Why does the volatility smile flatten as maturities increase?

First, I can't find a purely "financial" explanation for this.

Also the only mathematical explanation I've found so far was using the large deviations theory, which is quite complex.

Is there a rather simple mathematical explanation ?

Thanks !

Answer

The central limit theorem guarantees, under fairly general assumptions, that the sum of returns becomes more normally distributed as the number of returns grows (technically, defining a return as $\mathrm{log}(S_{t+\Delta t}/S_t)$, $\sum_i ^n \mathrm{log}(S_{t+\Delta t i}/S_{t+\Delta t (i-1)} \to \mathcal{N}(\cdot,\cdot)$ as $n \to \infty$). Thus, as $T$ gets larger, the Black Scholes assumption of normally distributed log returns becomes more and more valid. This is exemplified by the flattening implied volatility smile.