This work mainly studies modeling and existence issues for martingale models of option markets with one stock and a collection of European call options for one fixed maturity and infinetely many strikes. In particular, we study Dupire's and Schweizer-Wissel's models, especially the latter one. These two types of models have two completely different pricing approachs, one of which is martingale approach (in Dupire's model), and other one is a market approach (in Schweizer-Wissel's model). After arguing that Dupire's model suffers from the several lacks comparing to Schweizer-Wissel's model, we extend the latter one to get the variations for the case of options on interest rate indexes and futures options. Our models are based on the newly introduced definitions of local implied volatilities and a price level proposed by Schweizer and Wissel. We get explicit expressions of option prices as functions of the local implied volatilities and the price levels in our variations of models. Afterwards, the absence of the dynamic arbitrage in the market for such models can be described in terms of the drift restrictions on the models' coefficients. Finally we demonstrate the application of such models by a simple example of an investment portfolio to show how Schweizer-Wissel's model works generally.