Vasicek model

It is a type of one-factor short-rate model as it describes interest rate movements as driven by only one source of market risk.

The model can be used in the valuation of interest rate derivatives, and has also been adapted for credit markets.

It was introduced in 1977 by Oldřich Vašíček,[1] and can be also seen as a stochastic investment model.

The model specifies that the instantaneous interest rate follows the stochastic differential equation: where Wt is a Wiener process under the risk neutral framework modelling the random market risk factor, in that it models the continuous inflow of randomness into the system.

, determines the volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow.

amounts to increasing the speed at which the system will stabilize statistically around the long term mean

This is clear when looking at the long term variance, which increases with

Vasicek's model was the first one to capture mean reversion, an essential characteristic of the interest rate that sets it apart from other financial prices.

Thus, as opposed to stock prices for instance, interest rates cannot rise indefinitely.

This is because at very high levels they would hamper economic activity, prompting a decrease in interest rates.

As a result, interest rates move in a limited range, showing a tendency to revert to a long run value.

represents the expected instantaneous change in the interest rate at time t. The parameter b represents the long-run equilibrium value towards which the interest rate reverts.

), the interest rate remains constant when rt = b.

The parameter a, governing the speed of adjustment, needs to be positive to ensure stability around the long term value.

The main disadvantage is that, under Vasicek's model, it is theoretically possible for the interest rate to become negative, an undesirable feature under pre-crisis assumptions.

In recent research both models were used for data partitioning and forecasting.

[2] We solve the stochastic differential equation to obtain Using similar techniques as applied to the Ornstein–Uhlenbeck stochastic process we get that state variable is distributed normally with mean and variance Consequently, we have and Under the no-arbitrage assumption, a discount bond may be priced in the Vasicek model.