Explanation

Under the assumption of a lognormal distribution in binomial interest rate trees, the probabilities for the rate going up or down are typically set to be equal (50% each).

Key points about lognormal distribution in binomial trees:

1. Equal probabilities: The up and down movements are assigned equal probabilities of 0.5 each

2. Lognormal property: This ensures that interest rates remain positive (they cannot go below zero), which is a realistic assumption for interest rates

3. Mathematical foundation: The lognormal distribution implies that the logarithm of interest rates follows a normal distribution, which leads to symmetric probabilities in the binomial tree

4. Common implementation: In models like Black-Derman-Toy, the assumption of equal probabilities simplifies the tree construction while maintaining the lognormal property

Alternative: Some models may use unequal probabilities, but under the specific assumption of lognormal distribution, equal probabilities are the standard approach.