Vasicek Bond Price Calculator

The Vasicek model (1977) is one of the earliest and most influential interest rate models. It describes how interest rates evolve over time using a mean-reverting stochastic process. The model assumes rates fluctuate randomly but tend to drift back toward a long-run average level (theta). This makes it useful for pricing bonds and interest rate derivatives. While simple and tractable, the model has the limitation that it can produce negative interest rates.

The Vasicek model assumes the short rate follows a normal (Gaussian) distribution. Since normal distributions extend to negative infinity, there's always some probability of negative rates. While this was historically seen as unrealistic, negative interest rates have actually occurred in several economies (Europe, Japan) since 2014, making this "flaw" somewhat prescient. Alternative models like Cox-Ingersoll-Ross (CIR) ensure rates stay positive by using a square-root diffusion process.

These three risk-neutral parameters fully characterize the Vasicek model: Kappa (mean reversion speed): How quickly rates return to equilibrium. Higher kappa = faster reversion. Typical values: 0.1-1.0 per year. Theta (long-run mean): The risk-neutral equilibrium interest rate level used for pricing. Rates tend toward this value over time. Sigma (volatility): The standard deviation of rate changes. Higher sigma = more volatile rates. Typical values: 0.5%-2% annually. Note: These are pricing parameters under the risk-neutral measure, not necessarily historical estimates.

Both are single-factor mean-reverting models, but they differ in how volatility scales with the rate level: Vasicek: sigma * dW (constant volatility) - rates can go negative CIR: sigma * sqrt(r) * dW (volatility proportional to sqrt(rate)) - rates stay positive The CIR model is more realistic for most interest rate applications but has slightly more complex bond pricing formulas. Both produce closed-form bond prices (affine term structure models).

In an affine term structure model, bond prices have the exponential-affine form: P(t,T) = exp(A(tau) - B(tau)*r). The yield is linear (affine) in the short rate: y = (B*r - ln(A))/tau. This mathematical structure allows closed-form solutions for bond prices, making the model analytically tractable. Vasicek, CIR, and many other popular term structure models share this property.

Calibration involves fitting model parameters to observed market data. For pricing purposes (risk-neutral Q-measure calibration), fit parameters so model bond prices match observed Treasury yields across maturities. This is a numerical optimization problem. Historical estimation (P-measure) uses time series of short rates but produces different parameters than risk-neutral calibration. For pricing, risk-neutral parameters are required. This calculator assumes parameters are already known, focusing on understanding how they affect bond prices.