Short Rate Models and Where They Are Used

Vasicek, CIR, and Hull-White trade tractability against realism, and each one prices a different corner of a rates book.

A short rate model describes the instantaneous interest rate as a single stochastic process and derives the entire term structure, along with the price of any interest rate derivative, from that one process. The choice of model, most commonly Vasicek, the Cox-Ingersoll-Ross model, or Hull-White, determines whether rates can go negative, how volatility scales with the level of rates, and how well the model can be made to match the market's own curve on any given day.

The Vasicek model specifies that the short rate drifts toward a long-run mean at a constant speed, with a constant volatility added as a random shock. That mean reversion is the model's central feature, since it captures the empirical tendency of interest rates to pull back toward a normal range after moving away from it, rather than wandering off indefinitely the way a stock price is typically modeled to do.

The tradeoff for that tractability is that Vasicek permits the short rate to go negative, since the random shock is normally distributed and has no lower bound. For most of financial history that was treated as a theoretical flaw rather than a practical one, but the period of negative policy rates across parts of Europe and Japan made it a genuinely realistic outcome, and desks in those markets had to decide whether a negative-rate-capable model was a feature or a risk to hedge against.

CIR modifies Vasicek by scaling the volatility of the short rate by the square root of the rate's own level, so volatility shrinks toward zero as the rate approaches zero. Under a standard parameter condition, that scaling keeps the rate from ever crossing into negative territory, which made CIR the more popular choice for markets and eras where negative rates were not considered plausible.

The cost of that property is a more complex distribution for the rate at any future date, which makes closed-form bond and option pricing formulas under CIR more involved than under Vasicek, though still tractable enough that both models remain standard material in fixed income coursework and interview preparation.