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cir class

Superclasses:

Cox-Ingersoll-Ross mean-reverting square root diffusion models

Description

cir objects derive from the sdemrd (SDE with drift rate expressed in mean-reverting form) class. Use the cir constructor to create 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.

This method allows you to simulate any vector-valued SDE of the form:

dXt=S(t)[L(t)Xt]dt+D(t,Xt12)V(t)dWt

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.

Construction

CIR = cir(Speed,Level,Sigma) constructs a default cir object.

CIR = cir(Speed,Level,Sigma,Name,Value) constructs a 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.

For more information on constructing a cir object, see cir.

Input Arguments

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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.

Note

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.

Speed represents the parameter S, specified as an array or deterministic function of time.

If you specify Speed as an array, it must be an NVARS-by-NVARS matrix of mean-reversion speeds (the rate at which the state vector reverts to its long-run average Level).

As a deterministic function of time, when Speed is called with a real-valued scalar time t as its only input, Speed must produce an NVARS-by-NVARS matrix. If you specify Speed as a function of time and state, it calculates the speed of mean reversion. This function must generate an NVARS-by-NVARS matrix of reversion rates when called with two inputs:

  • A real-valued scalar observation time t.

  • An NVARS-by-1 state vector Xt.

Data Types: double | function_handle

Level represents the parameter L, specified as an array or deterministic function of time.

If you specify Level as an array, it must be an NVARS-by-1 column vector of reversion levels.

As a deterministic function of time, when Level is called with a real-valued scalar time t as its only input, Level must produce an NVARS-by-1 column vector. If you specify Level as a function of time and state, it must generate an NVARS-by-1 column vector of reversion levels when called with two inputs:

  • A real-valued scalar observation time t.

  • An NVARS-by-1 state vector Xt.

Data Types: double | function_handle

Sigma represents the parameter V, specified as an array or a deterministic function of time.

If you specify Sigma as an array, it must be an NVARS-by-NBROWNS matrix of instantaneous volatility rates or as a deterministic function of time. In this case, each row of Sigma corresponds to a particular state variable. Each column corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty.

As a deterministic function of time, when Sigma is called with a real-valued scalar time t as its only input, Sigma must produce an NVARS-by-NBROWNS matrix. If you specify Sigma as a function of time and state, it must return an NVARS-by-NBROWNS matrix of volatility rates when invoked with two inputs:

  • A real-valued scalar observation time t.

  • An NVARS-by-1 state vector Xt.

Data Types: double | function_handle

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

For more information on using optional name-value arguments, see cir.

Properties

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Drift rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The drift rate specification supports the simulation of sample paths of NVARS state variables driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic processes.

The drift class allows you to create drift-rate objects (using the drift constructor) of the form:

F(t,Xt)=A(t)+B(t)Xt

where:

  • A is an NVARS-by-1 vector-valued function accessible using the (t, Xt) interface.

  • B is an NVARS-by-NVARS matrix-valued function accessible using the (t, Xt) interface.

The drift object's displayed parameters are:

  • Rate: The drift-rate function, F(t,Xt)

  • A: The intercept term, A(t,Xt), of F(t,Xt)

  • B: The first order term, B(t,Xt), of F(t,Xt)

A and B enable you to query the original inputs. The function stored in Rate fully encapsulates the combined effect of A and B.

When specified as MATLAB double arrays, the inputs A and B are clearly associated with a linear drift rate parametric form. However, specifying either A or B as a function allows you to customize virtually any drift rate specification.

Note

You can express drift and diffusion classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components A and B as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: F = drift(0, 0.1) % Drift rate function F(t,X)

Attributes:

SetAccessprivate
GetAccesspublic

Data Types: struct | double

Diffusion rate component of continuous-time stochastic differential equations (SDEs), specified as a drift object or function accessible by (t, Xt.

The diffusion rate specification supports the simulation of sample paths of NVARS state variables driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic processes.

The diffusion class allows you to create diffusion-rate objects (using the diffusion constructor):

G(t,Xt)=D(t,Xtα(t))V(t)

where:

  • D is an NVARS-by-NVARS diagonal matrix-valued function.

