# 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.

Introducing this restriction, it is clear that the situation you describe violates this property (unless the risk-neutral probability is also 0 in one of the events, in which case the measures are equivalent and the result is correct!)

Edit: When it comes to the impact of the price of the stock, there is now only one price it can have, i.e. the disounted price of whatever its price will be with probability 1. Note that this is exactly the price that results in a risk-neutral probability of 1.

• So will that mean that under this circumstance, the binomial model is not applicable? – Mriganka Basu Roy Chowdhury Apr 9 at 8:16
• No, the binomial model is applicable. Have a look at my edit. Basically, what you are doing is assuming the probabilities to be fixed and then changing the stock price to something which is inconsistent with those probabilities. If the probability of one scenario is 1, then there is only one arbitrage-free stock price (i.e. the discounted future value). This stock price will then result in a risk-neutral probability of 1. The point is that if there is only one possible scenario the stock price is fixed, too! – AdB Apr 9 at 8:34
• Sorry to ask yet another question, but then, why don't the risk-neutral measure probabilities depend on the real-world probabilities (because as above, the real-world and the risk-neutral probabilities are equivalent)? Is it because we assume a specific form of the real-world (i.e., the Binomial model) ? – Mriganka Basu Roy Chowdhury Apr 28 at 13:42
• Well, they actually do implicitly! The stock price reflects the real-world probabilities, and we use the stock price to infer the risk-neutral probabilities. That is why you cannot change the stock price without also changing the real-world probabilities. Again, try to compute the risk-neutral probabilities using the discounted value of the outcome happening with probability 1. This should result in a risk-neutral probability of 1! – AdB Apr 29 at 8:55