Vasicek model

In finance, the Vasicek model is a mathematical model describing 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 1977 by Oldrich Vasicek.

The model specifies that the instantaneous interest rate follows the stochastic differential equation:

:$dr\_t\; =\; a(b-r\_t),\; dt\; +\; sigma\; ,\; dW\_t$

where "W_t" is a Wiener process modelling the random market risk factor. The standard deviation parameter, $sigma$, determines the volatility of the interest rate. This model is an Ornstein-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 ensure stability 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 the Hull-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