Chapter 20, Problem 13PS

Summary Introduction

(a)

To calculate:

If the high water mark is $\$66$ and asset value is $\$62$ , then ascertain the annual incentive fee as per the Black-Scholes formula.

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Answer to Problem 13PS

The annual incentive fee is $\$2.3366$ .

Explanation of Solution

Given:

  $\begin{array}{c}\text{Net asset value}\left({\text{S}}_0\right)=\$62\\ \text{High water mark}\left(\text{X}\right)=\$66\\ \text{Risk free rate}=4\%\\ \text{Standard Deviation}=0.50\\ \text{Incentive fee}=20\%\\ \text{Time}=1\text{ year}\end{array}$

The black-scholes formula is as follows:

  $\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)$

  $\begin{array}{c}\text{Here,}\\ {\text{S}}_0=\text{Current stock price}\\ \text{N}\left({\text{d}}_1\right)\text{andN}\left({\text{d}}_2\right)=\text{Cumulative normal distribution}\\ \text{X}=\text{Exercise price}\\ \text{r}=\text{Annualised risk free rate}\\ \text{s}=\text{Annualised standard deviation}\\ \text{T}=\text{Time to expiry}\end{array}$

For calculating the ${\text{d}}_1{\text{ and d}}_2$ , the formula is as follows:

  $\begin{array}{l}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ \\ {\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\end{array}$

  $\begin{array}{c}\text{Here,}\\ \text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)=\text{Natural logarithmic value of}\left(\frac{{\text{S}}_0}{\text{X}}\right)\\ \text{e}=\text{Exponential having the constant of 2}\text{.71828}\\ \text{δ}=\text{Annual dividend yield}\\ \text{σ}=\text{Annualised standard deviation}\end{array}$

Now the calculation of ${\text{d}}_1$ of call option:

  $\begin{array}{c}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ =\frac{\text{In}\left(\frac{\$62}{\$66}\right)+\left(0.04-0+\frac{{0.50}^2}{2}\right)1}{0.50\sqrt{1}}\\ =\frac{\text{In}\left(0.94\right)+\left(0.04+0.125\right)1}{0.50}\\ =\frac{-0.0619+0.165}{0.50}\\ =0.2062\end{array}$

The value of $\text{N}\left({\text{d}}_1\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_1\right)$ is $0.5817$

Now, the calculation of ${\text{d}}_2$ of call option:

  $\begin{array}{c}{\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\\ =0.2062-0.50\sqrt{1}\\ =0.2062-0.50\\ =-0.2938\end{array}$

The value of $\text{N}\left({\text{d}}_2\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_2\right)$ is $0.3845$

By substituting the values, value of call option is:

  $\begin{array}{c}\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)\\ =\$62\times 0.5817-\$66\times \frac{1}{{\text{e}}^{\text{rT}}}\times 0.3845\\ =\$36.0654-\$66\times \frac{1}{{2.71828}^{0.04\times 1}}\times 0.3845\\ =\$36.0654-\$66\times 0.9608\times 0.3845\\ =\$11.6832\end{array}$

Thus, the value of call option is $\$11.6832$

The computation of value of incentive fee is as follows:

  $\begin{array}{c}\text{Annual incentive fee}=\text{Percentage of incentive}\times \text{Call value}\\ =0.20\times \$11.6832\\ =\$2.3366\end{array}$

Thus, the annual incentive fee is $\$2.3366$ .

Summary Introduction

(b)

To calculate:

If the high water mark is zero and asset value is $\$62$ , then ascertain the annual incentive fee on total return as per the Black-Scholes formula.

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Answer to Problem 13PS

The annual incentive fee is $\$2.6505$ .

