We use Neural Networks to approximate contingent claim prices defined by a no-arbitrage Partial Differential Equation allowing for state dependent volatility, the smile effect for example. The Neural Networks' weights are determined by means of the Galerkin technique as to satisfy the no-arbitrage Partial Differential Equation. A general solution procedure is developed for European Contingent Claim pricing assuming that the volatility of the asset price can be described by means of a step function. Following this procedure we are able both to estimate endogenously the volatility and to define a price consistent with the noarbitrage condition. An illustrative example is provided using S&P 500 options data.
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