Assume the risk-free interest rate is 10% and is equal to fund's benchmark, the portfolio net asset value is $100, and the fund's standard deviation is 20%. Also assume time horizon of 1 year.
-What is the Black-Scholes value of the call option on management incentive fee?

Question

Quizplus · Accepted Answer

To calculate the Black-Scholes value of the call option on management incentive fee, we need the following inputs:
1. Strike Price: There is no information given about the strike price, so we assume it to be equal to the net asset value of the portfolio, which is $100.
2. Time to expiration (in years): The time horizon is given as 1 year.
3. Risk-free interest rate: Given as 10%.
4. Stock price volatility: Given as 20%, which is the standard deviation of the portfolio.

We do not have any information about the expected return or dividend yield of the portfolio, which are the other inputs needed for the Black-Scholes model. However, we can assume the portfolio return to follow a log-normal distribution, as is commonly done in option pricing models.

Using the Black-Scholes formula, the value of the call option on management incentive fee is:
C = S*N(d1) - Ke^(-rt)*N(d2)
where,
S = Portfolio net asset value = $100
K = Strike price = $100
r = Risk-free interest rate = 10%
T = Time to expiration = 1 year
sigma = Stock price volatility (annualized) = 20%

d1 = [ln(S/K) + (r + sigma^2/2)T]/(sigma*sqrt(T))
= [ln(100/100) + (0.1 + 0.2^2/2)*1]/(0.2*sqrt(1))
= 0.40

d2 = d1 - sigma*sqrt(T)
= 0.40 - 0.2*sqrt(1)
= 0.20

N(d1) = 0.6554 (from standard normal distribution table)
N(d2) = 0.5793 (from standard normal distribution table)

Plugging in the values, we get:
C = 100*0.6554 - 100*e^(-0.1*1)*0.5793
= $8.20

Assume the Risk-Free Interest Rate Is 10% and Is Equal