Stochastic Differential Equations for Interview Preparation

Every pricing model is written as a stochastic differential equation, and reading the notation correctly is the first skill interviewers check.

A stochastic differential equation describes how a random process evolves over an infinitesimal time step through a drift term and a diffusion term, and quant research interviews test them because every pricing model, from Black-Scholes-Merton to interest rate models, is specified as one. The general form writes the differential of X as a drift function of time and the state, multiplied by dt, plus a volatility function of time and the state, multiplied by the differential of Brownian motion.

The equation, the differential of X sub t equals mu of t and X sub t times dt plus sigma of t and X sub t times dW sub t, is shorthand for an integral equation: X at time t equals X at time zero plus the integral from zero to t of mu, plus the integral from zero to t of sigma with respect to W. The first integral is an ordinary Riemann integral and behaves the way integration always does. The second is an Ito integral, defined as a limit of sums evaluated using the left endpoint of each partition, a convention that matters because it keeps the integral non-anticipating and preserves its martingale property.

Interviewers frequently ask why the Ito integral evaluates the integrand at the left endpoint rather than the midpoint, as the alternative Stratonovich convention does. The answer is that the left endpoint convention makes the integral a martingale with respect to the filtration generated by the Brownian motion, because the integrand at each step is known before the corresponding increment occurs, matching how a trader actually rebalances a hedge using information available at the start of each interval rather than information from the future.

A stochastic differential equation has a unique strong solution under conditions on mu and sigma known as Lipschitz continuity and linear growth, which bound how fast the drift and volatility functions can change and how large they can grow relative to the state. Most models used in practice, including geometric Brownian motion and the Ornstein-Uhlenbeck process, satisfy these conditions comfortably, which is why existence and uniqueness are rarely discussed once a model has been chosen, though a candidate should know the conditions exist and be able to name them if asked directly.

Geometric Brownian motion, the equation behind Black-Scholes-Merton, has drift mu times X and volatility sigma times X, and its solution follows from applying Ito's lemma to the logarithm of X, producing a closed form involving an exponential of a drift-adjusted Brownian motion. That derivation, and the specific adjustment that appears because of the second-order term in Ito's lemma, is covered fully in Geometric Brownian Motion and Its Limitations.