Brownian Motion Properties That Quant Interviews Test

Brownian motion is defined by independent increments, continuous paths and quadratic variation equal to time, and interviewers test each in turn.

Standard Brownian motion is the stochastic process that sits underneath almost every model used in derivatives pricing, and quant interviewers test it early because a candidate who cannot state its properties precisely cannot be trusted with anything built on top of it. A process W is standard Brownian motion if W at time zero equals zero, its increments over disjoint time intervals are independent, each increment is normally distributed with mean zero and variance equal to the length of the interval, and its paths are continuous in time with probability one.

Independence of increments means the change in the process from time s to time t tells you nothing about the change from time t to time u, for s less than t less than u. This is the property that makes Brownian motion a building block for a Markov process: conditional on the current value, the future evolution does not depend on how the process got there. The normal distribution of each increment, with variance equal to elapsed time rather than a fixed number, gives the process its characteristic square root of time scaling. The standard deviation of the move over an interval of length t is the square root of t, not proportional to t itself.

Continuity of paths matters because it rules out jumps, a distinction that becomes important once a candidate is asked to contrast Brownian motion with a Poisson-driven jump process later in the same interview. Continuity is also what permits the construction of stochastic integrals against the process in the way Ito's calculus requires, and without it the entire apparatus of stochastic differential equations would need a different foundation.

The property that surprises candidates who have only seen ordinary calculus is that Brownian paths are continuous everywhere but differentiable nowhere. Informally, the increment over a small time step h has a standard deviation proportional to the square root of h, so the ratio of the increment to h behaves like one divided by the square root of h, which grows without bound as h shrinks toward zero. There is no well-defined instantaneous rate of change at any point on the path.

This single fact is why stochastic calculus needs its own set of rules rather than borrowing ordinary calculus directly, and it is the reason Ito's lemma carries a second-order term that an ordinary Taylor expansion would treat as negligible. The full derivation and its use in pricing are covered in Ito's Lemma and Stochastic Calculus Derivation.