CRR details

(a) Show that with $u=e^{\sigma\sqrt{\Delta t}}$, $d=1/u$ and $r>q$, the no-arbitrage condition $d<e^{(r-q)\Delta t}<u$ holds iff $\Delta t<\sigma^2/(r-q)^2$. Evaluate the bound for $\sigma=20\%$, $r-q=5\%$. (b) Derive the expansion (8.13) of $p$ in powers of $\sqrt{\Delta t}$. (c) Using Table 8.1, estimate the constant $C$ in error$,\approx C/n$ for even $n$, and use Richardson extrapolation on $n=500$ and $n=1000$ to improve the estimate of the ATM call. Why does extrapolating an even and an odd $n$ fail?