A financial institution wishes to unwind$/$liquidate a position of $160$ million shares in a stock over $8$ days. The dollar bid$$offer spread p as a function of daily trading volume q is p$($q$)=$ a $+$ becq where a $=0\mathrm{.}28,$ b $=0\mathrm{.}15,$ and c $=0\mathrm{.}06$ and q is measured in millions of shares. The standard deviation of the stock price change per day is $$1\mathrm{.}25.$

a$)$ What is the optimal strategy that minimizes the $90\%$ confidence level VaR after trading costs have been considered? Hint: use excel solver to identify the optimal trading strategy for the financial institution. $(6$ marks$)$

b$)$ What is the average time the institution waits before selling? $(3$ marks$)$

c$)$ How does the average time from part b$)$ change as the confidence level changes to $95\%?(3$ marks$)$

d$)$ Define the concept of a 'liquidity black hole' and identify the primary causes that can lead to this phenomenon in financial markets. Additionally, examine how algorithmic trading might contribute to the formation of liquidity black holes, citing specific scenarios that could trigger this event. $(6$ marks$)$