Extending previous work on non-equilibrium option pricing theory (Eur. Phys. J. 14 (2000) 383–394), a mean field approach is developed to understand the curvature of (implied by Black–Scholes (BS)) volatility surfaces (curves) as a function of moneyness (strike price divided by price). The previously developed hypothesis of a finite arbitrage time during which fluctuations around the equilibrium state (absence of arbitrage) are allowed to occur is generalized as follows. Instead of a unique arbitrage time independant of moneyness, a distribution of arbitrage times will be assumed, where the mean arbitrage time will be a function of moneyness. This hypothesis is motivated by the fact that the trading volume is the largest for at-the-money options. Assuming now the arbitrage time to be inversely proportional to trading volume naturally leads to our generalized hypothesis on the mean arbitrage time. Consequences on plain vanilla option prices will be studied.
