# 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*is an_{t}`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*is an_{t}`NBrowns`

-by-`1`

vector of (possibly) correlated zero-drift/unit-variance rate Brownian components.

## Creation

### Description

creates a default `BM`

= bm(`Mu`

,`Sigma`

)`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.

creates a `BM`

= bm(___,`Name,Value`

)`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

## Properties

## 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

## More About

## 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**

## See Also

`drift`

| `diffusion`

| `sdeld`

| `simulate`

| `interpolate`

| `simByEuler`

| `nearcorr`

### Topics

- Simulating Equity Prices
- Simulating Interest Rates
- Stratified Sampling
- Price American Basket Options Using Standard Monte Carlo and Quasi-Monte Carlo Simulation
- Base SDE Models
- Drift and Diffusion Models
- Linear Drift Models
- Parametric Models
- SDEs
- SDE Models
- SDE Class Hierarchy
- Quasi-Monte Carlo Simulation
- Performance Considerations