The Chicago Board Options Exchange $($CBOE$)$ Volatility Index $($VIX$)$ is a widely recognised measure of market volatility. It reflects the expectations in terms of price changes on the S&P $500$ index and it is used by investors to measure the level of volatility when making investment decisions.

After reviewing the present quotations of the VIX, answer the following questions:

$($a$)$

Why does the VIX offer predictions over the short term only?

$[20$ marks$]$

$($b$)$

If the current value of VIX is about $18,$ what do you expect is the market situation? $[20$ marks$]$

$($c$)$

Now consider the VIX rising to $32$ overnight. What conclusions can you derive? Offer possible scenarios. $[20$ marks$]$

$($d$)$

Does the VIX have any direct influence on the market prices? Discuss. $[20$ marks$]$

$($e$)$

When should you consider investing in the VIX? If so$,$ how would you do that in practice?