Denote by $r$ the risk$-$free interest rate, assumed to be constant. The value of a European

asset$-$or$-$nothing put with strike $K$ and maturity $T$ on a lognormally distributed underlying

asset with volatility $$ and paying dividends continuously at rate $q$ is

$P_{\text{A}\text{o}\text{N}}=S{e}^{-qT}N(-d_1),$

where

$d_1=\frac{ln(\frac{S}{K})+(r-q+\frac{{}^2}{2})T}{\sqrt[\phantom{2}]{T}}.$

Find the limit of its value as volatility goes to $0$ and to infinity. Finally, find the Delta of

the asset$-$or$-$nothing put.