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cev - Construct constant elasticity of variance models (objects of class CEV)

Synopsis

CEV = cev(Return, Alpha, Sigma)

CEV = cev(Return, Alpha, Sigma, 'Name1', Value1, 'Name2', Value2, ...)

Class

CEV

Description

This constructor creates and displays CEV objects, which derive from the SDELD (SDE with drift rate expressed in linear form) class. Use CEV objects to simulate sample paths of NVARS state variables driven by NBROWNS Brownian motion sources of risk over NPERIODS consecutive observation periods, approximating continuous-time stochastic processes.

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

(11-4)

where:

• X_t is an NVARS-by-1 state vector of process variables.

• μ is an NVARS-by-NVARS (generalized) expected instantaneous rate of return matrix.

• D is an NVARS-by-NVARS diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector raised to the corresponding power of α.

• V is an NVARS-by-NBROWNS instantaneous volatility rate matrix.

• dW_t is an NBROWNS-by-1 Brownian motion vector.

Input Arguments

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.

The required input parameters are:

Return	Return represents the parameter μ. If you specify Return as an array, it is a NVARS-by-NVARS 2-dimensional matrix that represents the expected (mean) instantaneous rate of return. If you specify Return as a function, it calculates the expected instantaneous rate of return. This function must generate an NVARS-by-NVARS matrix when invoked with two inputs: A real-valued scalar observation time t. An NVARS-by-1 state vector Xt.
Alpha	Alpha determines the format of the parameter D. If you specify Alpha as an array, it represents an NVARS-by-1 column vector of exponents. If you specify it as a function, Alpha must return an NVARS-by-1 column vector of exponents when invoked with two inputs: A real-valued scalar observation time t. An NVARS-by-1 state vector Xt.
Sigma	Sigma represents the parameter V. If you specify Sigma as an array, it represents an NVARS-by-NBROWNS 2-dimensional matrix of instantaneous volatility rates. In this case, each row of Sigma corresponds to a particular state variable. Each column of Sigma corresponds to a particular Brownian source of uncertainty, and associates the magnitude of the exposure of state variables with sources of uncertainty. If you specify it as a function, Sigma must generate 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.

Note   Although the constructor does not enforce restrictions on the signs of these input arguments, each argument is usually specified as a positive value.

Optional Input Arguments

Specify optional inputs as matching parameter name/value pairs as follows:

• Specify the parameter name as a character string, followed by its corresponding value.

• You can specify parameter name/value pairs in any order.

• Parameter names are case insensitive.

• You can specify unambiguous partial string matches.

Valid parameter names are:

StartTime	Scalar starting time of the first observation, applied to all state variables. If you do not specify a value for StartTime, the default is 0.
StartState	Scalar, NVARS-by-1 column vector, or NVARS-by-NTRIALS matrix of initial values of the state variables. If StartState is a scalar, cev applies the same initial value to all state variables on all trials. If StartState is a column vector, cev applies a unique initial value to each state variable on all trials. If StartState is a matrix, cev applies a unique initial value to each state variable on each trial. If you do not specify a value for StartState, all variables start at 1.
Correlation	Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes). Specify Correlation as an NBROWNS-by-NBROWNS positive semidefinite matrix, or as a deterministic function C(t) that accepts the current time t and returns an NBROWNS-by-NBROWNS positive semidefinite correlation matrix. A Correlation matrix represents a static condition. As a deterministic function of time, Correlation allows you to specify a dynamic correlation structure. If you do not specify a value for Correlation, the default is an NBROWNS-by-NBROWNS identity matrix representing independent Gaussian processes.
Simulation	A user-defined simulation function or SDE simulation method. If you do not specify a value for Simulation, the default method is simulation by Euler approximation (simByEuler).

Output Arguments

CEV

Object of class CEV with the following displayed parameters: 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 Return: Access function for the input argument Return, callable as a function of time and state Alpha: Access function for the input argument Alpha, callable as a function of time and state Sigma: Access function for the input argument Sigma, callable as a function of time and state

Remarks

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 X_t, and return an array of appropriate dimension. Even if you originally specified an input as an array, cev treats it as a static function of time and state, thereby guaranteeing that all parameters are accessible by the same interface.

Examples

• Creating Constant Elasticity of Variance (CEV) Models

• Implementing Multidimensional Equity Market Models, Implementation 3: Using SDELD, CEV, and GBM Objects

See Also

drift, diffusion, sdeld

bm

cir

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