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?
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.
