Why Some Binomial Short-Rate Trees Use Equal Branch Probabilities
Summary
The document explains why a binomial tree used to approximate a short-rate model may assign equal probabilities to its up and down branches. In the described approach, the model first specifies short-rate dynamics under the pricing measure, including drift and volatility, and is then calibrated to market prices. The tree discretizes the Brownian increment, whose symmetric distribution motivates equal branch probabilities; node values are chosen to match the target drift and volatility.
This differs from deriving risk-neutral probabilities from a traded asset's no-arbitrage condition in a basic binomial pricing model. The short rate is not itself a traded asset or a pricing-measure martingale, though contingent claims can be valued using discounted payoff expectations. The answer also notes an alternative tree construction that fixes the increment and varies probabilities to match prices. Thus equal probabilities belong to a particular discretization method, not a universal rule for short-rate trees.
Key ideas
- Short-rate models commonly specify dynamics under the pricing measure before constructing a tree.
- Symmetric Brownian increments motivate equal probabilities in one binomial discretization.
- Tree node values are adjusted to match the model's drift and volatility.
- The short rate is not a traded asset and need not be a pricing-measure martingale.
- Alternative tree methods can vary probabilities to fit prices instead of fixing them at one half.
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Full text
# Risk neutral probability in binomial short rate model assumed to be 0.5? # 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? ## Answer by Alexey Kalmykov (score 2) https://quant.stackexchange.com/a/4358 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.
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