Reconciliation of Black-Scholes Variants

Teaching Notes

Mingze Gao, PhD

Macquarie University


May 25, 2020

This note is just to show that the different variants of Black-Scholes formula in textbook and tutorial solutions are in fact the same.

Variant 1

This is the one shown in our formula sheet, and is also the traditional presentation of Black-Scholes model.

\[ \begin{equation} C=SN(d_1)-N(d_2)Ke^{-r_f t} \end{equation} \tag{1}\]

\[ \begin{equation} d_1=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}} \end{equation} \tag{2}\]

\[ \begin{equation} d_2=d_1 - \sigma \sqrt{t} \end{equation} \tag{3}\]

Variant 2

This one comes from textbook, and looks slightly different in that \(PV(K)\) replaces \(K\) in the natural logarithm.

\[ \begin{equation} C=SN(d_1)-N(d_2)PV(K) \end{equation} \tag{4}\]

\[ \begin{equation} d_1=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2} \end{equation} \tag{5}\]

\[ \begin{equation} d_2=d_1 - \sigma \sqrt{t} \end{equation} \tag{6}\]

However, it’s in fact easy to show that \(d_1\) in Equation 5 is the same as in Equation 2: Under continuous compounding, \(PV(K)=Ke^{-r_f t}\):

\[ \begin{align} d_1 &=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2}\newline &=\frac{ln(\frac{S}{Ke^{-r_f t}})}{\sigma \sqrt{t}} +\frac{\frac{\sigma^2}{2}t}{\sigma \sqrt{t}}\newline &=\frac{ln(\frac{S}{Ke^{-r_f t}})+\frac{\sigma^2}{2}t}{\sigma \sqrt{t}}\newline &=\frac{ln(\frac{S}{K})+r_f t+\frac{\sigma^2}{2}t}{\sigma \sqrt{t}}\newline &=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}} \end{align} \]

Therefore, the two variants are effectively the same under continuous compounding.  

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