The Vasicek model is a mathematical model used in finance to describe the movement of interest rates over time. It was developed by Oldrich Vasicek in 1977. It describes the evolution of interest rates by assuming that the short-term interest rate follows a mean-reverting stochastic process.

It is an example of Ornstein-Uhlenbeck process, the instantaneous interest-rate r follows the following stochastic differential equation:

dr(t) = a.(b – r(t)).dt + sigma.dW(t)

Where W(t) is a Wiener process (Brownian motion) introducing randomness into the model.

The model involves several constant parameters:

- b: the long-term mean or equilibrium to which the interest rate reverts. It is the level that the interest rate tends to approach over time.

- a: the speed of reversion. This parameter, positive, determines the speed at which the interest rate reverts to the mean reversion level. A higher level indicates a faster reversion to the mean.

- sigma: the instantaneous volatility, measures the randomness or fluctuations in the interest rate movement. Higher volatility means larger and more frequent fluctuations.

a.(b-r(t)).dt is the strength of mean-reversion. If the value of the instantaneous interest rate r(t) is higher (resp. lower) than its long-term average it is negative (resp. positive). It is playing the role of a force stabilizing the random value around its long-term mean. The higher the speed of reversion or the distance between the current value of r and its long-term mean, the stronger it is.

The half-life of the mean-reversion is the average time it take to be half-way back to the mean. The higher the speed of reversion the smaller the half-life.

This equation can be easily resolved by applying the Itô lemma to r(t).exp(a.t).

r(t) follows a gaussian distribution with a simple expressions for its mean and its variance. It admits a stationary probability distribution, the long-term average of r(t) being b and the long-term variance sigma^2 / (2 x a). The variance increases with the instantaneous volatility sigma but it decreases with the speed of reversion a.

There is an analytic formula for the price of a zero-coupon bond in the Vasicek model under the no-arbitrage assumption.

## Presentation

## To Go Further

We present in this post two methods to calibrate the Vasicek model to historical data by least squares and maximum likelihood estimation with Python code available: Calibration of the Vasicek Model to Historical Data with Python Code

We present here the Cox-Ingersoll-Roll model, an alternative to the Vasicek model which does not allow negative interest rates while preserving analytical solution for bond pricings: The Cox-Ingersoll-Ross (CIR) Model

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