# Why do we simply assume the risk neutral probabilities to be "0.5"?

I am aware that there was a question similar to this but my question is a little different.

Firstly, in context of binomial short rate, why do we simply assume the risk neutral probabilities p=1-p=0.5? Is this some kind of common practice or helps somehow?

Taken from a paper, "Let us also assume that under the risk-neutral probability measure (not necessarily the actual measure), that the probability of an up or down move is the same, equal to 0.5." Is there a rationale behind this?

Secondly and more importantly,

I have an introductory textbook, where basic characteristics of one factor interest rate models are explained like no-drift, flat volatility term structure specific to models. But here as well, 0.5 is taken to be risk neutral probability. But the problem is as soon as you change it from 0.5 to anything else, the so called "characteristics" of the models fall apart. So how can we generalise the behaviour of a model on the basis of a specific probability, i.e, 0.5? Isn't that wrong to say that a model has flat volatility term structure when it only exhibits so, if we were to choose risk neutral probabilities as 0.5?

• A continuous time model can be approximated in various ways by discrete time models. What is important it that the mean and variance (2 parameters) converge towards the desired value as the step size becomes smaller. (Admittedly the rate of convergence is also important, but is sometimes neglected on a first look). Some authors, such as Cox Ross Rubinstein proposed schemes with $p\ne q \ne 0.5$. Others such as Jarrow Rudd developed approx schemes with $p=q=0.5$ and there are many other choices available. All converge towards the same c.t. process and so are in some sense equivalent. Commented Jun 28, 2023 at 18:07
• @nbbo2 Are you saying the terminal distribution for any "p" values would be distributed in the same manner with same mean and variance, even though they might diverge at earlier time steps? (and we can overlook this for the fact that they ultimately converge?). And hence, in the long run, the characteristics of the models hold? Commented Jun 28, 2023 at 18:53