Option pricing impact of alternative continuous-time dynamics
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 stock-price processes is derived that behave almost equivalently to a geometric Brownian motion in that they are probabilistically identical along finite subsets of the time interval which has been selected. Then the problem of option pricing in a continuous-time framework is considered. The most interesting result is that notwithstanding the above mentioned discrete-time indistinguishability from the Black and Scholes model, the stock-price processes of our family lead to completely arbitrary option prices, ranging the whole interval of prices in-between the no-arbitrage lower and upper bounds. The authors' result is similar in spirit to that of \textit{L. C. G. Rogers} and \textit{S. E. Satchell} [``Does the behaviour of the assert tell us anything about the option price formula? A cautionary tale'' (Preprint Univ. of Bath, 1996)]. However, their approach is much more constructive in that they provide, under the original measure, easy and explicit dynamics for the stock price that are theoretically consistent with option price whatever is observed in reality.
scientific article; zbMATH DE number 1524264