Option Pricing under different uncertainty regimes

We define three uncertainty regimes. The first regime, we call the "classical" uncertainty regime. This is the regime used in the Black-Scholes model. The "non-classical" uncertainty regime is defined either in terms of so called "Heisenberg Uncertainty" or in terms of "e-Uncertainty". Intuitively, both types of "non-classical" uncertainty are augmented versions of classical uncertainty. For instance, with Heisenberg uncertainty it can be shown that the 'Reduction of Compound Lotteries' axiom in the von Neumann-Morgenstern expected utility model fails. "e -Uncertainty" can be shown to be a mixture of Heisenberg Uncertainty and classical uncertainty. We consider two models under "Heisenberg uncertainty". The first model introduces the relationship between the Schrödinger equation and a particular Brownian motion. Using Black-Scholes methodology, we find that although the portfolio change does not involve a Wiener process we can not say that the portfolio"s return is the risk-free rate. In the second model we augment the Brownian motion of the first model and we use again Black-Scholes methodology to derive an option price. The portfolio return is again not risk free and we obtain stochastic call and put prices. Finally, we provide for an appreciation of an option pricing model under so called " e-Uncertainty". d Finance 2004