Background In the 60s, economists that the pricing of options were independent of pricing of underlying assets. Nowadays, we can see that, if the underlying assets were obeying of a Brownian Motion, there is no additional degree of freedom that options can bring: that knowing the stocks will tell you exactly through a DiffEQ how the option will evolve. The idea, then, is that you can replicate options: by dynamically buying and selling pairs of securities in the same way as the option, your new portfolio can track the option exactly. Of course, there is a certain amount of volatility associated with Brownian Motion markets. Unfortunately, there is no one fixed volatility which can be used to model all options; you can fit a volatility given all strike prices—creating an implied volatility surface. Otherwise, you can also model volatility as a random variable, a stochastic process modeled by stochastic volatility. Reading pg 350-352: diffusion are described by stochastic differential equations Option Pricing A Vanilla Call Given some current price S, option price K, time to maturity T; the payoff increases linearly after the option matures. How much should the option be changed for the right to buy the option after T days? We can use the option info to calculate the implied volatility.