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 $15\%$ of annual returns, but its

current high$-$water mark is $$88,$ and net asset value is $$80.$

Required:

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.$)$

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

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?

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

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?

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

d$.$ Recalculate the incentive fee value for part $($b$)$ if an increase in fund leverage increases volatility to $59\%.$

Note: Do not round intermediate calculations. Round your answer to $2$ decimal places.

Answer is complete but not entirely correct.