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hwv

Construct HWV model

Synopsis

HWV = hwv(Speed, Level, Sigma)

HWV = hwv(Speed, Level, Sigma, 'Name1', Value1, 'Name2', Value2, ...)

Class

hwv

Description

This constructor creates and displays hwv objects, which derive from thesdemrd (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 Hull-White/Vasicek stochastic processes with Gaussian diffusions.

This method allows you to simulate vector-valued Hull-White/Vasicek processes of the form:

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

where:

  • 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.

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.

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.

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). 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.

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. 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.

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. 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 it as a function of time and state, Sigma 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.

Note

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

Optional Input Arguments

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 vector, 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 character vector 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).

Output Arguments

HWV

Object of class hwv with the following displayed parameters:

  • StartTime: Initial observation time

  • StartState: Initial state at StartTime

  • Correlation: Access function for the Correlation input, 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

Examples

Creating Hull-White/Vasicek (HWV) Gaussian Diffusion Models

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, hwv 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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