Brownian motion models
Creates and displays Brownian motion (sometimes called arithmetic
Brownian motion or generalized Wiener process)
bm
objects that derive 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:
where:
Xt 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.
dWt is an
NBrowns
-by-1
vector of (possibly)
correlated zero-drift/unit-variance rate Brownian components.
creates a default BM
= bm(Mu
,Sigma
)BM
object.
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.
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
interpolate | Brownian interpolation of stochastic differential equations |
simulate | Simulate multivariate stochastic differential equations (SDEs) |
simByEuler | Euler simulation of stochastic differential equations (SDEs) |
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, bm
treats it as a static function of time and state, by that means guaranteeing that all
parameters are accessible by the same interface.
[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.
diffusion
| drift
| interpolate
| nearcorr
| sdeld
| simByEuler
| simulate