New Skin for the Old Ceremony: Eight Different Derivations of the Black-Scholes Formula

Posted: 18 Dec 1996

See all articles by Jesper Andreasen

Jesper Andreasen

Saxo Bank

Bjarke Jensen

University of Aarhus

Rolf Poulsen

University of Copenhagen - Department of Statistics and Operations Research

Date Written: November 15, 1996

Abstract

The paper surveys eight different derivations that all lead to the celebrated Black and Scholes (1973) formula. Describing these derivations leads us through many of the techniques applied in continuous-time asset pricing. The paper can therefore also be seen as an introduction to continuous-time finance. From pure arbitrage reasoning we have six different derivations: (i) The classical hedge argument that leads to the fundamental partial differential equation for option prices, (ii) the martingale approach where we derive the Black-Scholes formula as a risk-adjusted expectation, (iii) the change of numeraire technique that enables us to solve for the option price without calculating a single integral, (iv) a stop-loss start-gain strategy argument, (v) the European option price also solves a forward partial differential equation where the variables are strike and maturity date whereas current time and spot price are kept fixed, and (vi) convergence of a binomial model. The two last derivations put the Black-Scholes formula in an equilibrium context. The Black-Scholes formula is shown to be consistent with: (vii) the continuous-time capital asset pricing model, and (viii) a single period representative investor economy, where the representative investor has constant relative risk-aversion and is endowed with lognormally distributed terminal wealth.

JEL Classification: G12

Suggested Citation

Andreasen, Jesper and Jensen, Bjarke and Poulsen, Rolf, New Skin for the Old Ceremony: Eight Different Derivations of the Black-Scholes Formula (November 15, 1996). Available at SSRN: https://ssrn.com/abstract=7994

Jesper Andreasen (Contact Author)

Saxo Bank ( email )

Philip Heymans Alle 15
Hellerup, 2900
Denmark

Bjarke Jensen

University of Aarhus ( email )

Bartholins Alle, Bygning 350
DK-8000 Aarhus C
Denmark
+45 8942 1563 (Phone)

Rolf Poulsen

University of Copenhagen - Department of Statistics and Operations Research ( email )

Universitetsparken 5
DK-2100
Denmark
+45 (353) 20685 (Phone)

Here is the Coronavirus
related research on SSRN

Paper statistics

Abstract Views
2,237
PlumX Metrics