A hedge fund with net asset value of $$90$ per share currently has a high$-$water mark of $$99.$ Suppose it is January $1,$ the standard deviation of the fund$$s annual returns is $46\%,$ and the risk$-$free rate is $3\%.$ The fund has an incentive fee of $14\%$ of annual returns, but its current high$-$water mark is $$99,$ and net asset value is $$90.$

Required:

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.

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.

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.

Recalculate the incentive fee value for part $($b$)$ if an increase in fund leverage increases volatility to $56\%.$

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