# What's the logic behind binomial model ups and downs?

I want to understand what is the underlying logic in the calculation of u and d in a binomial model.

$$u = \exp\Bigl(\sigma \sqrt{\Delta t} \Bigr), \quad d = \exp\Bigl(-\sigma \sqrt{\Delta t} \Bigr)$$

I don't know if i'm explaining myself correctly, but why are those formulas used to calculate the two possible values for an asset price given the volatility and a time step? what's the mathematical/statistical logic behind it?

## 1 Answer

one of the most fundamental results states that the binomial model converges towards the Black Scholes model if the step size $$\Delta t$$ converges to zero.

The Black Scholes model is an option pricing model where the underlying is given by

$$S_T = S_0 \cdot \exp \Bigl(\sigma W_T - \frac 12 \sigma^2 T \Bigr).$$

By choosing $$u = \exp(\sigma \sqrt{\Delta t}), \quad d = \exp(-\sigma \sqrt{\Delta t})$$

the price process converges in law (i.e. weak convergence) to the process $$S_T$$. More details can be found on page three of this document.