The focus of the present paper is on the simultaneous dynamics of all the liquidity traded derivative instruments written on the underlying stocks. For the sake of simplicity, only a single underlying index or stock is considered. It is assumed that European call options of all strikes and maturities are traded, their prices are observable and they can be bought and sold in any quantity. Arbitrage-free dynamics of equity market is modeled by Itô stochastic differential equations. The main focus of the paper is a characterization of the absence-of-arbitrage by means of a drift condition and a spot consistency condition in line with the original Heath-Jarrow-Morton absence-of-arbitrage condition. Markov spot models are characterized as those with bounded variation local volatility dynamics. A parametric family is introduced for fitting the initial local volatility surface to observed option prices, and it is demonstrated that it is well suited to the analysis od actual financial data.