- Vasicek model
In finance, the

**Vasicek model**is amathematical model describing the evolution ofinterest rate s. It is a type of "one-factor model" (short rate model ) as it describes interest rate movements as driven by only one source ofmarket risk . The model can be used in the valuation ofinterest rate derivative s, and has also been adapted for credit markets. It was introduced in1977 byOldrich Vasicek .The model specifies that the

instantaneous interest rate follows thestochastic differential equation ::$dr\_t\; =\; a(b-r\_t),\; dt\; +\; sigma\; ,\; dW\_t$

where "W

_{t}" is aWiener process modelling the random market risk factor. Thestandard deviation parameter, $sigma$, determines thevolatility of the interest rate. This model is anOrnstein-Uhlenbeck stochastic process.**Discussion**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 $a(b-r\_t)$ 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 ($dW\_t\; =\; 0$), the interest remains constant when "r

_{t}= b". The parameter "a", governing the speed of adjustment, needs to be positive to ensurestability around the long term value. For example, when "r_{t}" is below "b", the drift term $a(b-r\_t)$ 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 theHull-White model .**Asymptotic Mean and Variance**We solve the stochastic differential equation to obtain

:$r(t)\; =\; r(0)\; e^\{-a\; t\}\; +\; b\; left(1-\; e^\{-a\; t\}\; ight)\; +\; sigma\; e^\{-a\; t\}int\_0^t\; e^\{a\; s\},dW\_s,!$

Using similar techniques as applied to the

Ornstein-Uhlenbeck stochastic process this has mean:$E\; [r\_t]\; =\; r\_0\; e^\{-a\; t\}\; +\; b(1\; -\; e^\{-at\})$

and variance

:$Var\; [r\_t]\; =\; frac\{sigma^2\}\{2\; a\}(1\; -\; e^\{-2at\}).$

Consequently, we have:$lim\_\{t\; o\; infty\}\; E\; [r\_t]\; =\; b$and:$lim\_\{t\; o\; infty\}\; Var\; [r\_t]\; =\; frac\{sigma^2\}\{2\; a\}.$

**ee also***

Ornstein-Uhlenbeck process.

*Hull-White model

*Cox-Ingersoll-Ross model **References***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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