# Vasicek Model

Vasicek Model was introduced by Oldrich Vasicek in the year of 1977. It is a mathematical model that describes the interest rate evolution. The Vasicek Model is a type of “one-factor model” that explains the movements of interest rate when it is only driven by the market risk.
The Vasicek Model is also applied for the interest rate derivative valuation. The theory of Vasicek Model is very well adapted by the credit markets. Vasicek Model is an Ornstein-Uhlenbeck stochastic process.

According to the Vasicek Model, the instantaneous interest rate complies with the following stochastic differential equation:

drt = a(b-rt) dt + σdWt

Here,

Wt = Wiener process that models the risk factor of random market.

σ = Standard deviation that determines the volatility of the rate of interest.

a(b – rt) is the drift factor that describes the expected change in the rate of interest at a particular time t.

b is the parameter that stands for the long run equilibrium value towards which the interest rate goes back.

When dWt = 0, it is assumed that the shock is absent, then the interest is constant as the value of rt equalizes to b.

The parameter ‘a’ governs the adjustment speed and has to be positive in order to maintain stability around the long-term value.

Vasicek Model was the first economic model to capture the value of mean reversion. Mean reversion is an important characteristic of the interest rate that is responsible to set it apart from other financial prices. Hence, the interest rates cannot rise indefinitely unlike stock prices, because it may affect the economic activity adversely causing an interest rate decrease. Interest rates cannot decrease indefinitely also and hence the interest rates move within a limited range, showing a tendency to revert to a long run value.

The main disadvantage of Vasicek’s model is that, under this theory the value of interest rate can be negative. Cox-Ingersoll-Ross model was developed by fixing this shortcoming of the Vasicek’s model. The Hull-White model is the further extension over Vasicek Model. 