Short-Rate Interest Rate Models › Key Parameters

The key parameters for the Vasicek and CIR models govern all aspects of rate dynamics and derivative pricing:

• $r(t)$ Instantaneous short rate. The fundamental state variable driving the term structure. All bond prices and yields are functions of this single variable, which is why these are called "one-factor" models.

• $\kappa$: Speed of mean reversion. Controls how quickly the rate returns to the long-run mean $\theta$. Larger values of $\kappa$ produce tighter distributions around $\theta$ and faster convergence of yield curves to their long-term levels.

• $\theta$: Long-run mean level. The equilibrium level toward which interest rates gravitate over time. This parameter reflects structural factors in the economy such as potential growth rates and inflation expectations.

• $\sigma$: Volatility of the short rate. Determines the magnitude of random shocks affecting the rate path. In Vasicek, this is an absolute volatility; in CIR, it is scaled by the square root of the rate level.

• $T$: Time to maturity. Used in bond and option pricing formulas. The effect of maturity on prices depends on the interplay between mean reversion and volatility.

• $K$: Strike price. The exercise price for bond options. Together with the forward price of the underlying, the strike determines the moneyness of the option.