# Reconciliation of Black-Scholes VariantsÂ¶

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 1Â¶

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

$$$C=SN(d_1)-N(d_2)Ke^{-r_f t}$$$
$$$d_1=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}}$$$
$$$d_2=d_1 - \sigma \sqrt{t}$$$

## Variant 2Â¶

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

$$$C=SN(d_1)-N(d_2)PV(K)$$$
$$$d_1=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2}$$$
$$$d_2=d_1 - \sigma \sqrt{t}$$$

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. Â