Showing papers by "Danske Bank published in 1996"

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.