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Reconciliation of Black-Scholes Variants

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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.

C=SN(d1)N(d2)Kerft\begin{equation} C=SN(d_1)-N(d_2)Ke^{-r_f t} \end{equation}
d1=ln(SK)+(rf+σ22)tσt\begin{equation} d_1=\frac{ln(\frac{S}{K})+(r_f+\frac{\sigma^2}{2})t}{\sigma \sqrt{t}} \end{equation}
d2=d1σt\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)PV(K) replaces KK in the natural logarithm.

C=SN(d1)N(d2)PV(K)\begin{equation} C=SN(d_1)-N(d_2)PV(K) \end{equation}
d1=ln(SPV(K))σt+σt2\begin{equation} d_1=\frac{ln(\frac{S}{PV(K)})}{\sigma \sqrt{t}}+\frac{\sigma \sqrt{t}}{2} \end{equation}
d2=d1σt\begin{equation} d_2=d_1 - \sigma \sqrt{t} \end{equation}

However, it's in fact easy to show that d1d_1 in eq. (5) is the same as in eq. (2): Under continuous compounding, PV(K)=KerftPV(K)=Ke^{-r_f t}:

d1=ln(SPV(K))σt+σt2=ln(SKerft)σt+σ22tσt=ln(SKerft)+σ22tσt=ln(SK)+rft+σ22tσt=ln(SK)+(rf+σ22)tσt=eq.(2)\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.