In
finance, the Vasicek model is a
mathematical model describing the evolution of
interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, ...
s. It is a type of one-factor
short-rate model
A short-rate model, in the context of interest rate derivatives, is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate, usually written r_t \,.
The short rate
Under a s ...
as it describes interest rate movements as driven by only one source of
market risk
Market risk is the risk of losses in positions arising from movements in market variables like prices and volatility.
There is no unique classification as each classification may refer to different aspects of market risk. Nevertheless, the most ...
. The model can be used in the valuation of
interest rate derivative
In finance, an interest rate derivative (IRD) is a derivative whose payments are determined through calculation techniques where the underlying benchmark product is an interest rate, or set of different interest rates. There are a multitude of diff ...
s, 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
A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
:
:
where ''W
t'' is a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
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,
, determines the
volatility of the interest rate and in a way characterizes the amplitude of the instantaneous randomness inflow. The typical parameters
and
, together with the initial condition
, completely characterize the dynamics, and can be quickly characterized as follows, assuming
to be non-negative:
*
: "long term mean level". All future trajectories of
will evolve around a mean level b in the long run;
*
: "speed of reversion".
characterizes the velocity at which such trajectories will regroup around
in time;
*
: "instantaneous volatility", measures instant by instant the amplitude of randomness entering the system. Higher
implies more randomness
The following derived quantity is also of interest,
*
: "long term variance". All future trajectories of
will regroup around the long term mean with such variance after a long time.
and
tend to oppose each other: increasing
increases the amount of randomness entering the system, but at the same time increasing
amounts to increasing the speed at which the system will stabilize statistically around the long term mean
with a corridor of variance determined also by
. This is clear when looking at the long term variance,
:
which increases with
but decreases with
.
This model is an
Ornstein–Uhlenbeck stochastic process. Making the long term mean stochastic to another SDE is a simplified version of the cointelation SDE.
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 do not usually 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
represents the expected instantaneous change in the interest rate at time ''t''. The parameter ''b'' represents the
long-run In economics, the long-run is a theoretical concept in which all markets are in equilibrium, and all prices and quantities have fully adjusted and are in equilibrium. The long-run contrasts with the short-run, in which there are some constraints a ...
equilibrium value towards which the interest rate reverts. Indeed, in the absence of shocks (
), the interest rate remains constant when ''r
t = b''. The parameter ''a'', governing the speed of adjustment, needs to be positive to ensure
stability
Stability may refer to:
Mathematics
*Stability theory, the study of the stability of solutions to differential equations and dynamical systems
** Asymptotic stability
** Linear stability
** Lyapunov stability
** Orbital stability
** Structural sta ...
around the long term value. For example, when ''r
t'' 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 under pre-crisis assumptions. This shortcoming was fixed in the
Cox–Ingersoll–Ross model
In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes 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 mark ...
, exponential Vasicek model,
Black–Derman–Toy model
In mathematical finance, the Black–Derman–Toy model (BDT) is a popular short-rate model used in the pricing of bond options, swaptions and other interest rate derivatives; see . It is a one-factor model; that is, a single stochastic facto ...
and
Black–Karasinski model, among many others. The Vasicek model was further extended in the
Hull–White model In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively str ...
. The Vasicek model is also a canonical example of the
affine term structure model, along with the
Cox–Ingersoll–Ross model
In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes 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 mark ...
. In recent research both models were used for data partitioning and forecasting.
Asymptotic mean and variance
We solve the stochastic differential equation to obtain
:
Using similar techniques as applied to the
Ornstein–Uhlenbeck stochastic process we get that state variable is distributed normally with mean
:
and variance
:
Consequently, we have
:
and
:
Bond pricing
Under the no-arbitrage assumption, a
discount bond
A zero coupon bond (also discount bond or deep discount bond) is a bond in which the face value is repaid at the time of maturity. Unlike regular bonds, it does not make periodic interest payments or have so-called coupons, hence the term zer ...
may be priced in the Vasicek model. The time
value of a discount bond with maturity date
is exponential affine in the interest rate:
:
where
:
:
See also
*
Ornstein–Uhlenbeck process
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle ...
.
*
Hull–White model In financial mathematics, the Hull–White model is a model of future interest rates. In its most generic formulation, it belongs to the class of no-arbitrage models that are able to fit today's term structure of interest rates. It is relatively str ...
*
Cox–Ingersoll–Ross model
In mathematical finance, the Cox–Ingersoll–Ross (CIR) model describes 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 mark ...
References
*
*
*
External links
The Vasicek Model Bjørn Eraker,
Wisconsin School of Business
The Wisconsin School of Business (WSB) is the business school of the University of Wisconsin–Madison, a public research university in Madison, Wisconsin and consistently ranks among the top business schools in the world. Founded in 1900, i ...
Yield Curve Estimation and Prediction with the Vasicek Model D. Bayazit,
Middle East Technical University
{{Stochastic processes
Interest rates
Fixed income analysis
Short-rate models
Financial models