Reconciliation of Black-Scholes Variants

Author
Affiliation

Mingze Gao, PhD

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.

Variant 1

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

(1)C=SN(d1)βˆ’N(d2)Keβˆ’rft

(2)d1=ln(SK)+(rf+σ22)tσt

(3)d2=d1βˆ’Οƒt

Variant 2

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

(4)C=SN(d1)βˆ’N(d2)PV(K)

(5)d1=ln(SPV(K))Οƒt+Οƒt2

(6)d2=d1βˆ’Οƒt

However, it’s in fact easy to show that d1 in is the same as in : Under continuous compounding, PV(K)=Keβˆ’rft:

d1=ln(SPV(K))Οƒt+Οƒt2=ln(SKeβˆ’rft)Οƒt+Οƒ22tΟƒt=ln(SKeβˆ’rft)+Οƒ22tΟƒt=ln(SK)+rft+Οƒ22tΟƒt=ln(SK)+(rf+Οƒ22)tΟƒt

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

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