PRICING EUROPEAN OPTIONS WITHOUT PROBABILITY
It is well known that in the case where the stock price St is governed by the equation dSt/St = μdt + σdWt, any European option satisfying weak regularity conditions has a fair price (the Black—Scholes formula and its generalizations). We consider the case where no probabilistic assumptions are made about St; instead, we assume that the derivative security D which pays a dividend of (dSt/St) (the squared relative increase in the price of St) each instant dt is traded in the market. We prove that the “regular” European options have fair prices provided that both St and Dt (the price process of D) are continuous and the fractal dimensions of the graphs of St and Dt satisfy certain inequalities. Intuitively our assumptions are much weaker than the usual assumption dSt/St = μdt + σdWt.