  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.
