Option pricing impact of alternative continuous-time dynamics
type: Item
instance of: scholarly article
title: Option pricing impact of alternative continuous-time dynamics
zbMATH Open document ID: 0956.60034
author: Damiano Brigo
author: Fabio Mercurio
published in: Finance and Stochastics
publication date: 2000-11-01T00:00:00Z
review text: Since the seminal papers by \textit{F. Black} and \textit{M. Scholes} [J. Polit. Econ. 81, 637-654 (1973)]\ and \textit{R. Merton} [Bell. J. Econ. Manage. Sci. 4, 141-183 (1973)]\ the stock-price dynamics for option pricing models has often been modeled through a continuous-time geometric Brownian motion. This assumption is still quite popular, especially among practitioners, even though the empirical distribution of stock returns is usually found more leptokurtic than the normal one. Indeed, the tradeoff between analytical tractability and empirical findings still makes the geometric Brownian motion a quite acceptable choice. However, as the authors prove in this paper, the geometric Brownian motion is not the only continuous-time process that possesses a lognormal marginal distribution and normal log-returns. In fact, the authors construct a family of processes with these characteristics and analyze the main implications of the existence of such a family as far as option pricing is concerned. The paper begins with a stochastic differential equation (SDE) à la Black and Scholes for the stock price, whose solution is the traditional geometric Brownian motion. It is then possible to define a second SDE with a different (and arbitrary given) volatility in the diffusion coefficient such that its solution has the same marginal lognormal density as that of the Black and Scholes geometric Brownian motion. Furthermore, the drift of this second SDE can be chosen in such a way that its solution has transition densities between two any instants of a prescribed finite subset of the time interval that match the transition densities of the Black and Scholes process. This new stock-price process behaves as follows: the standard deviation (per unit time) of its instantaneous log-returns is an arbitrary constant, whereas the standard deviation (per unit time) of any-finite-time log-returns coincides with that of the original Black and Scholes geometric Brownian motion. A family of …
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 91B24
zbMATH DE Number: 1524264
zbMATH Keywords: stock-price dynamics
zbMATH Keywords: Black and Scholes model
zbMATH Keywords: option pricing
zbMATH Keywords: discrete \(\Delta\)-volatility
zbMATH Keywords: discrete time versus continuous time
MaRDI profile type: Publication
full work available at URL: s007800050009
OpenAlex ID: W2062274972
DOI: 10.1007/S007800050009
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description: scientific article; zbMATH DE number 1524264