A hedge fund with net asset value of $$80$ per share currently has a high water mark of $$88.$ Suppose it is January $1,$ the standard

deviation of the fund's annual returns is $49\%,$ and the risk$-$free rate is $5\%.$ The fund has an incentive fee of $15\%.$

a$.$ What is the value of the annual incentive fee according to the Black$-$Scholes formula? $($Treat the risk$-$free rate as a continuously

compounded value to maintain consistency with the Black$-$Scholes formula.$)($Do not round intermediate calculations. Round your

answer to $3$ decimal places.$)$

Annual incentive fee

per share

b$.$ What would the annual incentive fee be worth if the fund had no high water mark and it earned its incentive fee on its total return?

$($Do not round intermediate calculations. Round your answer to $3$ decimal places.$)$

Annual incentive fee

per share

c$.$ What would the annual incentive fee be worth if the fund had no high water mark and it earned its incentive fee on its return in

excess of the risk$-$free rate? $($Do not round intermediate calculations. Round your answer to $3$ decimal places.$)$

d$.$ Recalculate the incentive fee value for part $($b$)$ assuming that an increase in fund leverage increases volatility to $59\%.($Do not round

intermediate calculations. Round your answer to $3$ decimal places.$)$