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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.