Black Monday under fatter models

The S&P 500 fell 20.47% on 19 Oct 1987. (a) Using daily $\sigma = 1\%$, compute the one-day probability of a fall at least this large under (i) the normal, (ii) a Student-$t$ with $\nu = 3$ scaled to standard deviation $\sigma$, (iii) a Student-$t$ with $\nu = 4$ scaled likewise. Express (ii) and (iii) as expected waiting times in years of 252 days. (b) Repeat (i) and (ii) with the 20-day pre-crash volatility of 1.85% and the log return of $-22.9\%$. (c) What do (a) and (b) say about which modelling choice, distribution or volatility, matters more for this event?