Non-equilibrium phenomena occur not only in physical world, but also in
finance. In this work, stochastic relaxational dynamics (together with path
integrals) is applied to option pricing theory. A recently proposed model (by
Ilinski et al.) considers fluctuations around this equilibrium state by
introducing a relaxational dynamics with random noise for intermediate
deviations called ``virtual'' arbitrage returns. In this work, the model is
incorporated within a martingale pricing method for derivatives on securities
(e.g. stocks) in incomplete markets using a mapping to option pricing theory
with stochastic interest rates. Using a famous result by Merton and with some
help from the path integral method, exact pricing formulas for European call
and put options under the influence of virtual arbitrage returns (or
intermediate deviations from economic equilibrium) are derived where only the
final integration over initial arbitrage returns needs to be performed
numerically. This result is complemented by a discussion of the hedging
strategy associated to a derivative, which replicates the final payoff but
turns out to be not self-financing in the real world, but self-financing {\it
when summed over the derivative's remaining life time}. Numerical examples are
given which underline the fact that an additional positive risk premium (with
respect to the Black-Scholes values) is found reflecting extra hedging costs
due to intermediate deviations from economic equilibrium.Comment: 21 pages, 4 figures, to appear in EPJ B, major changes (title,
abstract, main text