- Vasicek model
In finance, the Vasicek model is a
mathematical modeldescribing the evolution of interest rates. It is a type of "one-factor model" ( 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 1977by Oldrich Vasicek.
The model specifies that the
instantaneous interest ratefollows the stochastic differential equation:
where "Wt" is a
Wiener processmodelling the random market risk factor. The standard deviationparameter, , determines the volatilityof the interest rate. This model is an Ornstein-Uhlenbeckstochastic process.
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. Similarly, interest rates can not decrease indefinitely. As a result, interest rates move in a limited range, showing a tendency to revert to a long run value.
The drift factor 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. Indeed, in the absence of shocks (), the interest remains constant when "rt = b". The parameter "a", governing the speed of adjustment, needs to be positive to ensure
stabilityaround the long term value. For example, when "rt" is below "b", the drift term becomes positive for positive "a", generating a tendency for the interest rate to move upwards (toward equilibrium).
The main disadvantage is that, under Vasicek's model, it is theoretically possible for the interest rate to become negative, an undesirable feature. This shortcoming was fixed in the
Cox-Ingersoll-Ross model. The Vasicek model was further extended in the Hull-White model.
Asymptotic Mean and Variance
We solve the stochastic differential equation to obtain
Using similar techniques as applied to the
Ornstein-Uhlenbeckstochastic process this has mean
Consequently, we have:and:
*cite book | author=Hull, John C. | title=Options, Futures and Other Derivatives| year=2003 | publisher = Upper Saddle River, NJ:
Prentice Hall| id = ISBN 0-13-009056-5
*cite journal | author=Vasicek, Oldrich | title=An Equilibrium Characterisation of the Term Structure | journal=Journal of Financial Economics| year=1977 | volume=5 | pages=177–188 | doi=10.1016/0304-405X(77)90016-2
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