There is a deeper relationship between the two risk-neutral measures. Take any event in the binomial model with a finite number of steps
and calculate the risk-neutral probability of it.
Take the same event in the Black Scholes model and calculate the risk-neutral probability of it. For most events, the two probabilities are different. Now let the number of ...
$q$ is not delta hedge.
$q$ is determined from the fact that $S_i$ is a martingale i.e. for $S_0$
$S_0=E(S_1)=quS_0+(1-q)dS_0$ (if no rates)
This equation gives the same $q$ , dependent only on $u$ and $d$ , if calculated for $S_0$ , $S_1$ etc , thus $q$ is the same for all steps.
Risk-neutral pricing is to help with relative value type questions: If I know the value of this what should the value of that be if it depends in some way on this. It doesn't help with absolute value type questions: Should I buy this or that, is the implied volatility too low or high etc. Those are generally "real world measure" type questions.
When you look at actual data from the stock market, the probability distribution that you have in mind and that would describe the likelihood of different scenarios occuring going forward is what we call the "physical" probability distribution.
Intuitively the risk-neutral probability distribution is a "distorted" version of the physical probability ...
Let's stick to a discrete market for simplicity. So, you have a finite number of states in this type of model.
The first fundamental theorem of asset pricing says that the absence of arbitrage in such markets imply the existence of (not necessarily unique) risk-neutral measure and vice-versa.
The reason it works in the second direction (the existence of a ...