Reconciliation of Black-Scholes Variants¶
This note is just to show that the different variants of Black-Scholes formula in textbook and tutorial solutions are in fact the same.
- \(S\): Underlying share price
- \(t\): Time to maturity
- \(\sigma\): Standard deviation of underlying share price
- \(K\): Exercise price
- \(r_f\): Risk-free rate
Variant 1¶
This is the one shown in our formula sheet, and is also the traditional presentation of Black-Scholes model.
\[
\begin{equation}
C=SN(d_1)-N(d_2)Ke^{-r_f t}
\end{equation}
\]
\[
\begin{equation}
d_1=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}}
\end{equation}
\]
\[
\begin{equation}
d_2=d_1 - \sigma \sqrt{t}
\end{equation}
\]
Variant 2¶
This one comes from textbook, and looks slightly different in that \(PV(K)\) replaces \(K\) in the natural logarithm.
\[
\begin{equation}
C=SN(d_1)-N(d_2)PV(K)
\end{equation}\]
\[
\begin{equation}
d_1=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2}
\end{equation}
\]
\[
\begin{equation}
d_2=d_1 - \sigma \sqrt{t}
\end{equation}
\]
However, it's in fact easy to show that \(d_1\) in eq. (5) is the same as in eq. (2): Under continuous compounding, \(PV(K)=Ke^{-r_f t}\):
\[
\begin{align}
d_1 &=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2}\newline
&=\frac{ln(\frac{S}{Ke^{-r_f t}})}{\sigma \sqrt{t}} +\frac{\frac{\sigma^2}{2}t}{\sigma \sqrt{t}}\newline
&=\frac{ln(\frac{S}{Ke^{-r_f t}})+\frac{\sigma^2}{2}t}{\sigma \sqrt{t}}\newline
&=\frac{ln(\frac{S}{K})+r_f t+\frac{\sigma^2}{2}t}{\sigma \sqrt{t}}\newline
&=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}}=eq. (2)
\end{align}
\]
Therefore, the two variants are effectively the same under continuous compounding.