In everyone's binomial trees online I see constant U and D. Even when I read Option Volatility and Pricing by Natenburg, all his diagrams use a constant U and D (where U is the upwards magnitude from one step to the next. For example 1.05). The whole premise of a binomial model is that, as the amount of steps increase, the option price derived from the model should get closer and closer to Black Scholes' (see below).
Binomial Vs Black Scholes
The problem is that, with a constant U and D, it doesn't. The value of the option continues to grow indefinitely with the number of steps. If we fix the time to maturity, and increase the number of steps in between now and maturity, we increase the range of potential prices the underlying can reach. This increases volatility and hence increases the value of the option.
Holding all else constant, increasing the number of steps only increases the value of the option, which makes me believe that I'm missing some other adjustment.
My question in: What adjustments to U and D, or to the model, do I need to make such that I can approximate the BS Model (as per the image above).
A constant U and D does not seem correct. I've attached another picture from the book by Natenburg below for reference. If I increase the number of steps from 3 to 20, the value of the option goes from 5.22 to 9.24.