Vasicek Interest Rate Model

Vasicek Interest Rate Model

The Vasicek model (1977) is the first equilibrium interest rate model. It describes how short rates evolve over time with mean reversion.

Stochastic Differential Equation

$dr = \kappa(\theta - r) \, dt + \sigma \, dW$

Where:

• $r$ = current short rate
• $\kappa$ = mean reversion speed
• $\theta$ = long-run mean (equilibrium rate)
• $\sigma$ = volatility
• $dW$ = Brownian motion increment ($\sim \mathcal{N}(0, dt)$)

Discrete-Time Approximation

For a small time step $\Delta t$ with a standard normal draw $\Delta W$:

$r_{t+\Delta t} = r_t + \kappa(\theta - r_t)\Delta t + \sigma\sqrt{\Delta t} \cdot \Delta W$

Analytical Zero-Coupon Bond Price

The Vasicek model has a closed-form bond price:

$P(0,T) = A(T) \cdot e^{-B(T) \cdot r_0}$

$B(T) = \frac{1 - e^{-\kappa T}}{\kappa}$

$A(T) = \exp\left[\left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B(T) - T) - \frac{\sigma^2 B(T)^2}{4\kappa}\right]$

Use math.exp() and math.sqrt() for implementation.