Explanation of Solution

Given:

  $\begin{array}{c}\text{Net asset value}\left({\text{S}}_0\right)=\$62\\ \text{Risk free rate}=4\%\\ \text{Standard Deviation}=0.50\\ \text{Incentive fee}=20\%\\ \text{Time}=1\text{ year}\end{array}$

The value of X is changed to $\$62$

The black-scholes formula is as follows:

  $\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)$

For calculating the ${\text{d}}_1{\text{ and d}}_2$ , the formula is as follows:

  $\begin{array}{l}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ \\ {\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\end{array}$

Now the calculation of ${\text{d}}_1$ of call option:

  $\begin{array}{c}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ =\frac{\text{In}\left(\frac{\$62}{\$62}\right)+\left(0.04-0+\frac{{0.50}^2}{2}\right)1}{0.50\sqrt{1}}\\ =\frac{\text{In}\left(1\right)+\left(0.04+0.125\right)1}{0.50}\\ =\frac{0+0.165}{0.50}\\ =0.33\end{array}$

The value of $\text{N}\left({\text{d}}_1\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_1\right)$ is $0.6293$

Now, the calculation of ${\text{d}}_2$ of call option:

  $\begin{array}{c}{\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\\ =0.33-0.50\sqrt{1}\\ =0.33-0.50\\ =-0.17\end{array}$

The value of $\text{N}\left({\text{d}}_2\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_2\right)$ is $0.4325$

By substituting the values, value of call option is:

  $\begin{array}{c}\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)\\ =\$62\times 0.6293-\$62\times \frac{1}{{\text{e}}^{\text{rT}}}\times 0.4325\\ =\$39.0166-\$62\times \frac{1}{{2.71828}^{0.04\times 1}}\times 0.4325\\ =\$39.0166-\$62\times 0.9608\times 0.4325\\ =\$13.2527\end{array}$

Thus, the value of call option is $\$13.2527$

The computation of value of incentive fee is as follows:

  $\begin{array}{c}\text{Annual incentive fee}=\text{Percentage of incentive}\times \text{Call value}\\ =0.20\times \$13.2527\\ =\$2.6505\end{array}$

Thus, the annual incentive fee is $\$2.6505$ .

Summary Introduction

(c)

To calculate:

If the high water mark is zero and asset value is $\$62$ , then ascertain the annual incentive fee on return in excess of risk-free rate as per the Black-Scholes formula.

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Answer to Problem 13PS

The annual incentive fee is $\$2.4179$ .

Explanation of Solution

Given:

  $\begin{array}{c}\text{Net asset value}\left({\text{S}}_0\right)=\$62\\ \text{Risk free rate}=4\%\\ \text{Standard Deviation}=0.50\\ \text{Incentive fee}=20\%\\ \text{Time}=1\text{ year}\end{array}$

The black-scholes formula is as follows:

  $\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)$

For calculating the ${\text{d}}_1{\text{ and d}}_2$ , the formula is as follows:

  $\begin{array}{l}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ \\ {\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\end{array}$

The value of X has been changed which is:

  $\begin{array}{c}\text{X}={\text{S}}_0\times {\text{e}}^{0.04}\\ =\$62\times {\text{e}}^{0.04}\\ =\$62\times {2.71828}^{0.04}\\ =\$64.5303\end{array}$

Now the calculation of ${\text{d}}_1$ of call option:

  $\begin{array}{c}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ =\frac{\text{In}\left(\frac{\$64.5303}{\$62}\right)+\left(0.04-0+\frac{{0.50}^2}{2}\right)1}{0.50\sqrt{1}}\\ =\frac{\text{In}\left(1.0408\right)+\left(0.04+0.125\right)1}{0.50}\\ =\frac{0.03999+0.165}{0.50}\\ =0.40998\end{array}$

The value of $\text{N}\left({\text{d}}_1\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_1\right)$ is $0.6591$

Now, the calculation of ${\text{d}}_2$ of call option:

  $\begin{array}{c}{\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\\ =0.40998-0.50\sqrt{1}\\ =0.40998-0.50\\ =-0.09002\end{array}$

The value of $\text{N}\left({\text{d}}_2\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_2\right)$ is $0.4641$

By substituting the values, value of call option is:

