Modelling Questions in Security Valuation

Abstract

The Standard model for the valuation of options and other derivative securities, which has been accepted by the financial markets, is the continuous-time Black-Scholes model. Stock prices are assumed to be given by a geometric Brownian motion $${S_t}{S_0}\exp \left( {\sigma {B_t} + \left( {\mu - \frac{{{\sigma ^2}}} {2}} \right)t} \right)\quad \left( {0 \leqslant t \leqslant T} \right)$$ , where (Bt)0≤t≤T denotes a standard Brownian motion and the constants σ > 0 and µ represent the volatility and the drift of the price process. It is assumed that securities can be and actually are traded continuously over time. In the real world, however, new stock prices appear at certain discrete time points only and prices remain constant in between. This raises the question of the relation between realistic discrete- time and idealized continuous- time models. More precisely we ask how close one class does approximate the other and in which sense. Various results have appeared recently (see e.g. [1], [3]) where approximation is discussed in terms of weak convergence of distributions. Willinger and Taqqu [4] study pathwise approximation. This is our goal here, too, since traders are used to look at the paths of stock prices as they appear on charts or on electronic boards and not at distributions.