  • Each diagonal element of D is the corresponding element of the state vector raised to the corresponding element of an exponent Alpha, which is an NVARS-by-1 vector-valued function.

  • V is an NVARS-by-NBROWNS matrix-valued volatility rate function Sigma.

  • Alpha and Sigma are also accessible using the (t, Xt) interface.

The diffusion object's displayed parameters are:

  • Rate: The diffusion-rate function, G(t,Xt).

  • Alpha: The state vector exponent, which determines the format of D(t,Xt) of G(t,Xt).

  • Sigma: The volatility rate, V(t,Xt), of G(t,Xt).

Alpha and Sigma enable you to query the original inputs. (The combined effect of the individual Alpha and Sigma parameters is fully encapsulated by the function stored in Rate.) The Rate functions are the calculation engines for the drift and diffusion objects, and are the only parameters required for simulation.

Note

You can express drift and diffusion classes in the most general form to emphasize the functional (t, Xt) interface. However, you can specify the components A and B as functions that adhere to the common (t, Xt) interface, or as MATLAB arrays of appropriate dimension.

Example: G = diffusion(1, 0.3) % Diffusion rate function G(t,X)

Attributes:

SetAccessprivate
GetAccesspublic

Data Types: struct | double

Starting time of first observation, applied to all state variables, specified as a scalar

Attributes:

SetAccesspublic
GetAccesspublic

Data Types: double

Initial values of state variables, specified as a scalar, column vector, or matrix.

If StartState is a scalar, the gbm constructor applies the same initial value to all state variables on all trials.

If StartState is a column vector, the gbm constructor applies a unique initial value to each state variable on all trials.

If StartState is a matrix, the gbm constructor applies a unique initial value to each state variable on each trial.

Attributes:

SetAccesspublic
GetAccesspublic

Data Types: double

User-defined simulation function or SDE simulation method, specified as a function or SDE simulation method.

Attributes:

SetAccesspublic
GetAccesspublic

Data Types: function_handle

Methods

simBySolutionSimulate approximate solution of diagonal-drift HWV processes

Inherited Methods

The following methods are inherited from thesde class.

interpolate

simulate

simByEuler

Instance Hierarchy

The following figure illustrates the inheritance relationships among SDE classes.

For more information, see SDE Class Hierarchy.

Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

Examples

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The Cox-Ingersoll-Ross (CIR) short rate class derives directly from SDE with mean-reverting drift (SDEMRD):

where is a diagonal matrix whose elements are the square root of the corresponding element of the state vector.

Create a cir object to represent the model: .

obj = cir(0.2, 0.1, 0.05)  % (Speed, Level, Sigma)
obj = 
   Class CIR: Cox-Ingersoll-Ross
   ----------------------------------------
     Dimensions: State = 1, Brownian = 1
   ----------------------------------------
      StartTime: 0
     StartState: 1
    Correlation: 1
          Drift: drift rate function F(t,X(t)) 
      Diffusion: diffusion rate function G(t,X(t)) 
     Simulation: simulation method/function simByEuler
          Sigma: 0.05
          Level: 0.1
          Speed: 0.2

Although the last two objects are of different classes, they represent the same mathematical model. They differ in that you create the cir object by specifying only three input arguments. This distinction is reinforced by the fact that the Alpha parameter does not display – it is defined to be 1/2.

Algorithms

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.

References

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.

Ait-Sahalia, Y. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, Vol. 54, No. 4, August 1999.

Glasserman, P. Monte Carlo Methods in Financial Engineering. New York, Springer-Verlag, 2004.

Hull, J. C. Options, Futures, and Other Derivatives, 5th ed. Englewood Cliffs, NJ: Prentice Hall, 2002.

Johnson, N. L., S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Vol. 2, 2nd ed. New York, John Wiley & Sons, 1995.

Shreve, S. E. Stochastic Calculus for Finance II: Continuous-Time Models. New York: Springer-Verlag, 2004.

Introduced in R2008a

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