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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.  

Last update: May 26, 2020

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