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

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