bm

Brownian motion (BM) models

Description

Creates and displays a Brownian motion (sometimes called arithmetic Brownian motion or generalized Wiener process) bm object that derives from the sdeld (SDE with drift rate expressed in linear form) class.

Use bm objects to simulate sample paths of NVars state variables driven by NBrowns sources of risk over NPeriods consecutive observation periods, approximating continuous-time Brownian motion stochastic processes. This enables you to transform a vector of NBrowns uncorrelated, zero-drift, unit-variance rate Brownian components into a vector of NVars Brownian components with arbitrary drift, variance rate, and correlation structure.

Use bm to simulate any vector-valued BM process of the form:

$$d{X}_t=\mu (t)dt+V(t)d{W}_t$$

where:

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

• μ is an NVars-by-1 drift-rate vector.

• V is an NVars-by-NBrowns instantaneous volatility rate matrix.

• dW_t is an NBrowns-by-1 vector of (possibly) correlated zero-drift/unit-variance rate Brownian components.

Creation

Syntax

BM = bm(Mu,Sigma)

BM = bm(___,Name,Value)

Description

BM = bm(Mu,Sigma) creates a default BM object.

Specify the 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.

BM = bm(___,Name,Value) creates a bm 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 BM 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

Input Arguments

Mu — Mu represents the parameter μ

array or deterministic function of time | deterministic function of time and state

Mu represents the parameter μ, specified as an array or deterministic function of time.

If you specify Mu as an array, it must be an NVars-by-1 column vector representing the drift rate (the expected instantaneous rate of drift, or time trend).

As a deterministic function of time, when Mu is called with a real-valued scalar time t as its only input, Mu must produce an NVars-by-NVars matrix. If you specify Mu as a function of time and state, it calculates the expected instantaneous rate of drift. This function must generate an NVars-by-1 column vector when invoked with two inputs:

• A real-valued scalar observation time t.

• An NVars-by-1 state vector X_t.

Data Types: double | function_handle

Sigma — Sigma represents the parameter V

array or deterministic function of time | deterministic function of time and state

Sigma represents the parameter V, specified as an array or 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 X_t.

Although the gbm constructor enforces no restrictions on the sign of Sigma volatilities, they are specified as positive values.

Data Types: double | function_handle

Properties

StartTime — Starting time of first observation, applied to all state variables

0 (default) | scalar

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

Data Types: double

StartState — Initial values of state variables

1 (default) | scalar, column vector, or matrix

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

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

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

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

Data Types: double

Correlation — Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes)

NBrowns-by-NBrowns identity matrix representing independent Gaussian processes (default) | positive semidefinite matrix | deterministic function

Correlation between Gaussian random variates drawn to generate the Brownian motion vector (Wiener processes), specified 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. If Correlation is not a symmetric positive semidefinite matrix, use nearcorr to create a positive semidefinite matrix for a correlation matrix.

A Correlation matrix represents a static condition.

As a deterministic function of time, Correlation allows you to specify a dynamic correlation structure.

Data Types: double

Simulation — User-defined simulation function or SDE simulation method

simulation by Euler approximation (simByEuler) (default) | function | SDE simulation method

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

Data Types: function_handle

Drift — Drift rate component of continuous-time stochastic differential equations (SDEs)

value stored from drift-rate function (default) | drift object or function accessible by (t, X_t)

This property is read-only.

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

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 drift of the form:

$$F(t,X_t)=A(t)+B(t)X_t$$

where:

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

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

The displayed parameters for a drift object are:

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

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

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

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, X_t) interface. However, you can specify the components A and B as functions that adhere to the common (t, X_t) interface, or as MATLAB arrays of appropriate dimension.

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

Data Types: object

Diffusion — Diffusion rate component of continuous-time stochastic differential equations (SDEs)

value stored from diffusion-rate function (default) | diffusion object or functions accessible by (t, X_t)

This property is read-only.

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

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 diffusion:

$$G(t,X_t)=D(t,X_t^{\alpha (t)})V(t)$$

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, X_t) interface.

The displayed parameters for a diffusion object are:

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

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

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

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, X_t) interface. However, you can specify the components A and B as functions that adhere to the common (t, X_t) interface, or as MATLAB arrays of appropriate dimension.

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

Data Types: object

Object Functions

interpolate	Brownian interpolation of stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, or SDEMRD models
simulate	Simulate multivariate stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, SDEMRD, Merton, or Bates models
simByEuler	Euler simulation of stochastic differential equations (SDEs) for SDE, BM, GBM, CEV, CIR, HWV, Heston, SDEDDO, SDELD, or SDEMRD models
simByMilstein	Simulate diagonal diffusion for BM, GBM, CEV, HWV, SDEDDO, SDELD, or SDEMRD sample paths by Milstein approximation
simByMilstein2	Simulate BM, GBM, CEV, HWV, SDEDDO, SDELD, SDEMRD process sample paths by second order Milstein approximation

Examples

Create a bm Object

Create a univariate Brownian motion (bm) object to represent the model: $d{X}_t=0.3d{W}_t$.

obj = bm(0, 0.3) % (A = Mu, Sigma)

obj = 
   Class BM: Brownian Motion
   ----------------------------------------
     Dimensions: State = 1, Brownian = 1
   ----------------------------------------
      StartTime: 0
     StartState: 0
    Correlation: 1
          Drift: drift rate function F(t,X(t)) 
      Diffusion: diffusion rate function G(t,X(t)) 
     Simulation: simulation method/function simByEuler
             Mu: 0
          Sigma: 0.3

bm objects display the parameter A as the more familiar Mu.

The bm class also provides an overloaded Euler simulation method that improves run-time performance in certain common situations. This specialized method is invoked automatically only if all the following conditions are met:

• The expected drift, or trend, rate Mu is a column vector.

• The volatility rate, Sigma, is a matrix.

• No end-of-period adjustments and/or processes are made.

• If specified, the random noise process Z is a three-dimensional array.

• If Z is unspecified, the assumed Gaussian correlation structure is a double matrix.

More About

Instance Hierarchy

There are inheritance relationships among the SDE classes.

The following figure illustrates the inheritance relationships.

For more information, see SDE Models, SDE Classes and Associated Simulation Methods, and SDE Class Hierarchy.

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

References

[1] Aït-Sahalia, Yacine. “Testing Continuous-Time Models of the Spot Interest Rate.” Review of Financial Studies, vol. 9, no. 2, Apr. 1996, pp. 385–426.

[2] Aït-Sahalia, Yacine. “Transition Densities for Interest Rate and Other Nonlinear Diffusions.” The Journal of Finance, vol. 54, no. 4, Aug. 1999, pp. 1361–95.

[3] Glasserman, Paul. Monte Carlo Methods in Financial Engineering. Springer, 2004.

[4] Hull, John. Options, Futures and Other Derivatives. 7th ed, Prentice Hall, 2009.

[5] Johnson, Norman Lloyd, et al. Continuous Univariate Distributions. 2nd ed, Wiley, 1994.

[6] Shreve, Steven E. Stochastic Calculus for Finance. Springer, 2004.

Version History

Introduced in R2008a

R2023b: Added simByMilstein2 method

Use the simByMilstein2 method to approximate a numerical solution of a stochastic differential equation.

R2023a: Added simByMilstein method

Use the simByMilstein method to approximate a numerical solution of a stochastic differential equation.