  $\begin{array}{c}\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)\\ =\$62\times 0.6591-\$64.5303\times \frac{1}{{\text{e}}^{\text{rT}}}\times 0.4641\\ =\$40.8642-\$64.5303\times \frac{1}{{2.71828}^{0.04\times 1}}\times 0.4641\\ =\$40.8642-\$64.5303\times 0.9608\times 0.4641\\ =\$12.0897\end{array}$

Thus, the value of call option is $\$12.0897$

The computation of value of incentive fee is as follows:

  $\begin{array}{c}\text{Annual incentive fee}=\text{Percentage of incentive}\times \text{Call value}\\ =0.20\times \$12.0897\\ =\$2.4179\end{array}$

Thus, the annual incentive fee is $\$2.4179$ .

Summary Introduction

(d)

To calculate:

If the high water mark is zero and asset value is $\$62$ , then ascertain the annual incentive fee on total return as per the Black-Scholes formula with an increase in the risk to $0.60$ .

Introduction:

The Black-scholes model is used for determining the price of European call option by using the variation of price in financial instruments. This model uses stock price, option price, and time for ascertaining the call option price.

Answer to Problem 13PS

The annual incentive fee is $\$3.1159$ .

Explanation of Solution

Given:

  $\begin{array}{c}\text{Net asset value}\left({\text{S}}_0\right)=\$62\\ \text{Risk free rate}=4\%\\ \text{Standard Deviation}=0.60\\ \text{Incentive fee}=20\%\\ \text{Time}=1\text{ year}\end{array}$

The value of X is changed to $\$62$

The black-scholes formula is as follows:

  $\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)$

For calculating the ${\text{d}}_1{\text{ and d}}_2$ , the formula is as follows:

  $\begin{array}{l}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ \\ {\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\end{array}$

Now the calculation of ${\text{d}}_1$ of call option:

  $\begin{array}{c}{\text{d}}_1=\frac{\text{In}\left(\frac{{\text{S}}_0}{\text{X}}\right)+\left(\text{r}-\text{δ}+\frac{{\text{σ}}^2}{2}\right)\text{T}}{\text{σ}\sqrt{\text{T}}}\\ =\frac{\text{In}\left(\frac{\$62}{\$62}\right)+\left(0.04-0+\frac{{0.60}^2}{2}\right)1}{0.60\sqrt{1}}\\ =\frac{\text{In}\left(1\right)+\left(0.04+0.18\right)1}{0.60}\\ =\frac{0+0.22}{0.60}\\ =0.3667\end{array}$

The value of $\text{N}\left({\text{d}}_1\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_1\right)$ is $0.6431$

Now, the calculation of ${\text{d}}_2$ of call option:

  $\begin{array}{c}{\text{d}}_2={\text{d}}_1-\text{σ}\sqrt{\text{T}}\\ =0.3667-0.60\sqrt{1}\\ =0.3667-0.60\\ =-0.2333\end{array}$

The value of $\text{N}\left({\text{d}}_2\right)$ will be computed by using the cumulative normal distribution table. This function can also be used in the Ms-excel.

Thus, the value of $\text{N}\left({\text{d}}_2\right)$ is $0.4078$

By substituting the values, value of call option is:

  $\begin{array}{c}\text{C}={\text{S}}_0\text{N}\left({\text{d}}_1\right)-{\text{Xe}}^{-\text{rT}}\text{N}\left({\text{d}}_2\right)\\ =\$62\times 0.6431-\$62\times \frac{1}{{\text{e}}^{\text{rT}}}\times 0.4078\\ =\$39.8722-\$62\times \frac{1}{{2.71828}^{0.04\times 1}}\times 0.4078\\ =\$39.8722-\$62\times 0.9608\times 0.4078\\ =\$15.5797\end{array}$

Thus, the value of call option is $\$15.5797$

The computation of value of incentive fee is as follows:

  $\begin{array}{c}\text{Annual incentive fee}=\text{Percentage of incentive}\times \text{Call value}\\ =0.20\times \$15.5797\\ =\$3.1159\end{array}$

Thus, the annual incentive fee is $\$3.1159$ .