$.$ Define or briefly explain the following relationships and concepts

$1.$ The relation between the BOPM and the B$-$S OPM

$2.$ Implied Variance

$3.$ Neutral Delta Strategy

$4.$ Neutral Delta Position with Positive and Negative Gamma

$$

$5.$ The value of Merton$$s continuous dividend$-$adjusted BOPM for puts over Merton$$s continuous dividend$-$adjust B$-$S OPM for puts.

$6.$ Why the mean is unimportant in the BOPM for large number of sub$-$periods and in the B$-$S OPM.

$7.$ The Black pseudo American model for pricing American call options.