We consider the problem of pricing and hedging general path dependent derivatives on a single asset, supposing that we already know the prices of the vanilla options. If we are to avoid introducing arbitrage possibilities, then this is the same as finding a model under which the discounted asset price is a martingale and for which every vanilla option has its price equal to the expected value of its discounted payout. It has been shown by Dupire ([1], [2]) that if we restrict ourselves to diffusions, then the local volatility surface can be determined by a simple equation which involves differentiating the option prices with respect to their maturity and strike price. We considerably extend this result of Dupire. First, we show that if we generalise the possible models for the asset price to include what we shall term comparable processes, then there exists a unique such model fitting the observed option prices. The option prices need not be differentiable - just that they are continuous with respect to the maturity. One problem with the method proposed by Dupire is that no matter how many options we may observe in practise, it is impossible to calculate the local volatility surface to within any degree of accuracy. However, we show that the model for the asset price does depend on the observed options in a continuous way, so the proposed method of pricing derivatives is stable. We show that if we use implicit finite differences to fit the observed option prices ever more closely, then the associated model for the asset price will always converge to the unique comparable martingale consistent with these option prices. This theorem requires no preconditions, and works for all possible comparable processes, not just diffusions. The same is true for implicit finite differences, as long the associated trinomial processes do not contain any negative probabilities. Fina~Jy, we extend the well known link between arbitrage and the existence of equivalent martingale measures. We show that if the market consists of non-negative continuous assets, then there exists an equivalent martingale measure if and only if it does not admit arbitrage in a carefully defined approximating sense. This extends a similar result by Delbaen [1], which only concerned bounded processes.