What asymptotic analysis does in quant models

Explain what is meant by asymptotic analysis, and describe how quants use it when building or approximating financial models.

It approximates a hard problem by expanding in a small (or large) parameter. One identifies a quantity that is small, such as time to expiry, volatility of volatility, a transaction cost or a mean-reversion timescale, and expands the solution in powers of it, keeping the leading terms. In finance this gives closed-form approximations, e.g. short-maturity implied-volatility expansions (SABR), small-cost hedging bands, or fast mean-reverting stochastic volatility corrections to Black-Scholes, which are quick to compute and reveal which effects dominate.