Reconciliation of Black-Scholes Variants

Teaching Notes
Author
Affiliation

Macquarie University

Published

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.

• $$S$$: 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.

$$$C=SN(d_1)-N(d_2)Ke^{-r_f t}$$ \tag{1}$

$$$d_1=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}}$$ \tag{2}$

$$$d_2=d_1 - \sigma \sqrt{t}$$ \tag{3}$

Variant 2

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

$$$C=SN(d_1)-N(d_2)PV(K)$$ \tag{4}$

$$$d_1=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2}$$ \tag{5}$

$$$d_2=d_1 - \sigma \sqrt{t}$$ \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.