Vasicek model
In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. 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, an
Overview
In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. 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, and can be also seen as a stochastic investment model.
Details
The model specifies that the instantaneous interest rate follows the stochastic differential equation:
d
r
t
= a ( b −
r
t
)
d t + σ
d
W
t
{\displaystyle dr_{t}=a(b-r_{t})\,dt+\sigma \,dW_{t}}
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,
σ
{\displaystyle \sigma }
, determines the volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters
b , a
{\displaystyle b,a}
and
Discussion
Vašíček'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 do not usually decrease much 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
a ( b −
r
t
)
{\displaystyle 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 (
d
W
t
= 0
{\displaystyle dW_{t}=0}
), 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. For example, when rt is below b, the drift term
a ( b −
r
t
)
{\displaystyle 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 Vašíček's model, it is theoretically possible for the interest rate to become negative, an undesirable feature under pre-crisis assumptions. 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. In recent research both models were used for data partitioning and forecasting.
Asymptotic mean and variance
We solve the stochastic differential equation to obtain
r
t
=
r
0
e
− a t
+ b
(
1 −
e
− a t
)
+ σ
e
− a t
∫
0
t
Bond pricing
Under the no-arbitrage assumption, a discount bond may be priced in the Vasicek model. In Vašíček's original derivation, the no-arbitrage argument implies that the expected return on any bond in excess of the instantaneous short rate is proportional to the bond's standard deviation, with the proportionality factor representing the market price of interest-rate risk. The time
t
{\displaystyle t}
value of a discount bond with maturity date
T
{\displaystyle T}
is exponential affine in the interest rate:
P ( t , T ) = A ( t , T )
e
− B ( t , T ) r ( t )
{\displaystyle P(t,T)=A(t,T)e^{-B(t,T)r(t)}}
where
B ( t , T ) =
1 −
e
− a ( T − t )
a
{\displaystyle B(t,T)={\frac {1-e^{-a(T-t)}}{a}}}
A ( t , T ) = exp
{
See also
Ornstein–Uhlenbeck process. Hull–White model Cox–Ingersoll–Ross model
References
Hull, John C. (2003). Options, Futures and Other Derivatives. Upper Saddle River, NJ: Prentice Hall. ISBN 978-0-13-009056-0. Damiano Brigo, Fabio Mercurio (2001). Interest Rate Models – Theory and Practice with Smile, Inflation and Credit (2nd ed. 2006 ed.). Springer Verlag. ISBN 978-3-540-22149-4. Jessica James, Nick Webber (2000). Interest Rate Modelling. Wiley. ISBN 978-0-471-97523-6.
External links
The Vasicek Model Archived 2010-02-16 at the Wayback Machine, Bjørn Eraker, Wisconsin School of Business Yield Curve Estimation and Prediction with the Vasicek Model, D. Bayazit, Middle East Technical University
Wikipedia contributors · Vasicek model · revision 1370144817 · CC BY-SA 4.0 · Plain text extracted and formatting adapted; tags derived from linked topics.