We obtain option pricing formulas for stock price models in which the drift and volatility terms are functionals of a continuous history of the stock prices. That is, the stock dynamics follows a nonlinear stochastic functional differential equation. A model with full memory is obtained via approximation through a stock price model in which the continuous path dependence does not go up to the present: there is a memory gap. A strong solution is obtained by closing the gap. Fair option prices are obtained through an equivalent (local) martingale measure via Girsanov's Theorem and therefore are given in terms of a conditional expectation. The models maintain the completeness of the market and have no arbitrage opportunities.
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