In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In considering a slightly more complicated model over a finite probability space, we see that Q once again makes its appearance. Finally, in the context of continuous time, we build a framework of stochastic calculus to model the trajectories of asset prices on a finite time interval. Under the absence of arbitrage once more, we see that Q makes its return as a Radon-Nikodym derivative of our initial probability measure. Finally, we use the properties of Q and a stochastic differential equation that models the dynamics of the assets of our market, known as the Ito formula, in order to derive the classic Black-Scholes Equation.