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Fundamental theorems of asset pricing-Continuous models

In this note we discuss the two fundamental theorems of asset pricing for the continuous time model, in particular the Black-Scholes model. The mathematical tool for discussion is martingale theory in continuous time. We will define the notion of martingales in continuous time, and show that under the risk neutral measure, the discounted underlying asset price is a martingale. By our pricing formula, the discounted value process of a non-American financial derivative is also a martingale under the risk neutral measure. This is used to prove the first fundamental theorem of asset pricing. Under the uniqueness of the risk neutral measure, we show the existence of the hedging portfolio for non-American financial derivatives.