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# The Vasicek interest rate model

11.04.2021

The Vasicek interest rate model (or simply the Vasicek model) is a mathematical method of modeling interest rate movements. The model describes the movement of an interest rate as a factor composed of market risk, time, and equilibrium value, where the rate tends to revert towards the mean of those factors over time. Essentially, it predicts where interest rates will end up at the end of a given period of time, given current market volatility, the long-run mean interest rate value, and a given market risk factor.

It is important to note that the equation can only test one market risk factor at a time. This stochastic model is often used in the valuation of interest rate futures and is sometimes used in solving for the price of various hard to value bonds.

The Vasicek interest rate model values the instantaneous interest rate using the following equation: drt=a(b−rt)dt+σdWt (1)

where:
W=Random market risk (represented by a Wiener process)
t=Time period
a(b−rt)=Expected change in the interest rateat time t (the drift factor)
a=Speed of the reversion to the mean
b=Long-term level of the mean
σ=Volatility at time t
The model specifies that the instantaneous interest rate follows the stochastic differential equation,

where d refers to the derivative of the variable following it.

One of the most attractive feature of the Vasicek Model is that there is close form solution for rt. We start solution of the Stochastic Differential Equation (1) by taking the derivative of which yields:
The most important feature which this model exhibits and explained by Vasicek is the mean reversion property, which means that if the interest rate is bigger than the long run mean (r<b then the coefficient a<0 makes the drift become negative so that the rate will be pulled down in the direction of b. Similarly, if the interest rate is smaller than the long run pulled down in the direction of b. Similarly, if the interest rate is smaller than the long run mean (r<b, then the coefficient a<0 makes the drift term become positive so that the rate will be pulled up in the direction of b. Therefore, the coefficient a is the speed of adjustment of the interest rate towards its long run level.

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