# Risk Neutral Probability vs. Probability of an Up/Down Move

So is RN the same as the probability of an up move? I notice how some questions don’t say that the RNP is 0.5, but say the probability of an up move is 0.5. Does this mean the RNP is 0.5? What if it says the Probability of an up move is 0.4. Does this mean the probability of a down move is 0.6. What then is the RNP? I am confused on this.

This question is about binomial trees.
In no particular order
(i)

What if it says the Probability of an up move is 0.4. Does this mean the probability of a down move is 0.6

Yes.
In the binomial tree, the price at the end of the step must be one of two states (‘up’ and ‘down’), so we must have
probability(up)+probability(down)=1.

(ii) The risk neutral probability.
If the percentage change in asset price for an up move is U and the percentage change in asset price for a down move is D, and r is the risk free rate, then for the risk neutral probability, the expected return from the asset will be the risk free rate:

(1+U\%)*\textrm{probability(up)}+(1+D\%)*\textrm{probability(down)}=1+r\%
If you expand this out and use the result that probability(up)+probability(down)=1, you get
U*\textrm{probability(up)}+D*\textrm{probability(down)}=r

If you are given the up and down percentage changes and the risk free rate, you can use this formula to find the risk-neutral probability/
There is a risk neutral probability of going up and associated risk neutral of going down. If the probability of going up is risk neutral then so is the probability of going down.

(iii) Note that I’ve called the moves ‘up’ and ‘down’.
The formula above is a weighted average, and all you can say is
U>r>D

(iv)

I notice how some questions don’t say that the RNP is 0.5,
but say the probability of an up move is 0.5.

If the question doesn’t mention the risk-neutral probability, don’t lose any sleep over it.
Just use the probabilities you are given to construct your binomial tree.

Thank you for the response - But I am not constructing a binomial tree. This question is referring to solving a 2-period option value using binomial option valuation model. You need the RNP. However, the questions do not sometimes give you the RNP. Instead it say the probability of an up move is 0.6. I know they want us to find the RNP using the information. What I don’t get is if the prob of an up move is the same as the RNP - If not, how do I get the RNP using this information so I can solve for option value using the 2-period formula which is n squared(C++) + 2n(1-n)(C±) + 1-n squared(C–). Nothing to do with a tree.

that would be constructing a binomial tree.

If you really want to find the RNP, there’s a formula:
RNP(up) =\frac{r\%-D\%}{U\%-D\%}
where U is the percentage change in asset price for an up move and D is the percentage change in asset price for a down move, and r is the risk free rate.