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We consider a generic market model with a single stock and with randomvolatility. We assume that there is a number of tradable options for that stockwith different strike prices. The paper states the problem of finding a pricingrule that gives Black-Scholes price for at-money options and such that themarket is arbitrage free for any number of tradable options, even if there aretwo Brownian motions only: one drives the stock price, the other drives thevolatility process. This problem is reduced to solving a parabolic equation.

Pricing Rule Based on Non-Arbitrage Arguments for Random Volatility and Volatility Smile