# What happens in the binomial model if the real-world probability is $0$

Consider a binomial model.

Suppose we know that the price of a stock will become a certain value at the next timestep. That is, one of the two outcomes has $$0$$ real-world probability.

Then it should not matter what the price of the stock is in that outcome, but indeed that does affect the price of today, through the risk neutral pricing formula.

How to resolve this (apparent) paradox?

• Normally, 0 probability events in the real and risk-neutral worlds coincide. – Raskolnikov Apr 8 at 18:12

If I understand your question correctly, another way to word it is: if an event that has probability 0 under the physical measure $$\mathbb{P}$$, how can it have a positive probability under the risk-neutral measure $$\mathbb{Q}$$?
The answer is simply: it cannot! According to the theory of risk-neutral pricing through no arbitrage arguments, we require that $$\mathbb{Q}$$ and $$\mathbb{P}$$ are equivalent measures. Simply put, if an event happens with probability 0 under one measure, it must also happen with probability 0 under the other.