Create HWV model
HWV = hwv(Speed, Level, Sigma)
HWV = hwv(Speed, Level, Sigma, 'Name1', Value1, 'Name2', Value2, ...)
This constructor creates and displays HWV objects, which derive from the SDEMRD (SDE with drift rate expressed in mean-reverting form) class. Use HWV 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 HWV stochastic processes with Gaussian diffusions.
Xt is an NVARS-by-1 state vector of process variables.
S is an NVARS-by-NVARS 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).
V is an NVARS-by-NBROWNS instantaneous volatility rate matrix.
dWtis an NBROWNS-by-1 Brownian motion vector.
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.
The required input parameters are:
|Speed||Speed represents the function S.
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). If you specify Speed as
a function, it calculates the speed of mean reversion. This function
must generate an NVARS-by-NVARS matrix
of reversion rates when called with two inputs: |
|Level||Level represents the function L.
If you specify Level as an array, it must be an
NVARS-by-1 column vector of reversion levels.
If you specify Level as a function, it must generate
an NVARS-by-1 column vector of reversion levels
when called with two inputs: |
|Sigma||Sigma represents the parameter V.
If you specify Sigma as an array, it must be an NVARS-by-NBROWNS matrix
of instantaneous volatility rates. 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. If you specify
it as a function, Sigma must return an NVARS-by-NBROWNS matrix
of volatility rates when invoked with two inputs: |
Specify optional input arguments as variable-length lists of matching parameter name/value pairs: 'Name1', Value1, 'Name2', Value2, ... and so on. The following rules apply when specifying parameter-name pairs:
Specify the parameter name as a character string, followed by its corresponding parameter 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, hwv applies the same initial value to all state variables on all trials.
If StartState is a column vector, hwv applies a unique initial value to each state variable on all trials.
If StartState is a matrix, hwv 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).|
|HWV||Object of class hwv with the following displayed
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, hwv treats it as a static function of time and state, thereby guaranteeing that all parameters are accessible by the same interface.
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.