Discrete Time vs Continuous Time Stock-price Dynamics and implications for Option Pricing

In the present paper we construct stock price processes with the same marginal lognormal law as that of a geometric Brownian motion and also with the same transition density (and returns’ distributions) between any two instants in a given discrete-time grid. We then illustrate how option prices based on such processes differ from Black and Scholes’, in that option prices can be either arbitrarily close to the option intrinsic value or arbitrarily close to the underlying stock price. We also explain that this is due to the particular way one models the stock-price process in between the grid time instants which are relevant for trading. ∗We are grateful to Aleardo Adotti, our current head at the Product Development Group of IMI/San Paolo, and to Renzo Avesani, our former head at Risk Management of Cariplo, for encouraging us in the prosecution of the most speculative side of research in mathematical finance. We are grateful also to Wolfgang Runggaldier and to two anonymous referees for their remarks and suggestions. Finally, the paper has been improved both in exposition and contents thanks to private communications with Prof. Hans Nieuwenhuis from the University of Groningen D. Brigo, F. Mercurio. Banca IMI – PDG internal report 2 The theoretical result concerning scalar stochastic differential equations with prescribed diffusion coefficient whose densities evolve in a prescribed exponential family, on which part of the paper is based, is presented in detail.