We explain some key mathematical ideas behind the no-arbitrage pricing of financial derivatives by replication, starting from a simple coin toss model and ending with the continuous-time limit of a multi-step coin-toss model using a geometric random walk model. In the limit, we obtain the classical Black-Scholes-Merton formula for pricing European call and put options.
Kostadinov, Boyan, "How Much Should You Pay For a Financial Derivative?" (2016). CUNY Academic Works.