Price = [ Uq + D(1-q) ] / e^(rt) The exponential there is just discounting by the risk-free rate.  This is the beginning of the equations you have mentioned.  The main thing to remember is that q is not the physical probability, it's the risk-neutral one.  I can't emphasize that enough.  If you have prespecified what U and D can be, then there is only one unknown in that equation: q. That means you can look at the stock price and solve for the risk neutral probability of the stock going up.   The reason this is useful is that you can same risk-neutral probability to price the associated option.  In the case of the option you don't know its price today (yet) but you do know how much money it will be worth if the stock moves up or down.  Use those values and the risk-neutral probability you computed from the stock to compute the option's price.  That's what's going on here.  To remember: the same risk-neutral probability measure prices everything out there.  That is, if you choose an asset, multiply each possibly outcome by its risk-neutral probability, and discount at the risk-free rate, you get its price.  In general we use prices of things we know to infer things about the risk-neutral probability measure in order to get prices we do not know.