Cox-Ingersoll-Ross mean-reverting square root diffusion model
Creates and displays cir
objects, which derive from the
sdemrd
(SDE with drift rate expressed in
mean-reverting form) class.
Use cir
objects to simulate sample paths of
NVARS
state variables expressed in mean-reverting drift-rate
form. These state variables are driven by NBROWNS
Brownian motion
sources of risk over NPERIODS
consecutive observation periods,
approximating continuous-time CIR stochastic processes with square root
diffusions.
You can simulate any vector-valued CIR process of the form:
where:
Xt is an
NVARS
-by-1
state vector of process
variables.
S is an
NVARS
-by-NVARS
matrix of mean
reversion speeds (the rate of mean reversion).
L is an
NVARS
-by-1
vector of mean
reversion levels (long-run mean or level).
D is an
NVARS
-by-NVARS
diagonal matrix,
where each element along the main diagonal is the square root of the
corresponding element of the state vector.
V is an
NVARS
-by-NBROWNS
instantaneous
volatility rate matrix.
dWt is an
NBROWNS
-by-1
Brownian motion
vector.
creates a default CIR
= cir(Speed
,Level
,Sigma
)CIR
object.
Specify required input parameters as one of the following types:
A MATLAB® array. Specifying an array indicates a static (non-time-varying) parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.
A MATLAB function. Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function.
You can specify combinations of array and function input parameters as needed.
Moreover, a parameter is identified as a deterministic function
of time if the function accepts a scalar time t
as its only input argument. Otherwise, a parameter is assumed to be
a function of time t and state
X(t) and is invoked with both input
arguments.
creates a CIR
= cir(___,Name,Value
)CIR
object with additional options specified by
one or more Name,Value
pair arguments.
Name
is a property name and Value
is
its corresponding value. Name
must appear inside single
quotes (''
). You can specify several name-value pair
arguments in any order as
Name1,Value1,…,NameN,ValueN
The CIR
object has the following Properties:
StartTime
— Initial observation time
StartState
— Initial state at time
StartTime
Correlation
— Access function for the
Correlation
input argument, callable as a
function of time
Drift
— Composite drift-rate function,
callable as a function of time and state
Diffusion
— Composite diffusion-rate
function, callable as a function of time and state
Simulation
— A simulation function or
method
Speed
— Access function for the input
argument Speed
, callable as a function of
time and state
Level
— Access function for the input
argument Level
, callable as a function of
time and state
Sigma
— Access function for the input
argument Sigma
, callable as a function of
time and state
interpolate | Brownian interpolation of stochastic differential equations |
simulate | Simulate multivariate stochastic differential equations (SDEs) |
simByEuler | Euler simulation of stochastic differential equations (SDEs) |
simByTransition | Simulate Cox-Ingersoll-Ross sample paths with transition density |
When you specify the required input parameters as arrays, they are associated with a specific parametric form. By contrast, when you specify either required input parameter as a function, you can customize virtually any specification.
Accessing the output parameters with no inputs simply returns the original input specification. Thus, when you invoke these parameters with no inputs, they behave like simple properties and allow you to test the data type (double vs. function, or equivalently, static vs. dynamic) of the original input specification. This is useful for validating and designing methods.
When you invoke these parameters with inputs, they behave like functions, giving the
impression of dynamic behavior. The parameters accept the observation time
t and a state vector
Xt, and return an array of appropriate
dimension. Even if you originally specified an input as an array, cir
treats it as a static function of time and state, by that means guaranteeing that all
parameters are accessible by the same interface.
[1] Ait-Sahalia, Y. “Testing Continuous-Time Models of the Spot Interest Rate.” The Review of Financial Studies, Spring 1996, Vol. 9, No. 2, pp. 385–426.
[2] Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, Vol. 54, No. 4, August 1999.
[3] Glasserman, P. Monte Carlo Methods in Financial Engineering. New York, Springer-Verlag, 2004.
[4] Hull, J. C. Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.
[5] Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, 2nd ed. New York, John Wiley & Sons, 1995.
[6] Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer-Verlag, 2004.
diffusion
| drift
| interpolate
| nearcorr
| sdeddo
| simByEuler
| simByTransition
| simulate