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[[File:Zins-Vasicek.png|thumb|A trajectory of the short rate and the corresponding yield curves at T=0 (purple) and two later points in time]]
In [[Mathematical finance|finance]], the '''Vasicek model''' is a [[mathematical model]] describing the evolution of [[interest rate]]s. It is a type of "one-factor model" (more precisely, 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 derivative]]s, and has also been adapted for credit markets, although its use in the credit market is in principle wrong, implying negative probabilities (see for example Brigo and Mercurio (2006), Section 21.1.1). It was introduced in 1977 by [[Oldrich Vasicek]] and can be also seen as a [[stochastic investment model]].

==Details==
The model specifies that the [[force of interest|instantaneous interest rate]] follows the [[stochastic differential equation]]:

:<math>dr_t = a(b-r_t)\, dt + \sigma \, dW_t</math>

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. The [[standard deviation]] parameter, <math>\sigma</math>, determines the [[Volatility (finance)|volatility]] of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters <math>b, a</math> and <math>\sigma</math>, together with the initial condition <math>r_0</math>, completely characterize the dynamics, and can be quickly characterized as follows, assuming <math>a</math> to be non-negative:

* <math>b</math>: "long term mean level". All future trajectories of <math>r</math> will evolve around a mean level b in the long run;
* <math>a</math>: "speed of reversion". <math>a</math> characterizes the velocity at which such trajectories will regroup around <math>b</math> in time;
* <math>\sigma</math>: "instantaneous volatility", measures instant by instant the amplitude of randomness entering the system. Higher <math>\sigma</math> implies more randomness

The following derived quantity is also of interest,

* <math>{\sigma^2}/(2 a)</math>: "long term variance". All future trajectories of <math>r</math> will regroup around the long term mean with such variance after a long time.

<math>a</math> and <math>\sigma</math> tend to oppose each other: increasing <math>\sigma</math> increases the amount of randomness entering the system, but at the same time increasing <math>a</math> amounts to increasing the speed at which the system will stabilize statistically around the long term mean <math>b</math> with a corridor of variance determined also by <math>a</math>. This is clear when looking at the long term variance,

:<math>\frac{\sigma^2}{2 a}</math>

which increases with <math>\sigma</math> but decreases with <math>a</math>.

This model is an [[Ornstein–Uhlenbeck process|Ornstein–Uhlenbeck stochastic process]]. Making the long term mean stochastic to another SDE is a simplified version of the [[cointelation]] SDE.<ref name="wilmottM.com">{{cite journal|authors=Mahdavi Damghani B.|title=The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model|journal=Wilmott Magazine|year=2013|url=http://onlinelibrary.wiley.com/doi/10.1002/wilm.10252/abstract}}</ref>

== Discussion ==

Vasicek's model was the first one to capture [[Mean reversion (finance)|mean reversion]], an essential characteristic of the interest rate that sets it apart from other financial prices. Thus, as opposed to [[common stock|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 below 0. As a result, interest rates move in a limited range, showing a tendency to revert to a long run value.

The drift factor <math>a(b-r_t)</math> represents the expected instantaneous change in the interest rate at time ''t''. The parameter ''b'' represents the long run [[steady state|equilibrium]] value towards which the interest rate reverts. Indeed, in the absence of shocks (<math>dW_t = 0</math>), the interest rate remains constant when ''rt = b''. The parameter ''a'', governing the speed of adjustment, needs to be positive to ensure [[stability theory|stability]] around the long term value. For example, when ''rt'' is below ''b'', the drift term <math>a(b-r_t)</math> 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]], exponential Vasicek model, [[Black–Derman–Toy model]] and [[Black–Karasinski model]], among many others. The Vasicek model was further extended in the [[Hull–White model]]. The Vasicek model is also a canonical example of the [[affine term structure model]], along with the [[Cox–Ingersoll–Ross model]].

== Asymptotic mean and variance ==
We solve the stochastic differential equation to obtain

:<math> r(t) = r(0) e^{-a t} + b\left(1- e^{-a t}\right) + \sigma e^{-a t}\int_0^t e^{a s}\,dW_s.\,\!</math>

Using similar techniques as applied to the [[Ornstein–Uhlenbeck process|Ornstein–Uhlenbeck]] stochastic process this has mean

:<math>\mathrm{E}[r_t] = r_0 e^{-a t} + b(1 - e^{-at})</math>

and variance

:<math>\mathrm{Var}[r_t] = \frac{\sigma^2}{2 a}(1 - e^{-2at}).</math>

Consequently, we have
:<math>\lim_{t \to \infty} \mathrm{E}[r_t] = b</math>
and
:<math>\lim_{t \to \infty} \mathrm{Var}[r_t] = \frac{\sigma^2}{2 a}.</math>

==See also==
* [[Ornstein–Uhlenbeck process]].
* [[Hull–White model]]
* [[Cox–Ingersoll–Ross model]]
* [[Cointelation]]

==References==
* {{cite book | author=Hull, John C. | title=Options, Futures and Other Derivatives| year=2003 | publisher = Upper Saddle River, NJ: [[Prentice Hall]] | 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 | issue=2 }}
* {{cite book | title = Interest Rate Models – Theory and Practice with Smile, Inflation and Credit| author = Damiano Brigo, Fabio Mercurio | publisher = Springer Verlag | year = 2001 | edition = 2nd ed. 2006 | isbn = 978-3-540-22149-4}}
* {{cite book | title = Interest Rate Modelling| author = Jessica James, Nick Webber | publisher = Wiley| year = 2000 | isbn = 0-471-97523-0}}

<references />

==External links==
* [http://www.quantcalc.net/ZCB_Vasicek.html Price of Zero Coupon Bond under Vasicek Model], Free Online Calculator, QuantCalc
* [http://www.dms.umontreal.ca/~dugas/act6230/articles/Vasicek77.pdf An Equilibrium Characterisation of the Term Structure], Oldrich Vasicek,(1977). Journal of Financial Economics 5: 177–188
* [http://www.marginalq.com/eraker/fixedIncome/vasicek-print.pdf The Vasicek Model], Bjørn Eraker, [[Wisconsin School of Business]]
* [http://www3.iam.metu.edu.tr/iam/images/2/25/Dervi%C5%9Fbayaz%C4%B1tthesis.pdf Yield Curve Estimation and Prediction with the Vasicek Model], D. Bayazit, [[Middle East Technical University]]

{{Bond market}}
{{Stochastic processes}}

[[Category:Finance theories]]
[[Category:Interest rates]]
[[Category:Mathematical finance]]
[[Category:Fixed income analysis]]
[[Category:Stochastic processes]]
[[Category:Short-rate models]]

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