# Risk neutral probability in binomial short rate model assumed to be 0.5?

This should be a basic question but I have not been able to find a satisfying explanation. In the simplest binomial model, the risk neutral probability is computed using the up/down magnitude and the risk-free rate. But why is the risk neutral probability simply "assumed" to be 0.5 in the context of short rate models?

I'll take a stab.

In short rate modelling we start by postulating dynamics under pricing measure $Q$ and then calibrating our model to market prices. General short rate dynamics under $Q$ can be described as

$dr(t)=b(t_0+t,r(t))dt+\sigma(t_0+t,r(t))dW(t)$

, where $W$ is $Q$ Brownian motion. Note that the short rate process itself is not a $Q$-martingale (short rate is not a traded asset). However, all contingent claims can be priced by taking $Q$-expectations of their discounted payoffs with respect to the short rate.

When we approximate this model with binomial tree, we don't need to find probabilities as we have already specified the $Q$-dynamics. We discretize Brownian motion increment $dW(t)$ which is symmetrically distributed around 0. Therefore, we use 0.5 as a probability in the binomial tree. Then all we need is to set nodes values that we match the drift $b$ and volatility $\sigma$ of $r(t)$. This approach is illustrated in the book "Term-Structure Models Using Binomial Trees" by Gerald W. Buetow Jr and James Sochacki.

But this there is another approach when you assign probabilities in your binomial model to match continuous prices process. See a book "Interest Rate Models: An Introduction" by Andrew J. G. Cairns, Chapter 10.2.3, page 163. There he fixes the increment and varies the probabilities.

• What confuses me is that I have seen two introductory level textbooks that simply assume risk-neutral probability of 0.5 as if it should be obvious, without resorting to SDE at all. That's why I thought there should be a simple/intuitive explanation(not that I don't appreciate your explanation).
– user2163
Oct 19, 2012 at 3:22
• @ezbentley Can you give me a reference? I'll take a look. Oct 19, 2012 at 11:47
• I forgot the exact books, but here they are also simply assuming 0.5 risk-neutral probability: link and link. My impression is that, in the stock binomial model, you assume up and down movement and calculate the risk-neutral probability. In the short rate model, you instead assume risk-neutral probability and match up and down movement to what's observed in the market. But I am not sure if this is the right way to view it.
– user2163
Oct 19, 2012 at 14:43